Mathematical Physics 1

By René Yves RASOANAIVO

African Virtual universityUniversité Virtuelle AfricaineUniversidade Virtual Africana

Mathematical Physics 1

African Virtual University �

NotICe

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I. MathematicalPhysics1 _____________________________________ 3

II. PrerequisiteCourseorKnowledge_____________________________ 3

III. Time ___________________________________________________ 3

IV. Materials_________________________________________________ 3

V. ModuleRationale __________________________________________ 4

VI. Content__________________________________________________ 4

6.1 Overview___________________________________________ 4 6.2 Outline_____________________________________________ 4 6.3 GraphicOrganizer____________________________________ 5

VII. GeneralObjective(s)________________________________________ 6

VIII. SpecificLearningObjectives__________________________________ 7

IX. Pre-assessment ___________________________________________ 8

X. TeachingandLearningActivities______________________________ 18

XI. GlossaryofKeyConcepts___________________________________ 53

XII. ListofCompulsoryReadings________________________________ 49

XIII. Completelistofmultimediaresources_________________________ 58

XIV. UsefulLinks _____________________________________________ 59

XV. SynthesisoftheModule____________________________________ 66

XVI. SummativeEvaluation______________________________________ 67

XVII. References ______________________________________________ 78

XVIII.MainAuthoroftheModule__________________________________ 79

Table OF CONTeNTS


I. Mathematical physics 1

By René Yves RASOANAIVO, Ph.D.

II. Prerequisites

IIn order to understand and follow the « Mathematical Physics » module, the student must master the following concepts :

• Concept of a function : real one-variable functions ; elementary functions

• Derivative of a function ; derivation techniques• Limit of a function• Graphical representation of a function

• Primitives;definiteintegrals;techniquesofintegration

III. Teaching Hours

The module is comprised of 4 learning units with the following number of teaching hours

1. Analysis Elements and Series : 30 H2. Derivation and Integration : 30 H3. Numerical Methods : 30 H4. Linear Algebra : 30 H

IV. Materials

• PersonalcomputerwithInternetconnection,MicrosoftOffice,Multime-dia materials

• For learning activities 2. , 3. and 4. Software : Microsoft Excel 2000 ; Maxima


V . Justification

Mathematics is regarded as a tool for the physical sciences. This module in Mathematical Physics 1 provides mathematical elements used in teach-ing physics. These features allow students to not only better understand the concepts of physics, but also to relate physical quantities between them. Many areas of physics such as mechanics, quantum physics, optics, thermodynam-ics use mathematical elements such as: derivation, differentiation, integration, solving a system of linear equations, calculus, and digital methods ... all developed in this module. A good mastery of the mathematical tools is necessary for a better explanation of certain laws of physics.

VI . Content

6 . 1 Summary

This module deals with the mathematical elements essential to understanding physics courses, namely the study of real functions, derivation and integration of a function with one and several real variables, the development of a function, some elementsofnumericalcalculationsand,finally,solvingasystemoflinearequations. Learningactivitiesofdifferentdifficultylevelsaredevelopedwithformalassess-ments. Moreover, online word and useful links enable students to study certain topics in detail. Finally, the students will also be able to use software such as “Microsoft Excel 2000” and “Maxima”.

6.2 General Overview

• Analysis : Analysis: Limit and continuity of a function, derivative, extrema determination;Integration(Riemann,Improper,Indefinite)

• Series:Infiniteseries,convergencetests,developmentofTaylorandMa-cLaurin series

• DifferentialCalculus: partial derivative, derivatives of implicit functions, total and exact differential

• Integration : Integration on open and closed intervals, multiple integrals.• Numericalmethods:Thegammafunction,numericalevaluationoffiniteandinfinitesum,numericalevaluationofanintegral:thetrapezoidalruleand Simpson’s rule

• LinearAlgebra: Matrix computations, determinant, solving a system of linear equations


6.3 Graphical Representation

5

Graphical Representation :

Analytical study of a function to a real variable: limit and continuity

Operations on a function to a real variable: derived, primitive, definite integral

Operations on a function with two real variables: partial derivatives, total differential

Line integral, double integral and Green's theorem

Representations of a function: Taylor Series and MacLaurin Series, Fourier Series

Numerical evaluation: Finite sum and infinite sum

Numerical evaluation of an integral: Trapezoidal rule, Simpson's Methods

Matrix algebra : sum, product, determinant, inverse

Methods of solving a system of linear equations: Cramer’s Formula, LU decomposition, Inverse matrix

method


VII. General objectives

Upon completion of this module, the student should understand and be able to apply the following concepts :

• Partial derivative• Total derivative• Calculating simple integrals• Calculating double and triple integrals• Numericalcalculationsonfinitesums• Numericalcalculationsoninfinitesums• Matrix operations• Finding solutions to a system of linear equations


VIII. Specific Objectives

Learning Activities Specific Objectives

1. Elements of analysis and series

• Recall the conditions of differentiability and continuity of a function

• Calculate the integral of a real variable function• Recall the series expansion of a function about a point

2. Derivation and integration

• Recall the total differential of a function with two real variables

• Calculate the partial derivative of a function with two real variables

• Calculate the value of a line integral along a closed curve and an open curve

• Calculate the value of a double integral over a given region

3. Numerical methods

• Remember the basic principles of numerical calculations• Numerically calculate a finite sum• Numerically calculate an infinite sum• Numerically calculate the value of a definite integral

4. Linear algebra

• Recall matrix operations• Calculate the determinant of a square matrix• Determine the inverse of a square matrix• Determine solutions to a system of linear and non-homoge-

neous equations


IX. Teaching and learning activities

Preliminary evaluation : Elements of analysis

Reasoning

This activity allows the student to situate themselves in relation to the level re-quired to begin the module and therefore identify the elements of mathematics he or she must review. The nineteen questions posed below affect three main areas ofanalysis,namelythederivative,theintegral,continuity,limitsandfinally,theextrema of a real variable function. They are designed to assess the prerequisites of a student.

In addition, these prerequisites are necessary for the professor to assess the level of students and help them prepare for the learning activities developed in this module. The professor can save time and students can thus be more motivated.

Questions

1. The derivative of the function f(x) = 3x2 – 2x + 1 is:

a. r 3x – 2 ; b. r 6x – 2; c. r 3x + 2

2. The derivative of the function f (x) = 1/ (x + 1) is :

a. r 1/ ( x + 1)2 ; b. r - 1/ ( x + 1)2 ; c. r x / ( x + 1)2

3. The derivative of the function f(x) = 1/ ( x2 – 2x + 1 ) is :

a. r 2 / ( x - 1)2 ; b. r -2 x / ( x - 1)3 ; c. r 2/ ( x - 1)5

4. The derivative of the function f(x) = tan ( x ) is written as f ’ ( x ) = 1 + tan2 ( x )

a. r True ; b. r False

5. The integral of the function f(x) = 3x3 + 2x2 – x + 1 is :

a. r 9 x 4 + 6 x3 – 2 x 2 + x + c; b. r (3/4) x 4 + (2/3) x3 – (1/2) x 2 + x + c

6. The integral of the function f ( x ) = 1/ x is :

a. r - 1/ x 2 + c ; b. r ln ( x ) + c ; c. r ln (x + c)

7. The integral of the function ln ( x) is F ( x ) = x ln (x) - x + c



8. The integral of f(x) = 3 cos( 2 x ) is F (x) = 3 sin ( 2 x ) + c


9. The limit of the function f (x ) = 2 x 3 – 3x + 1 when x approaches 1 is :

a. r 1 ; b. r 0 ; c. r - 1

10. The limit of the function f (x ) = cos (x)/ x when x approaches 0 is :

a. r 0 ; b. r 1 ; c. r∞

11. The limit of the function f (x ) = sin (x)/ x when x approaches 0 is :

a. r 0 ; b. r 1 ; c. r∞

12. The limit of the function f (x ) = ex / x when x approaches 0 is :

a. r 0 ; b. r 1 ; c. r∞

13. The limit of the function f (x ) = tan ( x ) when x approaches (π/2) + 0 is equalto+∞


14. The function f ( x ) = 2 x2 + 1 has a maximum at x = 0


15. Does the function f ( x ) = x3 + x have maxima and minima ?

a. r Yes ; b. r No

16. How many minima does the function f ( x ) = sin (x) contain in the interval

[ 0, 3 π ] ?

a. r 1 ; b. r 2 ; c. r 3

17. The function f (x) =x2 − 2x + 1x2 + 2x + 2

has extreme values at the following points

a (1,0)and (-1.5,5) ; b (0,1)and (1,0) ; c (-0.25,5) and (-1.5 ,5)

18. Find the mean value of x2 between x = 1 and x = 4

a. r 5 b. r 7 c. r 16

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19.The root mean square value(rms) of i = I o sin pt over the range t = 0 to

t =2π

pis

a.I o

a, b.

I o2

2 c.

I o

2

Correct Answers

1. The derivative of the function f(x) = 3x2 – 2x + 1

The derivative of f (x) = xn is written f ' (x) = n xn−1

If we apply this formula we obtain : f ’( x ) = 6x – 2

The correct answer is b.

(If you checked boxes a. or c., you have obviously forgotten the formula for the derivative)

2. The derivative of the function f(x)=1/(x+1) :

The formula to use is :

If f(x) = f (x) =

uv

, then f '(x) =

u'v − uv '

v2

In particular, if u = 1 and v = x+1 , then f '(x) =

−1

(x + 1)2


3. The derivative of the function f(x)=1/(x2–2x+1) :

Note that the function can also be written as :

f (x) =1

(x − 1)2

Therefore the derivative is :

f '(x) =−2

(x − 1)3The correct answer is b.

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If you did not answer this question correctly, you should practice your deriva-tives.

4. The derivative of the function f(x) = tan(x) :

We apply the formular found in question 2. The result is :

f '(x) = 1+ tan2(x) =

1

cos2(x)


5. The integral of the function f(x)=3x3+2x2–x+1 :

For each term in the polynomial, we apply the formula :

xndx = xn+1

n + 1∫ + c .

Where : F(x) =

34

x4 +23

x3 −12

x2 + x + c


6. The integral of the function f(x)=1/x :

F(x) = ln(x) + c

*This is one of the integrals of elementary functions that you must remember.


7. The integral of the function ln(x)

follows by performing the following operation :

ln(x) dx∫

Integrate by parts :

u dv∫ = u v - v du∫We substitute : u = ln (x) and dv = dx , thus : v = x and du = dx / x


Thus :

ln(x) dx∫ = x ln(x) − dx = x ln(x) − x + c∫


8. The integral of the function f(x)=3cos(2x)

Recall integrals of trigonometric functions.

As before, we must perform :

cos(ax) dx∫

Perform a change of variables by substituting u = a x :

cos(ax)dx∫ =

1a

cos(ax)d(ax) =1a∫ cos(u)du∫

Finally, we obtain :

cos(ax) dx∫ =

1a

sin(ax) + c d'où F(x) = 32

sin(2x) + c


9. The limit of the function f(x)=2x3–3x+1 as x approaches1:

This is a continuous function ∀ x , thus the limit is obtained by replacing x by 1 in the equation of f(x) :

limx→1

f (x) = f (1) = 0



10. The limit of the function f(x)=cos(x)/x as x approacheszero :

Recall cos (0) = 1, thus f (0) =

10, thezero in thedenominator signifies that

the

functionisnotdefinedforx=0.

However, we can say that : limx→0

f (x) = ∞

The correct answer is c.

11. Thelimitofthefunctionf(x)=sin(x)/xasxapproacheszero.

If we replace x by 0 in f(x), we obtain : f (0) =

sin(0)0

= 00

;

Theresultisnotdefined.

However, the sinusoidal function can still be represented by a series :

Thus :

From this expression : limx→0

f (x) = 1


12. The limit of the function f(x)=ex/xasxapproacheszero:

We replace x by 0 in f ( x ) :



13. The limit of the function f(x)=tan(x) as x approaches(π/2)+0

Recall: sin( )

tan( ) cos( )

=x

xx

Since cos (π/2) = 0 and sin (π/2) = 1, we have : 1

tan( )2 0

π=

Thusthefunctiontan(x)isnotdefinedwhenx=π/2 .

/ 2lim tan( ) →

= ± ∞x

xπ

, as long as x = π/2 + 0 or x = π/2 - 0 thus :

/ 2 0lim tan( ) +

→ −= ∞

xx

π, the left limit

/ 2 0lim tan( ) -

→ += ∞

xx

π , the right limit

This becomes clear if you plot the curve tan (x) vs. x, as the graph shows that the tangent function is indeed asymptotic to the abscissa: x = π (2k+1)/2, for k = 0,1,2,….


14. The function f(x)=2x2+1 has a maximum at x = 0

Recall that the function f( x ) has a maximum or minimum at certain abcissas x

kifthederivativeiszeroatthesepoints.However,wemustfirstcalculatethe

derivative to verify this :

f ’ ( x ) = 4 x

Thisderivativeiszeroifx=0,andthefunctionhasanextremumatthepoint

x = 0.

To know whether this extremum is a maximum or a minimum, we must evaluate the sign of the second derivative of f(x ), thus the sign of f ‘’ ( x )

Inthiscase,f’’(x)=4,thusitispositive;thissignifiesthattheextremumisamaximum.

The correct answer is a.

You should draw the function to verify the result.


15. Does the function f(x)=x3+x have maxima and minima ?

Start by calculating the derivative of f ( x ) : f ’( x ) = 3x2 + 1

Thisfunctioncannotbezerosoitdoesnotexhibitmaxima.

The correct answer is b

16. What are the minima of the function f(x)=sin(x) in the interval [ 0, 3 π ] ?

Calculate the derivative of f(x ) : f ’ ( x) = cos ( x ),

and cos (x) = 0 for xk = π (2k+1)/2

xo = π /2 , x

1 = 3π /2, x

2 = 5π /2, x

3 = 7 π /2, …..

Furthermore ,

The function this has 2 maxima and a minimum in the interval [ 0, 3 π ]


17 . What are the coodinates of the points at which the function f (x) =x2 − 2x + 1x2 + 2x + 2has extreme values

Thefunctionneverbecomesinfinityordiscontuiousbecausethedenominator which is (x+1)2 +1 doesnot vanish.

We let y = f (x) so that y =1−

2x

+1x2

1+2x

+2x2

,

y → 1 as x → ∞ ,ywillnotthenbecomeinfinitewithx

dydx

=2(x − 1) 2x + 3( )(x2 + 2x + 2)2 . For extreme values

dydx

= 0 ;the turning points are

x = 1


and x =−32

.

x = 1 gives a minimum value y = 0 while x =−32

gives a maximum value

y=5. The extreme points are (1,0)and (-1.5,5)

The correct answer is a

18 . What is the mean value of x2 between x = 1 and x = 4

Let y = f (x) .Themeanvalueofybetweenthepointsaandbisdefined

by y =

ydxa

b

∫b − a

⇒ y =1

4 − 1x2

1

4

∫ dx =13

x3

3⎡

⎣⎢

⎤

⎦⎥

1

4

=64 − 1

9=7


19. Find the rms of i = I 0 sin pt over the range t = 0 to t =2π

p

N.B.the mean square value of a function is the 2nd power of the function

r .sm.s =1

b − ai 2

a

b

∫ dx⎛

⎝⎜⎞

⎠⎟

Mean square value=1

2πp

I o2 sin2 ptdt

0

2 π

p

∫ =

pI o2

2π

1− cos2 pt2

dt =pI 2

4πt −

sin2 pt2 p

⎡

⎣⎢

⎤

⎦⎥

0

2 π

∫0

2 πp

=pI 2

4π

2π

p− 0⎡

⎣⎢

⎤

⎦⎥

=I 0

2

2 Hence r.m.s=I 0

2

2= I 0

2

The correct answer is c


Comments for students

If you have more than 75%, your interest in mathematics is obvious, and I encourageyoutopersevereinthefieldsinceI’msurewe’lldogoodworktogether.Youwillseethatthestudyofopticsisaveryexcitingfield. If you have between 50% and 75%, your result is very encouraging, and mathematics is not unknown to you. We have a lot of work to do throughout thiscourse,andIcanassureyouthatthisisaveryexcitingfieldthatyouhavechosen. Good luck.

If you have between 35% and 50%, this is far from a perfect mark. However, itseemstomethatyouhavethewilltosucceedinthisfield.Itisthiscommit-mentthatisneeded.Thefieldyouhavechosenisveryexciting,butitisalsoalot of hard work. To begin, there is some catching up that you need to do, and then we can succeed together. If you have less than 35%, you will have some serious work to do, since in addition to this module, you should review your previous mathematics course.


X. Learning activities

Learning activity # 1

Title : Elements of analysis and series

Teaching hours : 30H

Instructions

For this activity, if you have atleast¾ofthepoints you’ve done a good job, you can continue.

If you have lessthanhalfthepoints you should review the readings proposed and repeat the activity.

If you have morethanhalfthepointsandlessthan¾ofthepoints, you did a good job, but you must strive for more.

Specific objectives

Following the completion of this module, the student should understand the following concepts :

• Continuity of a real variable function• Derivability of a real variable function• Calculating simple integrals• Convergence tests of a numerical series• Developing the series of a function (Taylor, Maclaurin)• Fourier series

Activity summary

This course addresses the essential elements of analysis in addition to elements already acquired by the student, namely the derivative of a function, the condi-tion of differentiability of a function, continuity of a function and the existence of extrema. In addition, exercises are designed to remind the student on how to test convergence of a series.


Required readings

• RASOANAIVO, R-Y. ( 2006). Eléments d’Analyse I. Ecole Normale Supérieure, Université d’Antananarivo, Madagascar

• RASOANAIVO, R-Y. (2006). Eléments d’Analyse II. Ecole Normale Supérieure, Université d’Antananarivo, Madagascar

Multimedia resources

Microsoft Excel

Useful links

MITOPENCOURSEWARE : http://ocw.mit.edu/Les-Mathematiques.net : http://www.les-mathematique;netOpenlearninginitiative :http://www.cmu.edu/oli/courses/

Activity description

This activity involves three exercises dealing with topics in the area of ana-lysis, before continuing this module. The three exercises below are carefully selected so that the student revises them and then addresses new subjects.


Evaluation

These 10 exercises, 8 of which are under the form of multiple choice questions (MCQ), and must be answered by checking the correct answer(s), on paper, and in order.

All the 10 exercises are each worth 10%.

Exercise 1

The value of the derivative of sin(x2–1) about the point x = 1 is :

a.q 1 b. q 0 c. q 2

Exercise 2

Consider the function f(x) = 1/(1 + x2)

a. q the function is derivable about the point x = 1

b. q the function is not derivable about the point x = - 1

c. q the function is continuous about the point x = 1

d. q the function does not have extrema

Exercise 3

Consider the function : f (x) = x3 − 2x2 + x + 1

1. The number of maxima is : a. q 0 b. q 1 c. q 2

2. The number of minima is : a. q 0 b. q 1 c. q 2

Exercise 4

Consider the function : f (x) =

x + 1x − 1

a. q

limx→∞

f (x) = − 1 b. q

limx→∞

f (x) = +1


c. q

limx→ − ∞

f (x) = − 1 d. q

limx→ − ∞

f (x) = 1

e. q

limx→ 1+

f (x) = − ∞ f. q

limx→ 1+

f (x) = +∞

g. q

limx→ 1-

f (x) = − ∞ h q

limx→ 1-

f (x) = +∞

Exercise 5

The series of general terms Wn = 2n / n ! is

a. q convergent b. q divergent c. q not convergent or divergent

Exercise 6

The interval of convergence of the series

S(x) = xn

nn=1

∞∑ is :

a. q [ -1 , 1[ b. q ] – 1 , 1 [ c. q ] – 1 , 1 ] d. q [ – 1 , 1 ]

Exercise 7

The value of the integral

I = dx

x2 + 10∞

∫ is equal to :

a. q π/2 b. q - π/2 c. q π


Exercise 8

(4 + 3x)−3 evaluated to 3 term in ascending powers of x gives

127x3 1−

4x

⎧⎨⎩

+323x2

⎫⎬⎭

a. q True b. q False

Exercise 9

Develop the following into a Fourier series :

f (x) =

0 ,−π < x < 0

1 ,0 < x < π

⎧⎨⎩

Exersice 10

The intesity of radaition from radioactive substance a tant time t is given by

I = I 0 e− kt where I 0 =initialintensity.Ifthehalflifeofthesubstanceis2000yr.Find how long it would take for the intensity of the substance to reduce to

110 I 0 .

Learning activities

- Eachstudentmustfirstreadthecourseonanalysisandseriesbeforedoingthe exercises.

- Thetutorwillorganizethegroupforcollaborativework.- Students will collaboratively discuss topics that were initially not understood

under the supervision of the tutor.- When the tutor decides that the students have achieved an appropriate level

of understanding, they may begin the exercises.- All groups address the same exercise at the same time under the supervision

of the tutor, who will set the duration.- Each group will appoint a leader who will put the names of all group mem-

bers on the exercise before sending it by email attachment to the professor of the course.


Correct answers

Exercise 1

The derivative of a function is a topic that frequently appears in physics pro-blems. This exercise gives an example

The formula to use is :

df [u(x)]dx

= dudx

dfdu

In this case, u( x ) = x2 - 1

Thus :

Where :

dfdx

= 2x cos (x2 − 1)

For x = 1 , we have

dfdx

= 2


Exercice 2

The conditions of derivability and continuity of a fonction about a point are important when manipulating functions. This exercise gives a student the chance to practice and solidify these skills.

The function considered here is : f (x) =

1

x2 + 1ItisafunctiondefinedforallxbelongingtoR.


Its derivative is : f '(x) =

−2x(x2 + 1)2

To obtain the correct answers, recall the condition of derivability, the condition of continuity, and the condition of existence of extrema :

A function f( x ) is continuous about the point x = xo,

limx→xo

f (x) = f (xo)

In this case, we have :

limx→1

(1

x2 + 1) =

12

limx→1

2x

(x2 + 1)2=

12

Whichsignifiesthat:

• The function is continuous and derivable at the abscissa x = 1 • The function has an extremum at the abscissa x = 0 since the derivative is zeroatthispoint

The correct answers are a. and c.

Exercise 3

The derivative of f (x) = x3 − 2x2 + x + 1 is written f '(x) = 3x2 − 4x + 1

It has two roots : x1 = 1 ; x2 = 1 / 3

Thus, f ( x ) has two extrema. To demonstrate this, we can draw up a table as follows :

x - ∞ 1/3 1 ∞f’(x) + 0 - 0 +

This table indicates the signs of the derivatives :


f ’ (x) < 0 for x ε [ 1/3 , 1 ] and f ’ ( x ) > 0

Recall that : f(x) is increasing in the region where f ‘ ( x ) > 0

f (x ) is decreasing in the region f ‘ ( x ) < 0

Whichsignifiesthatf(x)hasamaximumattheabscissax=1/3andthemini-mum at the abscissa x = 1


Exercise 4

The function : f (x) =

x + 1x − 1

Isnotdefinedforx=1,

ThusitisdefinedinthedomainD=]-∞,1[U]1,∞[

Determine the limits at the extremities for these intervals :

Tofindthelimitsasxapproachesinfinity:

f (x) =1+

1x

1−1x

; (1+1x

)(1−1x

) ; 1−1

x2

Thus :

limx→±∞

f (x) = 1

The correct answers are b. and d.


To understand the behaviour of f(x) at the two ends of x = 1, we calculate the right and left limits of f ( x ) .

Right limit :

let x –1 = ε > 0, then

limx→ 1+

f (x) = limε→0

(1+2ε

) = +∞

Left limit :

let 1- x = ε > 0 , then

limx→ 1-

f (x) = limε→0

(1−2ε

) = −∞

The correct answers are f. and g.

Exercice 5

This exercise consists of convergence tests for series. The student should recall the tests developed during the module. The general term in this case is:

Wn =

2n

n!

Apply D’ Alembert’s test :

lim

n→∞

Wn+1Wn

= limn→∞

2n + 1

= 0

Since the limit obtained is less than 1, the series is convergent.



Exercise 6

The interval of convergence of the series

S(x) =xn

nn=1

∞∑ :

Apply D’Alembert’s test :

limn→∞

xn+1

n + 1

n

xn= lim

n→∞

n + 1n

x

Thus the series converges if | x | < 1

Also, note that :

We can easily show that S(1) is a divergent series, however S(-1) is conver-gent.

WhichsignifiesthattheintervalofconvergenceofS(x)is[-1,1[


Exercise 7

The value of the integral

I = dx

x2 + 10∞

∫

Performachangeofvariables:x=tan(θ)

Thus :

1+ x2 = 1+ tan2(θ) =1

cos2(θ)and

dx =dθ

cos2(θ)

⇒ dx

x2 + 1= dθ


when x = 0, θ=0;andwhenx=∞,θ = π/2

The integral becomes : I = dθ =

π20

π/2∫


Exercice 8

Expand (4 + 3x)−3 in ascending powers of x up to 3 terms

1(4 + 3x)3 =

143 (1+ 3x

4)3= 1

64 1+ 3x4( )−3

If3x4

∠1

Ie x ∠43

=1

641+ −3( ){ 3x

4⎛⎝⎜

⎞⎠⎟

+−3( ) −3 − 1( )

23x4

⎛⎝⎜

⎞⎠⎟

2

+ .....

=1

641−

9x4

⎧⎨⎩

+27x2

8⎫⎬⎭

to 3 terms


Exercise 9

Develop a Fourier series :

f (x) =

0 ,−π < x < 0

1 ,0 < x < π

⎧⎨⎩

Write:

f (x) =ao2

+ ann=1

∞∑ cos(nt) + bn

n=0

∞∑ sin(nt)


ao =

1π

f (t)dt =1π−π

π∫ dt = 1

0π

∫

an =

1π

f (t)dt =1π−π

π∫ cos(nt)dt = 0

0π

∫

bn =

1π

f (t)sin(nt)dt =1π−π

π∫ sin(nt)dt

0π

∫

Thus, we obtain :

f (x) =12

+2π

sin(2n + 1)x2n + 1

n=0

∞∑

Exercise 10

The intensity of a radiation of a radioactive substance is given by I = I 0 e− kt

.findhowlongittakestheintensityofradiationtoreduceto 110 I 0 ,if the hal-

flifeofthesubstanceis2000yrs.

Given I = I 0 e− kt

Athalflife I = 12 I 0 ; 1

2 I 0 = I 0 e− k 2000 ⇒ 12 = e−2000 k ⇒ e2000 k = 2

Taking logs to base, e, 2000k = In2 ⇒ k = In22000

If T is the time after which the intensity falls to 110 I 0 then

110 I 0 = I 0 e− kT ⇒ 1

10 = e− kT ⇒ e− kT = 10 ;


Taking logs to base,e

kT = In10 ⇒ T = In10k =

2000 In10In2

=2000x2.3026

0.6931= 6645yrs

Self-evaluation

The students should note the mistakes that they have made while consulting the correct answers to the exercises. They should review parts of the course that were not well understood in order to prepare for future evaluations.

Professor’s Guide

The professor will correct group assignments, and will then place it in a study area accessible to the students. The corrections should be accompanied by ade-quate feedback. The grades obtained by each group are given to the members ofthatgroup,andwillcountfor20%ofthefinalgradeofthemodule.


Learning activity # 2 :

Title:Differentiation:partialderivative,derivativeofafunctionofafunction,totalderivative.Curvilinearintegrals.Doubleintegrals.Green’sTheorem

Teaching hours

30 H

Instructions




Specific objectives

• Upon completion of this module, the student should be able to :• Determine the partial derivative of a function with two real variables• Determine the total derivative of a function with two real variables• Calculate the value of a line integral along an open curve• Calculate the value of a double integral over a given region

Activity summary

This activity deals with calculations of total or exact derivatives, and accurate assessment of a line integral along two different open curves connecting two pointsAandBofdifferentnature,andfinallytheevaluationofadoubleinte-graloverawell-definedgeometricregion.

Required readings

RASOANAIVO, R. Y. ( 2006). Eléments d’Analyse II, Ecole Normale Supé-rieure, Université d’Antananarivo, Madagascar. Cours inédit


Microsoft Excel ; Maxima


Useful links

MITOPENCOURSEWARE : http://ocw.mit.edu/

Les-Mathematiques.net : http://www.les-mathematique;net

Openlearninginitiative : http://www.cmu.edu/oli/courses/


Theactivityis,first,toassesstheknowledgeofthestudentonthecalculationsof partial derivatives and exact derivatives (Exercise 1) and then allow the student to implement their knowledge in the calculation of a line integral along a curve (Exercise 2). Finally, the student will complete an exercise on the eva-luation of a double integral (Exercise 3).


Evaluation

The 4 exercises are given under the form of multiple choice questions. The stu-dent should answer by checking the correct answer(s), and in the correct order.

Each exercise is worth 25% of the total mark.

Exercise 1

Determine the total differential of the function u(x,y) given below :

.u(x,y)=excos(y)

aq du = ex cos(y) dx + ex sin(y) dy

bq du = ex cos(y) dx - ex sin(y) dy

cq du = ex sin(y) dx - ex cos (y) dy

1.2.u(x,y)=ln(x2+y2)

a. q du =

2xx2 + y2

dx +

2 yx2 + y2

dy

b. q du =

2 yx2 + y2

dx +

2xx2 + y2

dy

c. q du =

2xx2 + y2

dx -

2 yx2 + y2

dy

Exercise 2

A 2-variable function u( x,y) is given such that du = 2y dx + 2x dy

1. Determine the correct expression for u(x,y) given the possible answers be-low

a. q u = xy b. q u = 2xy c. q u = 2(x+y)

2. We want to calculate the value of the integral : I = duC∫ , where C is an open

curve represented by the diagram below:

C is a circular arc with center O and radius equal to 1 ( in blue ).

Choose the correct answer :

a. q I = 0 b. q I = 1 c. q I = π


2.2 C is a line segment connecting points A (1,0) and B(0,1)

Choose the correct answer :

a. q I = 2 / 3 b. q I = 0 c. q I = 1/ 4

Exercise 3

O

x

C

A

B

Consider the curvilinear integral : I =

rF.d

rr

(C)∫Where :

( C ) is following a path on which the integral must be performed

drr is the elementary displacement on ( C )

rF = (x − y)

ri + (x + y)

rj

are unit vectors on the axes Ox and Oy of a Cartesian coordi-nate system Oxy.

If ( C ) is given by line segments connecting A(1,1), Q(3,1) and B(3,3), the value of the integral is :

a. q I = 10 b. q I = 2 c. q I = 12

Exercise 4

We would like to evaluate the following double integral :

2 3

0 1( )

xI x y dydx

−= +∫ ∫


4.1 The geometric nature of the integration region is :

q a rectangular surface

q a triangular surface

q a circular surface with radius = 1

The value of the integral is :

a. q 11/ 3 b. q 14 / 3 c. q 5 / 3

Exercise 5

5.1 The envelope of the line mamxy += ,where m is the parameter is

xay 22 4=


5.2 The envelope of the family of lines 1sincos =+ �� yx is 122 =+ yx


Learning activities

• Eachstudentmustfirstreadthecourseonnumericalanalysisbeforedoingthe exercises.

• Thetutorwillorganizethegroupforcollaborativework.

• Students will collaboratively discuss topics that were initially not understood under the supervision of the tutor.

• When the tutor decides that the students have achieved an appropriate level of understanding, they may begin the exercises.

• All groups address the same exercise at the same time under the supervision of the tutor, who will set the duration.

• Each group will appoint a leader who will put the names of all group mem-bers on the exercise before sending it by email attachment to the professor of the course.


Correct answers

Exercise 1

This exercise includes calculations of partial derivatives of two basic functions to two real variables, namely the cosine, logarithm and exponential. In general, the total differential of a function u (x, y) is written:

du(x, y) =

∂u∂x

dx + ∂u∂y

dy

Therefore, to obtain the correct answer, one must know how to calculate the partial derivatives and, above all, remember the derivatives of logarithmic, cosine and exponential functions

1.1 If u(x,y) = ex cos (y ) , we obtain :

1.2 If u(x,y) = ln ( x2 + y2 ) , we obtain :

The correct answers are b. in 1.1, and a. in 1.2.

A student who did not answer these correct should review the sections of the module on derivation.


Exercise 2

This exercise focuses on an example of calculating a line integral. The integral is in fact an exact differential, so the result should be independent of the curve betweenthetwopointspreviouslydefined.

1. Initially,thefirstquestionasksthestudenttoidentifythefunctionu(x,y)from the expression of its exact derivative. To get the correct answer, we must identify:

From these equations we can conclude : u(x,y) = 2xy


2. This exercise engages an exceptional ability in the learner (e). Wemustfirstestablishaparametricequationofthecurveinquestionandusethe result to express differential of the function of the parameter considered.

1. Consider the integration on the curve C. given by a semi-circle with radius equal to 1:

The parametric equation of C is : x = cos ( θ ) and y = sin ( θ ), θ ε [ 0, π / 2 ]

Where : dx = - sin ( θ ) d θ and dy = cos ( θ ) d θ

We obtain : d u = (cos 2 ( θ ) - sin 2 ( θ ) ) d θ

du = (cos2(θ) − sin2(θ))dθ = 0

0

π / 2

∫C∫

2. Consider the integration on the curve C given by a line segment connecting A and B :

The parametric equation of the segment is : x = t and y = - t + 1 , where t ε [ 0, 1 ]

Where : d u = ( - 2t + 1) d t

Thus :

du = (−2t + 1)dt = 0

0

1

∫C∫

The correct answers are a. in question 1 and b. in question 2.


Exercise 3

Evaluation of a curvilinear integral: I =

rF.d

rr

(C)∫Where :

(C)isapathonwhichtheintegralmustbeanalyzed

drr is the elementary displacement on ( C )

rF = (x − y)

ri + (x + y)

rj

Calculate the integral : rF.d

rr = (x − y)dx + (x + y)dy

This, the integral becomes : I = I1 + I

2 , such that :

In integral I1 , dx = 0 on the segment QB and y = 1 on AQ, thus :

I1 = (x − 1)dx =x2

2− x

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥1

3∫

1

3

= 2

In the integral I2 , dy = 0 on the segment AC and x = 3 on QB, thus :

I2 = (3+ y)dy = 3y +y2

2

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥1

3∫

1

3

= 10

Finally , we have : I = I1 + I

2 = 2 + 10 = 12



Exercise 4

4.1 A sketch of the curve shows that the region is bounded by y-axis(x=0), line x+y=3 joining points (0,3)and (1,2) and the line y=1


4.2 We evaluate the double integral by the following method :

I = (x + y)dydx

1

3− x

∫0

2

∫

= xy +12

y2⎡

⎣⎢⎤

⎦⎥0

2

∫1

3− x

= 3x[ − x2

0

2

∫ +92

− 3x +x2

2− x −

12

⎤

⎦⎥

= −[ x2

20

2

∫ − x + 4]

= −x3

6−

12

x2 + 4x⎡

⎣⎢

⎤

⎦⎥

0

2

=143


Exercise 5

5.1 Wefindtheenvelopeoftheline y = mx + am as follows ;

Rearranging the equation mx2 − my + a = 0 We let f = mx2 − my + a = 0

∂f∂m

= 2mx − y = 0; Solving for x and y ;

y = 2mx , substituting for y in the equation of the line gives

m2 x − 2m2 x + a = 0 ⇒ x = am2


Substutiting for x in the equation of the line gives

m2 am2 − my + a = 0 ⇒ y =

2am

or; m =2ay

∴ x =ay2

4a2 ⇒y2

4a= x

Equation of the envelope y2 = 4ax


5.2 Tofindtheenvelopeof x cosθ + ysinθ = 1 we let

f = x cosθ + ysinθ = 1

∂f∂θ

= −xsinθ + ycosθ = 0 ⇔ ycosθ = xsinθ ⇔ y = x tanθ

Substuting for y inthe original equation gives x cosθ − x tanθ sinθ = 1

⇔ x cos2 θ − xsin2 θ = cosθ ⇒ x = cosθ, y = sinθ ⇒ x2 + y2 = 1.The correct answer is a

Self-evaluation


Professor’s Guide



Learning activity # 3

Title : Numerical calculations : evaluation of a finite sum and an infinite sum,evaluationofanintegral.

Teaching hours

30 H

Instructions




Specific objectives

Upon completion of this module, the student should be able to :

• Apply fundamental principles to numerical calculations• Use the iteration method• Identify sources of error in numerical calculations• Estimate the error in approximations• Interpret the algorithm of a numerical calculation• Numericallycalculateafinitesumandaninfinitesum• Identify sources of error in numerical integrations• Applythetrapezoidalformula• Apply Simpson`s formulas• Identify the proper formulas for these cases

Summary

This activity deals with the numerical evaluation of integrals. Key elements are particularly developed namely the method of iteration, the estimation of truncation error, and the study of algorithms commonly used to evaluate an integralnumerically(trapezoidalformulaandSimpson’sformulas).


Required readings

RASOANAIVO, R. Y. ( 2006). Méthodes Numériques. Ecole Normale Su-périeure, Université d’Antananarivo, Madagascar.


Microsoft Excel , Maxima

Useful links

MITOPENCOURSEWARE : http://ocw.mit.edu/




Theactivityis,first,toassessthestudent’sknowledgeonthefundamentalsof numerical calculations, and then to allow the student to implement them innumericalcalculationsofadefiniteintegral.Simpleexercisesarethusfirstproposed to measure the level of understanding of the student (Exercise 1 and Exercise 2) and secondly, to put into practice the knowledge acquired (Exer-cise 3).


Evaluation

The students must complete all of the work in groups. The mark of the group is identical for all group members. Exercise 1 counts for 45% of the marks, exercise 2 for 30% of the points, and exercise 3 for 25% of the marks.

The student should check the correct answer(s).

Exercise 1

The equation x2 – 2x – 3 = 0 has a positive root and a negative root.

Which of the following iteration formulas leads to the positive root ?

a. q x = 2x + 3 b. q x =

3x − 2

Exercise 2

We would like to integrate the function y(x), whose numerical values are given in the table below, x varying from 1.0 to 2.5 :

x 1.1 1.5 1.9 2.3

y 1.6 5.2 6.3 4. 8

Theresultofcalculationsbasedonthetrapezoidalformulais:

a.q 11,07 b. q 5,88 c. q 7,19

Exercise 3

The following table gives numerical values of a function y(x) :

x 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8y 1.54 1.66 1.81 1.97 2.15 2.35 2.57 2.88 3.11

Calculate the integral:

I = y(x)dx

1.01.8

∫

Using the 1/3 Simpson’s formula.


Learning activities

• Eachstudentmustfirstreadtherequiredlecturesbeforedoingtheexerci-ses.


• Students will collaboratively discuss topics that were initially not unders-tood under the supervision of the tutor.


• All groups address the same exercise at the same time under the supervi-sion of the tutor, who will set the duration.

• Each group will appoint a leader who will put the names of all group members on the exercise before sending it by email attachment to the professor of the course.


Correct answers

Exercise 1

This exercise is an example of the iteration method frequently used in numeri-cal calculations. It consists of studying an algorithm for determining the roots of a function f (x). The equation in question is x2 - 2x - 3 = 0 To answer the question, we must perform the calculations.

Take : x

n+1= 2xn +3 , n = 0,1,2,….. and x

0 = 1

N 1 2 3 4

xn

2.2360 2.7334 2.9098 2.9697

Take :

xn+1 =

3xn − 2

, n = 0, 1, 2,….. and x0 = 1

N 1 2 3 4

xn

-3 -0.6 -1.1538 -0.9512

Theseresultsshowthatitisthefirstformulathatgivesthepositiveroot,whosenumerical value is equal to 3.


Exercise 2

Recallthetrapezoidalformula:

I = (h/2) [ y1 + 2 ( y

2 + y

3 +…..+ y

n-1 ) + y

n ]

h being the integration step

In the table below, h = 0,4

I = (0,4/2) [1,6 + 2 ( 5,2+6,3 ) + 4,8 ] = 5,88



Exercise 3

The following table gives numerical values for a function y(x) :

x 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8y 1.54 1.66 1.81 1.97 2.15 2.35 2.57 2.88

3.11

Calculate the integral:

I = y(x)dx

1.01.8

∫

By using the 1/3 Simpson formula.

When you need to integrate a function with only a few known points, you shouldalwaysstartbyanalyzingthearrayofvalues,andparticularlywhetherthe abscissa or the base points are equidistant and then count the number of intervals. The method used depends on the outcome of this preliminary analy-sis. The 1/3 Simpson formula is applicable in this case because it has an even number of intervals.

Recall the 1/3 Simpson formula :

I = (h/3) [[ y1 + 4 ( y

2 + y

3 +…..+ y

n-1 ) + y

n ]

In this case, the integration step is h = 0,1.

If we replace the yi values correctly, we should obtain I = 1,7714

Self-evaluation


Professor’s Guide



Learning activity 4

Title:Determinant:propertiesandevaluation.Matrixalgebra.Solvingasystemoflinearequations

Teaching hours

30 H

Instructions




Specific objectives

Upon completion of this activity, the student should be able to :

• Perform matrix operations

• Calculate the determinant of a matrix

• Calculate the inverse of a matrix

• Solve a system of linear equations

Summary

This activity involves three exercises related to solving a system of non-homo-geneous linear equations. The mastering of matrix operations and knowledge of the properties of the determinant of a matrix are essential when one must determinesolutionstoasystemofequations.Thefirsttwoexercisesaddresssome matrix operations and calculations for determining whether a matrix is invertible or not. The last exercise proposed is an application of knowledge obtained earlier in the activity.

Required readings

RASOANAIVO, R.-Y. ( 2006). Eléments d’algèbre linéaire, Ecole Normale Supérieure, Université d’Antananarivo, Madagascar


Multemedia resources

Microsoft Excel ; Mathematica

Useful links

MITOPEN COURSEWARE : http://ocw.mit.edu/




This activity includes exercises in the form of multiple choice questions. In most cases, the student is required to perform calculations on paper. Initially, thestudentmustperformmatrixoperationsandrealizethatsomeoperationsare not possible (Exercise 1). Then he / she will perform calculations to de-termine and infer the relationship with the inversion of a matrix (Exercise 2). Finally, the student will practice solving a system of linear equations.


Evaluation

All of the exercises must be completed in groups. The mark of the group will be applicable to all members of the group. Exercise 1 will count for 40% of the marks, exercise 2 for 30% of the marks, and exercise 3 for 30% of the marks.

Exercise 1

Consider the following matrices :

A =

1 2 12 0 01 −1 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

; B = 1 00 2-1 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

1. Which of the following operations can be performed ? :

a. q. A B b. q B BT c. q A BT d. q BT A e. q BA

2. We are given matrices B = A + AT and C = A - AT .

a. q B is symmetric b. q B is asymmetric ;

c. q C is symmetric d. q C est asymmetric

Exercise 2

Consider the matrices A and B:

A = 3 −1 31 2 32 −2 −1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

;

B =

3 −1 3

1 2 3

2 4 6

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

a. q A is invertible b. q A is not invertible

c. q B is invertible d. q B is not invertible


Exercise 3

Solve the following system of linear equations :

3x − y = 12

x + 2 y = 11

1.ThedeterminantΔisequalto:

a. q 5 b. q 7 c. q 11

2. The solutions are :

a.q x = 5, y = 3 b. q x = 13/7, y = 3 c. q x = 3 , y = 5

Learning activities

• Eachstudentmustfirstreadtheappropriatelecturesbeforedoingtheexercises.


• Students will collaboratively discuss topics that were initially not unders-tood under the supervision of the tutor.


• All groups address the same exercise at the same time under the supervi-sion of the tutor, who will set the duration.

• Each group will appoint a leader who will put the names of all group members on the exercise before sending it by email attachment to the professor of the course.


CORRECT ANSWERS

Exercise 1

This exercise will show whether or not the student has mastered matrix opera-tions.

1. The product of two matrices is possible only if the number of columns in the firstmatrixequalsthenumberofrowsinthesecond.Togettherightanswer,the student must check whether this rule is respected. Thus, answers a, b. and d. are correct, while c. and e. are incorrect.

The product AB est is possible since :

v A is a 3 x 3 matrix : 3 roms et 3columns

v B is a 3 x 2 matrix: 3rows et 2 columns.

The number of columns in A is equal to the number of rows in B

2. Toobtainthecorrectanswer,thestudentmustrecallthedefinitionofsym-metric and asymmetric matrices :

The correct answers are a. and d. since :

B = ( A + At ) is symmetric, however C = (A – At ) is asymmetric.

Exercise 2

This exercise asks the student to calculate the determinant of 2 matrices, since thereisnoinverseifthedeterminantiszero.

From calcuations done on paper, the student should discover :

• ThedeterminantofAisequalto-13,whichisdifferentfromzero• ThedeterminantofBisequaltozero

However,withoutperformingcalculations,itshouldberealizedthatthelasttwolinesofmatrixBareproportional,indicatingadeterminantofzero.

The correct answers are a. and d.


Exercise 3

This exercise provides an opportunity for the student to apply their knowledge in solving a system of two equations. To determine the values of x and y that satisfy both equations, the student should remember the procedure developed inthecourse,whichistofirstcalculatethedeterminantΔandthenusethefollowing formulas for x and y:

x =

1Δ

12 −1

11 2 ; y=

1Δ

3 12

1 11 où Δ =

3 -1

1 2

Whichgives:Δ=7,x=5andy=3


Self-evaluation


Professor’s Guide



XI. Key Terms of the Module

Series

In physics, we often group a number of terms together, or a «series» of terms. Aseriescaneitherhaveafinitenumberofterms,oraninfinitenumberofterms.Inthelattercase,itisan«infiniteseries». Note that a series can be used to represent either a constant or a function f (x). For the latter case, one speaks of a «whole series».

Convergence

The convergence of a series is linked to the fact that the sum of the terms of theseriesgivesafinitevaluewhenitreachesastablenumberoftermsintheseries, and otherwise we say that the series diverges.

Partial derivative

In physics, we frequently encounter a function that depends on several varia-bles. For example, in thermodynamics, the temperature T can depend on both the pressure P and volume V. If you want to know the temperature variation depending on the pressure, we must calculate the derivative of T according to P, without affecting V This situation leads to the partial derivative of T with respect to P, and V is considered constant in this operation. We write:

Total differential

Thetotaldifferentialofafunctiontoseveralrealvariablesistheinfinitesimalchangeofthefunctionwheneachofitsvariablesundergoesaninfinitesimalchange. Example:

df (x, y) =

∂f∂x

dx +∂f∂x

dy

The notation df(x,y) represents the exact differential of f(x,y)


Exact differential

Thetotaldifferentialissaidtobeexactifthefollowingconditionissatisfied:

∂∂y

(∂f∂x

) =∂∂x

(∂f∂y

)

Defined integral

Anintegralisknownas“defined”whenthelimitsofintegrationarespecified,otherwisewespeakofanintegralthatis“indefinite.”Theresultofadefiniteintegral is the value of the surface below the curve considered.

Curvilinear integral

The curvilinear integral is an integration operation along a curve. Example :

(i) f (x, y)d

rr

(C )∫ ; (ii) xdy

(C)—∫

( i ) : integration along an open curve ( C )

( ii ) : integration along a closed curve ( C )

Matrix

A matrix is an array of values containing n rows and m columns, denoted gene-rally n x m We can distinguish several kinds of matrices:

a. Square matrix: the number of rows equals the number of columns, n x nb. Column matrix: the number of columns is equal to 1, n x 1c. Row matrix: the number of rows is equal to 1, 1 x m

Approximate function

In physics, the results of experiments in the laboratory are often numerical values of the measured quantity, Y, according to some parameter x. The inter-pretation of this result requires some knowledge of the correlation between Yandx.Byanumericalmethod,theprocedureistofindafunctionFforan“approximate”sizeofY.Thefunctionthusdefinediscalledanapproximatefunction.


Base points

These are discrete values, xi, of the x parameter which depends on a physical quantity Y measured in the laboratory (see: Function Estimation). The pair (Yi, xi)definesapointinacoordinatesystemOxy.


XII . list of required Readings

Reading #1 :

Completereference : RASOANAIVO, R-Y. (2006). Eléments d’Analyse I. Ecole Normale Supérieure, Université d’Antananarivo, Madagascar

Summary: The course contains chapters dealing with aspects of mathematics, some already known by students, and learning to know the limit of a real varia-ble function, continuity of functions and calculus. In particular, the concepts of a differential and a total exact differential are introduced. Discussions based on well-chosen examples allow further analysis of these elements. Justification: This reading helps to consolidate the prerequisites acquired by the students. It provides information to solve some exercises in the module.

Reading #2 :

Completereference : RASOANAIVO, R.-Y. ( 2006). Eléments d’Analyse II. Ecole Normale Supérieure, Université d’Antananarivo, Madagascar

Summary : This course is devoted to the study of series: power series, Taylor series, Maclaurin series and Fourier series. The various tests of convergence are explained by the test of D’Alembert, Gauss’s test, etc ... The applications are illustrated with solved exercises. Justification: This reading allows students to have extensive knowledge of mathematical tools often used in science.

Reading # 3 :

Completereference : RASOANAIVO, R.-Y. ( 2006). Eléments d’Algèbre Linéaire. Ecole Normale Supérieure, Université d’Antananarivo, Madagascar

Summary: This course outlines the various properties of matrices and matrix operations: the addition and the product of two matrices. In addition, calcula-tions of determinants are illustrated by examples. These tools will be used to solve a system of linear equations which will be in the last chapter. Justification: This reading informs the student on some elements of algebra that are essential for solving many exercises in the module. In this sense, the reading is essential.


Reading # 4 :

Completereference : RASOANAIVO, R. Y.( 2006). Méthodes numériques. Ecole Normale Supérieure, Université d’Antananarivo, Madagascar

Summary: This course covers elements of numerical methods beginning with theiterationmethod,followedbymethodsofnumericalevaluationofafinitesumandaninfinitesum.Inaddition,thefinalchapterpresentssometechniquesofnumericalevaluationofanintegral:thetrapezoidalmethodandSimpson’smethods. Justification: The aspects developed in this course are required to solve some exercises in the assessment module. Reading by the student is therefore neces-sary.


XIII. Complete list of multimedia resources

Complete resource #1 : Microsoft ® Excel 2000

Description

It is a software designed and developed by engineers at Microsoft Corporation, which not only make statistical calculations, but also to plot curves represen-ting digital functions or functions with known analytical expressions. The software, through several integrated functions, is an effective learning tool, available in all computers that run Windows. Justification: Plotting functions.

Complete resource #2 : Maxima

Description

Maxima is a free algebra computer software under GPL licence package Mac-syma. This software can be downloaded directly by opening (see link # 6). Justification Maxima can perform calculations both analytically and numerically. So it is an essential learning tool for students.


XIV. Complete list of useful links

Useful link # 1

Title:WEBMATHS

URL : www.Webmaths.com/

Screencapture

Description: This site is free and offers courses in mathematics, algebra and analysis and can be viewed or downloaded free. In addition, the site provides a forum through which the student’s ideas can be exchanged. Justification: This site allows students not only to expand their knowledge but also to get in touch with other student in the world and discuss math problems with them in the Help Forum


Useful link # 2

Title Les-mathematiques.net :

URL : http://www.les-mathematiques.net/

Screencapture

Description :

This site is free, and offers mathematicians the chance to communicate within a forum. Courses are available and downloadable, and useful links are also suggested. Justification: The site offers a special opportunity for students to follow the evolution of ma-thematics by participating in a forum or to read the articles written by various teachers. The links suggested in this site allow students to update their records.


Useful link # 3 :

Title: MITOPENCOURSEWARE:

URL : http://ocw.mit.edu

Screencapture

Description: This site is free and provides a list of mathematics courses developed by MIT professors in the U.S., which are freely downloadable. The students may choose the topic they want to focus on. Justification:The additional courses are needed for the students for the following reasons: first,theywillreadotherapproachesorotherexplanationsfromteachers.Theywill have the opportunity to deepen their understanding of the concepts contai-ned in the module.


Useful link # 4 :

Title: Openlearninginitiative:

URL:http://www.cmu.edu/oli/courses/

Screencapture

Description: This site is free and provides a list of mathematics courses developed at Carne-gie Mellon, which are freely downloadable. The students may choose the topic they wish to focus on. Justification: Thissiteofferscoursesinvariousfields:mathematics,chemistry,statistics,biology, etc ... that can be used as online resources for students.


Useful link # 5

Title:Mathematicsportal

URL:http://en.wikipedia.org/wiki/Mathematics_portal

Screencapture

Description: Wikipedia is a free encyclopedia site and open to everyone, with a range of scientificarticlesthatstudentscanconsult. Justification: The portal leads to a world of mathematics where the students can findarticlesorcoursesthatinterestthem,asshowninthetablebelow.


Général

Géométrie Analyse

• Mathématiciens célèbres

• Histoire des mathématiques

• Philosophie des mathématiques

• Notation (mathématiques)

• Panorama des mathématiques

• Géométrie

• Géométrie différentielle

• Géométrie algébrique

• Géométrie analytique

• Géométrie synthétique

• Analyse réelle

• Analyse complexe

• Analyse fonctionnelle

• Analyse non standard

• Équation aux dérivées partielles

Algèbre Statistiquesetprobabilités Théoriedesnombres

• Algèbre

• Algèbre abstraite

• Algèbre linéaire

o Algèbre multilinéaire

o Algèbre tensorielle

• Théorie des ensembles

• Théorie de Galois (théorie des groupes)

• Algèbre commutative (théorie des anneaux)

• Statistiques

oLe portail des statistiques et des probabilités

oÉcart type

oMoyenne

• Probabilités

oLe portail des statistiques et des probabilités

oThéorie des probabilités

oLoi de probabilité

• Théorie des nombres

• Théorie analytique des nombres

• Liste des matières de la théorie des nombres

• Arithmétique

oArithmétique modulaire

• Nombre

• Théorie des corps de classes

• La théorie des fonctions L


Useful link # 6

Title:Maxima

URL: http://michel.gosse.free.fr/telechargement/index.htlm

Screencapture

Description It is a free software with several versions: Linux version and Windows version, whosesizeisaround9MB,sodownloadingshouldnotposeanyparticularproblems. Justification Downloadingthissoftwarecanhelpthestudentstoeliminatethedifficultiesencountered in certain mathematical operations.


XV. Synthesis of the module

This module is designed for future teachers of physics in high school. The content is composed of four units, namely «Elements and Analysis Series», «Calculus and Integration,» Numerical Methods» and» Linear Algebra «. This is to meet predetermined objectives such as mastering the tools needed to monitor the physics courseduringtheirtraining,andfinallytoteachscienceinasecondaryschool.

Each unit is a learning activity in which the student uses their knowledge in the context of a formative evaluation, that is to say:

• Astudentmustfirstreadthecoursesbeforedoingtheexercises.

• Thetutororganizescollaborativeworkfortheretobeexchangesbetweenstudents.

• Provisions of relevant and useful links are available to students.

• Students are required to respond to questions while complying with ins-tructions.

• Some correct answers are provided for students.

The unit entitled “Elements of Analysis and Series’, includes explanations of limits andcontinuityofarealvariablefunction,indefiniteanddefiniteintegrals,seriesandconvergencetests,andfinallytheseriesexpansionofarealfunction(Taylorseries and MacLaurin series). Some of these mathematical elements are already known by students, but it is used to expand their knowledge and enhance their skills. The unit entitled “Calculus and Integration” introduces students with new mathematical tools. The learning activity is concentrated on the calculation of partial derivatives, the calculation of curvilinear integrals and double integrals. The unit entitled “Numerical Methods”, is a key area of mathematics impor-tant for future scientists. Students have the opportunity to interpret numeri-cal values to estimate errors due to approximations, as well as manipulating algorithms.Covered also are the numerical ratings of a finite sum and aninfinitesuminvolvingsystematic iterationmethodswidelyused innumericalcalculations. Moreover, the numerical evaluations of integrals provide inte-resting examples of algorithms, such as the trapezoid formula andSimpson.


Finally, the unit entitled “Linear Algebra” covers matrix calculations, the calculation of the determinant of a matrix and, finally, solving a system of linear equations. Indeed, many physical problems lead to a system of linear equations. Conventional methods are explained, in particular, the method of Gaussian elimination and the use of the inverse of a matrix. The explanations of the contents of this module are developed in works specially designed for this module. Other relevant courses are in open sites listed under “Useful Links”.


XVI. evaluation Summative

The student should check the correct answer(s) to the multiple choice ques-tions.

1. Consider :

limx→0

(1− cos(x)

x2) = L

a. q L = 0 b. q L = 1 c. q L = ½

2. Consider :

limx→∞

1−2x

⎛

⎝⎜⎞

⎠⎟

x= L

a. q L = -2 b. q L = e-2 c. q L = ln(2)

3. The nth derivative of f(x) = sin(x) is written f(n) (x) = sin (x + n π/2)


4. Consider :

S(x) = (−1)n(n + 1)xn

n=0

∞∑


The series is absolutely convergent for :

a. q x ∈] − ∞,+∞[ b. q x ∈] − 1,+1[ c. q x ∈] − 1,+1]

d.q x ∈[−1,+1[ e. q x ∈] − 1,+1[

5. Consider :

f (x) =

12

ln(1+ x1− x

)

The radius of convergence R of the representative series of f(x) is :

a. q R = 1/2 b. q R=1 c. q R=2

6. Consider :

I = x

11,4

∫ dx

a. CalculateInumericallyusingthetrapezoidalmethodusinganintegrationstep of

h = 0,1.

b. Compare with the analytical result

7. Analytically calculate the integral :

I =

2π

sin(x)

x0π

∫ dx

8. Consider the matrix :

A =

−1 4 0

−2 6 1

3 −8 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟


ThedeterminantΔofAisequalto:

a. qΔ=1 b.qΔ=4 c.q R= - 4


A =

0 1 1

1 0 2

2 1 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

a. Calculate the inverse of A.

b. Solve :

x + z = −1

x + 2z = 0

2x + y + z = 1

10. Develop the Fourier series of the following function :

f (x) =

x , 0<x<π

−x , -π<x<0

⎧⎨⎩

Correct answers

1. Consider :

limx→0

(1− cos(x)

x2) = L

We start with the development of the function cos(x) :



2. Consider :

limx→∞

1−2x

⎛

⎝⎜⎞

⎠⎟

x= L

Calculate : ln f(x)

ln(f (x)) = x ln(1−

2x

)

By developing the MacLaurin series of ln(1+x), we obtain :

ln(1−

2x

) = (−2x

) −12

(−2x

)2 +13

(−2x

)3 − .....

Where :

ln(f (x)) = −2 −12

(22

x) −

13

(23

x2) − .....

Bytakingthelimitasxapproachesinfinity,weobtainthefollowingresult:

lim

x→∞ln (f (x)) = − 2 ⇒ lim

x→∞f(x) = e-2



3. The nth derivative of f(x) = sin(x) is written f(n) (x) = sin (x + n π/2)

Recall that: ( sin (x ) )’ = cos(x) ; (cos (x) )’ = - sin (x ) ;

sin ( x + π/2) = cos (x ) ; sin ( x + π) = - sin (x )

Apply a reasoning by induction :

For n = 1, f (1) (x) = ( sin (x ) )’ = cos (x) = sin(x + π/2) , thus it is true

For n = 2, f (2) (x) = [ f (1) (x) ]’ = ( cos(x) )’ = - sin (x) = sin(x + π) , thus it is true

Demonstrate that, if it is true for n, it is also true for (n + 1)

f( n+1) (x) = [ f ( n ) (x)] ’ = cos (x + n π/2) = sin [ (x + n π/2) + π/2 ] = sin (x + (n+1) π/2 )

4. Consider :

S(x) = (−1)n (n + 1) xn

n=0

∞∑

We apply D’Alembert’s test , that being we consider :

Un+1Un

Since we have an alternating series, we take the limit of the absolute value of theaboverelationasnapproachesinfinity.

lim

n→∞|Un+1Un

|

If this limit is less than 1, then the series is absolutely convergent. Verify :

lim

n→∞|Un+1Un

| = limn→∞

n+1n

| x | = |x|


We determine that the series converges when |x | < 1.

The series is not conclusive for x = ± 1. However, if we calculate S(± 1), the sumoftheseriesincreasesindefinitelysowecanconcludethattheseriesdiverges for x = ± 1.

The correct answer is e.

5. The radius of convergence of the series representing the function:

f (x) =

12

ln(1+ x1− x

)

Recall that :

f(x) = (1/2) [ ln(1+x) – ln(1-x) ]

However :

Indeed, the interval of convergence is valid simultaneously for both sets of ln (1 + x), and ln (1-x) must be the open interval] -1, 1 [.

The correct answer is e.


6. Calculate the integral :

I = x

11,4

∫ dx

a. Applythetrapezoidalmethodwithanintegrationsteph=0,1.

Construct a table of the integrant of f(x) :

x 1 1,1 1,2 1,3 1,4f(x) 1 1,0488 1,0954 1,1401 1,1832

In this particular case, the formula is written:

I = (h/2) [ f1 + 2 ( f

2 + f

3 + f

4 ) + f

5 ], where f

i are the values of f(x) at the abs-

cissas xi .

By replacing fi by their respective values, we obtain : I = 0,4375

b. The analytical calculation gives :

I = [

23

x3/2 ]11,4 = 0,4376

The two results are almost identical.

7. Analytical calculation of the integral :

I =

2π

sin(x)

x0π

∫ dx

We use the series representing the function sin(x)


The integration gives :

I =

2π

[x −13

(x3

3!) +

15

(x5

5!) −

17

(x7

7!) + ...]0

π

Thefinalresultisobtainedbyreplacingxbyπ. If we retain 5 terms of the series, we obtain:

I = 1,1794 which is the correct answer

8. Calculating the determinant of a matrix :

A =

−1 4 0

−2 6 1

3 −8 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

We will develop the answer by following the last column. The answer should not depend on this choice.

Δ = (−1)

−1 4

3 8= 4

Thus, the correct answer is b.


A =

0 1 1

1 0 2

2 1 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

c. Calculate the inverse of A.

Calculatethedeterminantfirst,toseeifitisinvertibleornot.


Δ = (−1)

1 2

2 1+ (1)

1 0

2 1= 4

Sincethedeterminantisnotzero,thematrixisinvertible

Remember that to determine the elements of inverse matrix, calculate the diffe-rent cofactors for each element of the transposed matrix AT of A:

AT =

0 1 1

1 0 1

2 2 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Thecofactorscorrespondtoelementsofthefirstcolumn:

For element 0, we have

0 2

1 1= −2 ; For element 1, we have

(−1)

1 2

2 1= 3

For element 2, we have

1 1

0 1= 1

The inverse matrix is written as :

A−1 =14

−2 0 2

3 −2 1

1 2 −1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Verify is sometimes necessary. If we compute the product AA-1 we should have the identity matrix.

d. Solution of:


x + z = −1

x + 2z = 0

2x + y + z = 1

Write the system of equations in matrix form :

0 1 1

1 0 2

2 1 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

x

y

z

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=

−1

0

1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Note that the 3 x 3 matrix is the same as «A » from the previous problem, and the inverse has already been calculated.

The answer is obtained by multiplying the above equation by A-1. We should obtain :

x

y

z

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=14

−2 0 2

3 −2 1

1 2 −1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−1

0

1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

=

1

−1 / 2

−1 / 2

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

The solutions are : x=1;y=-1/2;z=-1/2

10. Development the function into a Fourier series :

f (x) =

x , 0<x<π

−x , -π<x<0

⎧⎨⎩

The function f(x) is an even function, the corresponding Fourier series is under the form :


f (x) =ao2

+ ann=1

∞∑ cos(nx)

where ao and a

n are given by :

ao =

1π

f(t)dt ; an-ππ

∫ = f(t) cos(-ππ

∫ nt)dt

These expressions allow the constants of the Fourier series to be obtained, by replacing f(t) by its expression :

ao =

1π

[ (−t) dt−π0

∫ + (t)0π

∫ dt ] = 2π

t dt = π0π

∫

Thus the result is :

4 1f (x) - cos[(2n 1)x]

22 (2n+1)

π= +

π ∑

XVII . References

ARFKEn, G. (1970). Mathematical Methods for Physicists, Academic Press, Inc


BRONSON, R. (1989). Matrix Operations, Schaum’s outline series, McGraw-Hill, Inc.

GERALD, C-F,(1980). Applied Numerical Analysis, Addison-Wesley Pu-blishing company

RASOANAIVO, R-Y. ( 2006). Eléments d’Analyse I , Ecole Normale Supérieure, Université d’Antananarivo, Madagascar

RASOANAIVO, R-Y. ( 2006). Eléments d’Analyse II, Ecole Normale Su-périeure, Université d’Antananarivo, Madagascar. Cours inédit

RASOANAIVO, R-Y. ( 2006). Eléments d’Algèbre Linéaire, Ecole Normale Supérieure, Université d’Antananarivo, Madagascar

RASOANAIVO, R-Y. ( 2006). Méthodes numériques, Ecole Normale Su-périeure, Université d’Antananarivo, Madagascar

RILEY, K-F, HOBSON, M-P et BENCE S-J, (2004). Mathematical Methods for physics and engineering, Cambridge University Press

KREYSZIK, E. (1972). Advanced Engineering Mathematics, John Wiley and Sons, Inc.

HURLy, J-F. (1980). Intermediate calculus, Saunders College

SCHEID, F. ( 1988). Numerical Analysis, Schaum’s outline series, McGraw-Hill, Inc.

• www.Webmaths.com/

• http://www.les-mathematiques.net/

• http://ocw.mit.edu

• http://www.cmu.edu/oli/courses/

• http://fr.wikipedia.org/Portail;Math%C3%A9matiques

• http://michel.gosse.free.fr/telechargement/index.htlm

1

MATHEMATICAL PHYSICS 1

Required Readings

Source: Wikipedia.org

2

Table of Contents Series (mathematics) ................................................................................................................................... 4

Basic properties ...................................................................................................................................... 4

Potential confusion ............................................................................................................................. 4

Convergent series ............................................................................................................................... 5

Examples ............................................................................................................................................. 6

Properties of series ................................................................................................................................. 7

Non-negative terms ............................................................................................................................ 7

Absolute convergence ........................................................................................................................ 7

Conditional convergence ................................................................................................................... 8

Convergence tests ................................................................................................................................... 8

Series of functions .................................................................................................................................. 9

Power series ...................................................................................................................................... 10

Laurent series ................................................................................................................................... 11

Dirichlet series .................................................................................................................................. 11

Trigonometric series ........................................................................................................................ 12

History of the theory of infinite series ................................................................................................ 12

Development of infinite series ......................................................................................................... 12

Convergence criteria ........................................................................................................................ 13

Uniform convergence ....................................................................................................................... 14

Semi-convergence ............................................................................................................................. 14

Fourier series .................................................................................................................................... 14

Generalizations ..................................................................................................................................... 15

Asymptotic series ............................................................................................................................. 15

Divergent series ................................................................................................................................ 15

Series in Banach spaces ................................................................................................................... 15

Summations over arbitrary index sets ........................................................................................... 16

Taylor series .............................................................................................................................................. 19

Definition .............................................................................................................................................. 20

Derivation ............................................................................................................................................. 21

Examples ............................................................................................................................................... 22

Convergence ......................................................................................................................................... 23

3

History ................................................................................................................................................... 24

Properties .............................................................................................................................................. 25

List of Maclaurin series of some common functions ......................................................................... 26

Calculation of Taylor series ................................................................................................................ 29

First example .................................................................................................................................... 29

Second example ................................................................................................................................ 30

Taylor series as definitions .................................................................................................................. 31

Taylor series in several variables ........................................................................................................ 32

Matrix calculus .......................................................................................................................................... 32

Notice ..................................................................................................................................................... 33

Notation ................................................................................................................................................. 33

Vector calculus ..................................................................................................................................... 33

Matrix calculus ..................................................................................................................................... 34

Identities ............................................................................................................................................... 35

Examples ............................................................................................................................................... 36

Derivative of linear functions .......................................................................................................... 36

Derivative of quadratic functions ................................................................................................... 36

Derivative of matrix traces .............................................................................................................. 37

Derivative of matrix determinant ................................................................................................... 37

Relation to other derivatives ............................................................................................................... 37

Partial derivative ....................................................................................................................................... 37

Introduction .......................................................................................................................................... 37

Definition .............................................................................................................................................. 39

Basic definition ................................................................................................................................. 39

Formal definition ............................................................................................................................. 40

Examples ............................................................................................................................................... 41

Notation ................................................................................................................................................. 42

Antiderivative analogue ....................................................................................................................... 43

4

Series (mathematics)

In mathematics, given an infinite sequence of numbers { an }, a series is informally the result of

adding all those terms together: a1 + a2 + a3 + · · ·. These can be written more compactly using

the summation symbol ∑. An example is the famous series from Zeno's dichotomy

The terms of the series are often produced according to a certain rule, such as by a formula, or by

an algorithm. As there are an infinite number of terms, this notion is often called an infinite

series. Unlike finite summations, infinite series need tools from mathematical analysis to be fully

understood and manipulated. In addition to their ubiquity in mathematics, infinite series are also

widely used in other quantitative disciplines such as physics and computer science.

[] Basic properties

Series can be composed of terms from any one of many different sets including real numbers,

complex numbers, and functions. The definition given here will be for real numbers, but can be

generalized.

Given an infinite sequence of real numbers { an }, define

Call SN the partial sum to N of the sequence { an }, or partial sum of the series. A series is the

sequence of partial sums, { SN }.

[] Potential confusion

When talking about series, one can refer either to the sequence { SN } of the partial sums, or to

the sum of the series,

i.e., the limit of the sequence of partial sums (see the formal definition in the next section) – it is

clear which one is meant from context. To make a distinction between these two completely

different objects (sequence vs. summed value), one sometimes omits the limits (atop and below

the sum's symbol), as in

http://en.wikipedia.org/wiki/Mathematics

http://en.wikipedia.org/wiki/Infinite_set

http://en.wikipedia.org/wiki/Sequence

http://en.wikipedia.org/wiki/Summation

http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Proposed_solutions

http://en.wikipedia.org/wiki/Formula

http://en.wikipedia.org/wiki/Algorithm

http://en.wikipedia.org/wiki/Mathematical_analysis

http://en.wikipedia.org/wiki/Real_numbers

http://en.wikipedia.org/wiki/Complex_numbers

http://en.wikipedia.org/wiki/Function_%28mathematics%29

5

in order to refer to the formal series, that may or may not have a definite sum.

[] Convergent series

A series ∑an is said to 'converge' or to 'be convergent' when the sequence SN of partial sums has

a finite limit. If the limit of SN is infinite or does not exist, the series is said to diverge. When the

limit of partial sums exists, it is called the sum of the series

The easiest way that an infinite series can converge is if all the an are zero for n sufficiently

large. Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.

Working out the properties of the series that converge even if all the terms are non-zero is the

essence of the study of series. Consider the example

It is possible to "visualize" its convergence on the real number line: we can imagine a line of

length 2, with successive segments marked off of lengths 1, ½, ¼, etc. There is always room to

mark the next segment, because the amount of line remaining is always the same as the last

segment marked: when we have marked off ½, we still have a piece of length ½ unmarked, so we

can certainly mark the next ¼. This argument does not prove that the sum is equal to 2 (although

it is), but it does prove that it is at most 2. In other words, the series has an upper bound. Proving

that the series is equal to 2 requires only elementary algebra, however. If the series is denoted S,

it can be seen that

Therefore,

Mathematicians extend the idiom discussed earlier to other, equivalent notions of series. For

instance, when we talk about a recurring decimal, as in

we are talking, in fact, just about the series

http://en.wikipedia.org/wiki/Convergent_series

http://en.wikipedia.org/wiki/Limit_of_a_sequence

http://en.wikipedia.org/wiki/Divergent_series

http://en.wikipedia.org/wiki/Real_number

http://en.wikipedia.org/wiki/Repeating_decimal

6

But since these series always converge to real numbers (because of what is called the

completeness property of the real numbers), to talk about the series in this way is the same as to

talk about the numbers for which they stand. In particular, it should offend no sensibilities if we

make no distinction between 0.111… and 1/9. Less clear is the argument that 9 × 0.111… =

0.999… = 1, but it is not untenable when we consider that we can formalize the proof knowing

only that limit laws preserve the arithmetic operations. See 0.999... for more.

[] Examples

A geometric series is one where each successive term is produced by multiplying the

previous term by a constant number. Example:

In general, the geometric series

converges if and only if |z| < 1.

The harmonic series is the series

The harmonic series is divergent.

An alternating series is a series where terms alternate signs. Example:

The series

converges if r > 1 and diverges for r ≤ 1, which can be shown with the integral criterion

described below in convergence tests. As a function of r, the sum of this series is

Riemann's zeta function.

A telescoping series

http://en.wikipedia.org/wiki/Real_numbers

http://en.wikipedia.org/wiki/Complete_space

http://en.wikipedia.org/wiki/0.999...

http://en.wikipedia.org/wiki/Geometric_series

http://en.wikipedia.org/wiki/If_and_only_if

http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29

http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29#Divergence_of_the_harmonic_series

http://en.wikipedia.org/wiki/Alternating_series

http://en.wikipedia.org/wiki/Series_%28mathematics%29#Convergence_tests

http://en.wikipedia.org/wiki/Riemann_zeta_function

http://en.wikipedia.org/wiki/Telescoping_series

7

converges if the sequence bn converges to a limit L as n goes to infinity. The value of the

series is then b1 − L.

[] Properties of series

Series are classed not only by whether they converge or diverge: they can also be split up based

on the properties of the terms an (absolute or conditional convergence); type of convergence of

the series (pointwise, uniform); the class of the term an (whether it is a real number, arithmetic

progression, trigonometric function); etc.

[] Non-negative terms

When an is a non-negative real number for every n, the sequence SN of partial sums is non-

decreasing. It follows that a series ∑an with non-negative terms converges if and only if the

sequence SN of partial sums is bounded.

For example, the series

is convergent, because the inequality

and a telescopic sum argument imply that the partial sums are bounded by 2.

[] Absolute convergence

Main article: Absolute convergence

A series

is said to converge absolutely if the series of absolute values


http://en.wikipedia.org/wiki/Absolute_convergence

http://en.wikipedia.org/wiki/Absolute_value

8

converges. It can be proved that this is sufficient to make not only the original series converge to

a limit, but also for any reordering of it to converge to the same limit.

[] Conditional convergence

Main article: Conditional convergence

A series of real or complex numbers is said to be conditionally convergent (or semi-

convergent) if it is convergent but not absolutely convergent. A famous example is the

alternating series

which is convergent (and its sum is equal to ln 2), but the series formed by taking the absolute

value of each term is the divergent harmonic series. The Riemann series theorem says that any

conditionally convergent series can be reordered to make a divergent series, and moreover, if the

an are real and S is any real number, that one can find a reordering so that the reordered series

converges with sum equal to S.

Abel's test is an important tool for handling semi-convergent series. If a series has the form

where the partial sums BN = b0 + ··· + bn are bounded, λn has bounded variation, and lim λn Bn

exists:

then the series ∑ an is convergent. This applies to the pointwise convergence of many

trigonometric series, as in

with 0 < x < 2π. Abel's method consists in writing bn+1 = Bn+1 − Bn, and in performing a

transformation similar to integration by parts (called summation by parts), that relates the given

series ∑ an to the absolutely convergent series

[] Convergence tests

http://en.wikipedia.org/wiki/Conditional_convergence

http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29

http://en.wikipedia.org/wiki/Riemann_series_theorem

http://en.wikipedia.org/wiki/Abel%27s_test

http://en.wikipedia.org/wiki/Integration_by_parts

http://en.wikipedia.org/wiki/Summation_by_parts

9

Main article: convergence tests

n-th term test: If limn→∞ an ≠ 0 then the series diverges.

Comparison test 1: If ∑bn is an absolutely convergent series such that |an | ≤ C |bn | for

some number C and for sufficiently large n , then ∑an converges absolutely as well. If

∑|bn | diverges, and |an | ≥ |bn | for all sufficiently large n , then ∑an also fails to converge

absolutely (though it could still be conditionally convergent, e.g. if the an alternate in

sign).

Comparison test 2: If ∑bn is an absolutely convergent series such that |an+1 /an | ≤

|bn+1 /bn | for sufficiently large n , then ∑an converges absolutely as well. If ∑|bn |

diverges, and |an+1 /an | ≥ |bn+1 /bn | for all sufficiently large n , then ∑an also fails to

converge absolutely (though it could still be conditionally convergent, e.g. if the an

alternate in sign).

Ratio test: If |an+1/an| approaches a number less than one as n approaches infinity, then

∑an converges absolutely. When the limit of the ratio is 1, convergence can sometimes be

determined as well.

Root test: If there exists a constant C < 1 such that |an|1/n

≤ C for all sufficiently large n,

then ∑an converges absolutely.

Integral test: if ƒ(x) is a positive monotone decreasing function defined on the interval [1,

∞) with ƒ(n) = an for all n, then ∑an converges if and only if the integral ∫1∞ ƒ(x) dx is

finite.

Cauchy's condensation test: If an is non-negative and non-increasing, then the two series

∑an and ∑2ka(2

k) are of the same nature: both convergent, or both divergent.

Alternating series test: A series of the form ∑(−1)n an (with an ≥ 0) is called alternating.

Such a series converges if the sequence an is monotone decreasing and converges to 0.

The converse is in general not true.

For some specific types of series there are more specialized convergence tests, for

instance for Fourier series there is the Dini test.

[] Series of functions

Main article: Function series

A series of real- or complex-valued functions

converges pointwise on a set E, if the series converges for each x in E as an ordinary series of

real or complex numbers. Equivalently, the partial sums

converge to ƒ(x) as N → ∞ for each x ∈ E.

http://en.wikipedia.org/wiki/Convergence_tests

http://en.wikipedia.org/wiki/N-th_term_test

http://en.wikipedia.org/wiki/Comparison_test


http://en.wikipedia.org/wiki/Comparison_test

http://en.wikipedia.org/wiki/Ratio_test

http://en.wikipedia.org/wiki/Root_test

http://en.wikipedia.org/wiki/Integral_test_for_convergence

http://en.wikipedia.org/wiki/Monotone_decreasing

http://en.wikipedia.org/wiki/Interval_%28mathematics%29

http://en.wikipedia.org/wiki/Integral

http://en.wikipedia.org/wiki/Cauchy%27s_condensation_test

http://en.wikipedia.org/wiki/Alternating_series_test


http://en.wikipedia.org/wiki/Monotone_decreasing

http://en.wikipedia.org/wiki/Fourier_series

http://en.wikipedia.org/wiki/Dini_test

http://en.wikipedia.org/wiki/Function_series

http://en.wikipedia.org/wiki/Pointwise_convergence

10

A stronger notion of convergence of a series of functions is called uniform convergence. The

series converges uniformly if it converges pointwise to the function ƒ(x), and the error in

approximating the limit by the Nth partial sum,

can be made small independently of x by choosing a sufficiently large N.

Uniform convergence is desirable for a series because many properties of the terms of the series

are then retained by the limit. For example, if a series of continuous functions converges

uniformly, then the limit function is also continuous. Similarly, if the ƒn are integrable on a

closed and bounded interval I and converge uniformly, then the series is also integrable on I and

can be integrated term-by-term. Tests for uniform convergence include the Weierstrass' M-test,

Abel's uniform convergence test, Dini's test, and the Cauchy criterion.

More sophisticated types of convergence of a series of functions can also be defined. In measure

theory, for instance, a series of functions converges almost everywhere if it converges pointwise

except on a certain set of measure zero. Other modes of convergence depend on a different

metric space structure on the space of functions under consideration. For instance, a series of

functions converges in mean on a set E to a limit function ƒ provided

as N → ∞.

[] Power series

Main article: Power series

Many functions can be represented as Taylor series; these are infinite series involving powers of

the independent variable and are also called power series. For example, the series

converges to ex for all x.

In general, a power series is any series of the form

Such a series converges on a certain open disc of convergence centered at the point c, and may

also converge at some of the points of the boundary. The radius of this disc is known as the

http://en.wikipedia.org/wiki/Uniform_convergence

http://en.wikipedia.org/wiki/Integral

http://en.wikipedia.org/wiki/Weierstrass_M-test

http://en.wikipedia.org/wiki/Abel%27s_uniform_convergence_test

http://en.wikipedia.org/wiki/Dini%27s_test

http://en.wikipedia.org/wiki/Cauchy_criterion

http://en.wikipedia.org/wiki/Measure_theory

http://en.wikipedia.org/wiki/Measure_theory

http://en.wikipedia.org/wiki/Almost_everywhere

http://en.wikipedia.org/wiki/Null_set

http://en.wikipedia.org/wiki/Modes_of_convergence

http://en.wikipedia.org/wiki/Metric_space

http://en.wikipedia.org/wiki/Power_series

http://en.wikipedia.org/wiki/Taylor_series

11

radius of convergence, and can in principle be determined from the asymptotics of the

coefficients an. The convergence is uniform on closed and bounded (that is, compact) subsets of

the interior of the disc of convergence: to wit, it is uniformly convergent on compact sets.

Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even

if they were not convergent. When calculus was put on a sound and correct foundation in the

nineteenth century, rigorous proofs of the convergence of series were always required. However,

the formal operation with non-convergent series has been retained in rings of formal power series

which are studied in abstract algebra. Formal power series are also used in combinatorics to

describe and study sequences that are otherwise difficult to handle; this is the method of

generating functions.

[] Laurent series

Main article: Laurent series

Laurent series generalize power series by admitting terms into the series with negative as well as

positive exponents. A Laurent series is thus any series of the form

If such a series converges, then in general it does so in an annulus rather than a disc, and possibly

some boundary points. The series converges uniformly on compact subsets of the interior of the

annulus of convergence.

[] Dirichlet series

Main article: Dirichlet series

A Dirichlet series is one of the form

where s is a complex number. For example, if all an are equal to 1, then the Dirichlet series is the

Riemann zeta function

Like the zeta function, Dirichlet series in general play an important role in analytic number

theory. Generally a Dirichlet series converges if the real part of s is greater than a number called

the abscissa of convergence. In many cases, a Dirichlet series can be extended to an analytic

http://en.wikipedia.org/wiki/Radius_of_convergence

http://en.wikipedia.org/wiki/Closed_set

http://en.wikipedia.org/wiki/Bounded_set

http://en.wikipedia.org/wiki/Compact_set

http://en.wikipedia.org/wiki/Compact_convergence

http://en.wikipedia.org/wiki/Leonhard_Euler

http://en.wikipedia.org/wiki/Formal_power_series

http://en.wikipedia.org/wiki/Abstract_algebra

http://en.wikipedia.org/wiki/Combinatorics


http://en.wikipedia.org/wiki/Generating_function

http://en.wikipedia.org/wiki/Laurent_series

http://en.wikipedia.org/wiki/Annulus_%28mathematics%29

http://en.wikipedia.org/wiki/Dirichlet_series

http://en.wikipedia.org/wiki/Dirichlet_series

http://en.wikipedia.org/wiki/Complex_number

http://en.wikipedia.org/wiki/Riemann_zeta_function

http://en.wikipedia.org/wiki/Analytic_number_theory

http://en.wikipedia.org/wiki/Analytic_number_theory

http://en.wikipedia.org/wiki/Analytic_function

12

function outside the domain of convergence by analytic continuation. For example, the Dirichlet

series for the zeta function converges absolutely when Re s > 1, but the zeta function can be

extended to a holomorphic function defined on with a simple pole at 1.

This series can be directly generalized to general Dirichlet series.

[] Trigonometric series

Main article: Trigonometric series

A series of functions in which the terms are trigonometric functions is called a trigonometric

series:

The most important example of a trigonometric series is the Fourier series of a function.

[] History of the theory of infinite series

[] Development of infinite series

Egyptians like Ahmes, after 2050 BCE, and Greek, Arab, and medieval scribes like Fibonacci

wrote rational numbers in binary base 10 Egyptian fraction numeration systems for 3700 years.

Egyptian fractions placed an Old Kingdom Eye of Horus numeration system. Positive rational

numbers were scaled by least common multiples in Ahmes' RMP 2/n tables and unit fraction

series answers to 87 problems. Irrational and higher order numbers like pi were approximated.

The traditional pi value of 256/81 was changed to 22/7 in Rhind Mathematical Papyrus (RMP)

38 that corrected for grain losses, scaled in hekat units.

Greek mathematician Archimedes produced the first known summation of an infinite series with

a method that is still used in the area of calculus today written in Egyptian fractions. He used the

method of exhaustion and an Egyptian fraction method proof to calculate the area under the arc

of a parabola with the summation of an infinite series, and gave a remarkably accurate

approximation of π.[1][2]

The modern idea of an infinite series expansion of a function was conceived in India by

Madhava in the 14th century, who also developed precursors to the modern concepts of the

power series, the Taylor series, the Maclaurin series, rational approximations of infinite series,

and infinite continued fractions. He discovered a number of infinite series, including the Taylor

series of the trigonometric functions of sine, cosine, tangent and arctangent, the Taylor series

approximations of the sine and cosine functions, and the power series of the radius, diameter,

circumference, angle θ, π and π/4. His students and followers in the Kerala School further


http://en.wikipedia.org/wiki/Analytic_continuation

http://en.wikipedia.org/wiki/General_Dirichlet_series

http://en.wikipedia.org/wiki/Trigonometric_series

http://en.wikipedia.org/wiki/Trigonometric_function


http://en.wikipedia.org/wiki/Ahmes

http://en.wikipedia.org/wiki/Fibonacci

http://en.wikipedia.org/wiki/Egyptian_fraction

http://en.wikipedia.org/wiki/Eye_of_Horus

http://en.wikipedia.org/wiki/Least_common_multiple

http://en.wikipedia.org/wiki/RMP_2/n_table

http://en.wikipedia.org/wiki/Unit_fraction

http://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus

http://en.wikipedia.org/wiki/Hekat

http://en.wikipedia.org/wiki/Greek_mathematics

http://en.wikipedia.org/wiki/Archimedes

http://en.wikipedia.org/wiki/Egyptian_fraction

http://en.wikipedia.org/wiki/Method_of_exhaustion

http://en.wikipedia.org/wiki/Area

http://en.wikipedia.org/wiki/Parabola

http://en.wikipedia.org/wiki/Pi

http://en.wikipedia.org/wiki/Series_%28mathematics%29#cite_note-0

http://en.wikipedia.org/wiki/Series_%28mathematics%29#cite_note-0

http://en.wikipedia.org/wiki/Finite_set

http://en.wikipedia.org/wiki/Indian_mathematics

http://en.wikipedia.org/wiki/Madhava_of_Sangamagrama



http://en.wikipedia.org/wiki/Maclaurin_series

http://en.wikipedia.org/wiki/Continued_fraction




http://en.wikipedia.org/wiki/Sine

http://en.wikipedia.org/wiki/Cosine

http://en.wikipedia.org/wiki/Tangent_%28trigonometric_function%29

http://en.wikipedia.org/wiki/Arctangent


http://en.wikipedia.org/wiki/Radius

http://en.wikipedia.org/wiki/Diameter

http://en.wikipedia.org/wiki/Circumference

http://en.wikipedia.org/wiki/%CE%98

http://en.wikipedia.org/wiki/%CE%A0

http://en.wikipedia.org/wiki/Kerala_School

13

expanded his works with various other series expansions and approximations, until the 16th

century[citation needed]

.

In the 17th century, James Gregory worked in the new decimal system on infinite series and

published several Maclaurin series. In 1715, a general method for constructing the Taylor series

for all functions for which they exist was provided by Brook Taylor. Leonhard Euler in the 18th

century, developed the theory of hypergeometric series and q-series.

[] Convergence criteria

The study of the convergence criteria of a series began with Madhava in the 14th century, who

developed tests of convergence of infinite series, which his followers further developed at the

Kerala School[citation needed]

.

In Europe, however, the investigation of the validity of infinite series is considered to begin with

Gauss in the 19th century. Euler had already considered the hypergeometric series

on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and

the questions of remainders and the range of convergence.

Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are

convergent their product is not necessarily so, and with him begins the discovery of effective

criteria. The terms convergence and divergence had been introduced long before by Gregory

(1668). Leonhard Euler and Gauss had given various criteria, and Colin Maclaurin had

anticipated some of Cauchy's discoveries. Cauchy advanced the theory of power series by his

expansion of a complex function in such a form.

Abel (1826) in his memoir on the binomial series

corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the

series for complex values of m and x. He showed the necessity of considering the subject of

continuity in questions of convergence.

Cauchy's methods led to special rather than general criteria, and the same may be said of Raabe

(1832), who made the first elaborate investigation of the subject, of De Morgan (from 1842),

whose logarithmic test DuBois-Reymond (1873) and Pringsheim (1889) have shown to fail

within a certain region; of Bertrand (1842), Bonnet (1843), Malmsten (1846, 1847, the latter

without integration); Stokes (1847), Paucker (1852), Chebyshev (1852), and Arndt (1853).

http://en.wikipedia.org/wiki/Wikipedia:Citation_needed

http://en.wikipedia.org/wiki/James_Gregory_%28astronomer_and_mathematician%29

http://en.wikipedia.org/wiki/Decimal

http://en.wikipedia.org/wiki/Maclaurin_series


http://en.wikipedia.org/wiki/Brook_Taylor


http://en.wikipedia.org/wiki/Hypergeometric_series

http://en.wikipedia.org/wiki/Q-series



http://en.wikipedia.org/wiki/Integral_test_for_convergence

http://en.wikipedia.org/wiki/Kerala_school_of_astronomy_and_mathematics


http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

http://en.wikipedia.org/wiki/Hypergeometric_series

http://en.wikipedia.org/wiki/Cauchy



http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

http://en.wikipedia.org/wiki/Colin_Maclaurin



http://en.wikipedia.org/wiki/Niels_Henrik_Abel

http://en.wikipedia.org/wiki/Binomial_series

http://en.wikipedia.org/wiki/Joseph_Ludwig_Raabe

http://en.wikipedia.org/wiki/Augustus_De_Morgan

http://en.wikipedia.org/wiki/Paul_David_Gustav_du_Bois-Reymond

http://en.wikipedia.org/wiki/Alfred_Pringsheim

http://en.wikipedia.org/wiki/Joseph_Louis_Fran%C3%A7ois_Bertrand

http://en.wikipedia.org/wiki/Pierre_Ossian_Bonnet

http://en.wikipedia.org/wiki/Carl_Johan_Malmsten

http://en.wikipedia.org/wiki/George_Gabriel_Stokes

http://en.wikipedia.org/wiki/Paucker

http://en.wikipedia.org/wiki/Chebyshev

http://en.wikipedia.org/wiki/Arndt

14

General criteria began with Kummer (1835), and have been studied by Eisenstein (1847),

Weierstrass in his various contributions to the theory of functions, Dini (1867), DuBois-

Reymond (1873), and many others. Pringsheim's memoirs (1889) present the most complete

general theory.

[] Uniform convergence

The theory of uniform convergence was treated by Cauchy (1821), his limitations being pointed

out by Abel, but the first to attack it successfully were Seidel and Stokes (1847-48). Cauchy took

up the problem again (1853), acknowledging Abel's criticism, and reaching the same conclusions

which Stokes had already found. Thomae used the doctrine (1866), but there was great delay in

recognizing the importance of distinguishing between uniform and non-uniform convergence, in

spite of the demands of the theory of functions.

[] Semi-convergence

A series is said to be semi-convergent (or conditionally convergent) if it is convergent but not

absolutely convergent.

Semi-convergent series were studied by Poisson (1823), who also gave a general form for the

remainder of the Maclaurin formula. The most important solution of the problem is due,

however, to Jacobi (1834), who attacked the question of the remainder from a different

standpoint and reached a different formula. This expression was also worked out, and another

one given, by Malmsten (1847). Schlömilch (Zeitschrift, Vol.I, p. 192, 1856) also improved

Jacobi's remainder, and showed the relation between the remainder and Bernoulli's function

Genocchi (1852) has further contributed to the theory.

Among the early writers was Wronski, whose "loi suprême" (1815) was hardly recognized until

Cayley (1873) brought it into prominence.

[] Fourier series

Fourier series were being investigated as the result of physical considerations at the same time

that Gauss, Abel, and Cauchy were working out the theory of infinite series. Series for the

expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had

been treated by Jakob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier

by Vieta. Euler and Lagrange simplified the subject, as did Poinsot, Schröter, Glaisher, and

Kummer.

Fourier (1807) set for himself a different problem, to expand a given function of x in terms of the

sines or cosines of multiples of x, a problem which he embodied in his Théorie analytique de la

Chaleur (1822). Euler had already given the formulas for determining the coefficients in the

series; Fourier was the first to assert and attempt to prove the general theorem. Poisson (1820-23)

http://en.wikipedia.org/wiki/Ernst_Kummer

http://en.wikipedia.org/wiki/Ferdinand_Eisenstein

http://en.wikipedia.org/wiki/Weierstrass

http://en.wikipedia.org/wiki/Ulisse_Dini

http://en.wikipedia.org/wiki/Uniform_convergence

http://en.wikipedia.org/wiki/Philipp_Ludwig_von_Seidel

http://en.wikipedia.org/wiki/George_Gabriel_Stokes


http://en.wikipedia.org/wiki/Carl_Johan_Malmsten

http://en.wikipedia.org/wiki/Schl%C3%B6milch

http://en.wikipedia.org/wiki/Faulhaber%27s_formula

http://en.wikipedia.org/wiki/Angelo_Genocchi

http://en.wikipedia.org/wiki/Josef_Hoene-Wronski

http://en.wikipedia.org/wiki/Arthur_Cayley


http://en.wikipedia.org/wiki/Jakob_Bernoulli

http://en.wikipedia.org/wiki/Johann_Bernoulli

http://en.wikipedia.org/wiki/Franciscus_Vieta

http://en.wikipedia.org/wiki/Joseph_Louis_Lagrange

http://en.wikipedia.org/wiki/Louis_Poinsot

http://en.wikipedia.org/w/index.php?title=Karl_Schr%C3%B6ter&action=edit&redlink=1

http://en.wikipedia.org/wiki/James_Whitbread_Lee_Glaisher

http://en.wikipedia.org/wiki/Ernst_Kummer

http://en.wikipedia.org/w/index.php?title=Th%C3%A9orie_analytique_de_la_Chaleur&action=edit&redlink=1

http://en.wikipedia.org/w/index.php?title=Th%C3%A9orie_analytique_de_la_Chaleur&action=edit&redlink=1

http://en.wikipedia.org/wiki/Sim%C3%A9on_Denis_Poisson

15

also attacked the problem from a different standpoint. Fourier did not, however, settle the

question of convergence of his series, a matter left for Cauchy (1826) to attempt and for Dirichlet

(1829) to handle in a thoroughly scientific manner (see convergence of Fourier series).

Dirichlet's treatment (Crelle, 1829), of trigonometric series was the subject of criticism and

improvement by Riemann (1854), Heine, Lipschitz, Schläfli, and DuBois-Reymond. Among

other prominent contributors to the theory of trigonometric and Fourier series were Dini,

Hermite, Halphen, Krause, Byerly and Appell.

[] Generalizations

[] Asymptotic series

Asymptotic series, otherwise asymptotic expansions, are infinite series whose partial sums

become good approximations in the limit of some point of the domain. In general they do not

converge. But they are useful as sequences of approximations, each of which provides a value

close to the desired answer for a finite number of terms. The difference is that an asymptotic

series cannot be made to produce an answer as exact as desired, the way that convergent series

can. In fact, after a certain number of terms, a typical asymptotic series reaches its best

approximation; if more terms are included, most such series will produce worse answers.

[] Divergent series

Main article: Divergent series

Under many circumstances, it is desirable to assign a limit to a series which fails to converge in

the usual sense. A summability method is such an assignment of a limit to a subset of the set of

divergent series which properly extends the classical notion of convergence. Summability

methods include Cesàro summation, (C,k) summation, Abel summation, and Borel summation,

in increasing order of generality (and hence applicable to increasingly divergent series).

A variety of general results concerning possible summability methods are known. The

Silverman–Toeplitz theorem characterizes matrix summability methods, which are methods for

summing a divergent series by applying an infinite matrix to the vector of coefficients. The most

general method for summing a divergent series is non-constructive, and concerns Banach limits.

[] Series in Banach spaces

The notion of series can be easily extended to the case of a Banach space. If xn is a sequence of

elements of a Banach space X, then the series Σxn converges to x ∈ X if the sequence of partial

sums of the series tends to x; to wit,

http://en.wikipedia.org/wiki/Augustin_Louis_Cauchy

http://en.wikipedia.org/wiki/Convergence_of_Fourier_series

http://en.wikipedia.org/wiki/Crelle

http://en.wikipedia.org/wiki/Rudolf_Lipschitz

http://en.wikipedia.org/wiki/Ludwig_Schl%C3%A4fli

http://en.wikipedia.org/wiki/DuBois-Reymond

http://en.wikipedia.org/wiki/Ulisse_Dini

http://en.wikipedia.org/wiki/Charles_Hermite

http://en.wikipedia.org/wiki/Georges_Henri_Halphen

http://en.wikipedia.org/wiki/Paul_%C3%89mile_Appell

http://en.wikipedia.org/wiki/Asymptotic_series

http://en.wikipedia.org/wiki/Asymptotic_expansion

http://en.wikipedia.org/wiki/Asymptotic_series

http://en.wikipedia.org/wiki/Divergent_series

http://en.wikipedia.org/wiki/Summability_method

http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation

http://en.wikipedia.org/wiki/Abel_summation

http://en.wikipedia.org/wiki/Borel_summation

http://en.wikipedia.org/wiki/Silverman%E2%80%93Toeplitz_theorem

http://en.wikipedia.org/wiki/Banach_limit

http://en.wikipedia.org/wiki/Banach_space

16

as N → ∞.

More generally, convergence of series can be defined in any abelian Hausdorff topological

group. Specifically, in this case, Σxn converges to x if the sequence of partial sums converges to

x.

[] Summations over arbitrary index sets

Definitions may be given for sums over an arbitrary index set I. There are two main differences

with the usual notion of series: first, there is no specific order given on the set I; second, this set I

may be uncountable.

[] Families of non-negative numbers

When summing a family {ai}, i ∈ I, of non-negative numbers, one may define

When the sum is finite, the set of i ∈ I such that ai > 0 is countable. Indeed for every n ≥ 1, the

set is finite, because

If I is countably infinite and enumerated as I = {i0, i1,...} then the above defined sum satisfies

provided the value ∞ is allowed for the sum of the series.

Any sum over non-negative reals can be understood as the integral of a non-negative function

with respect to the counting measure, which accounts for the many similarities between the two

constructions.

[] Abelian topological groups

Let a : I → X, where I is any set and X is an abelian Hausdorff topological group. Let F be the

collection of all finite subsets of I. Note that F is a directed set ordered under inclusion with

union as join. Define the sum S of the family a as the limit

http://en.wikipedia.org/wiki/Abelian_group

http://en.wikipedia.org/wiki/Hausdorff_space

http://en.wikipedia.org/wiki/Topological_group


http://en.wikipedia.org/wiki/Counting_measure

http://en.wikipedia.org/wiki/Abelian_group

http://en.wikipedia.org/wiki/Hausdorff_space


http://en.wikipedia.org/wiki/Finite_set

http://en.wikipedia.org/wiki/Subset

http://en.wikipedia.org/wiki/Directed_set

http://en.wikipedia.org/wiki/Ordered_set

http://en.wikipedia.org/wiki/Inclusion_%28mathematics%29

http://en.wikipedia.org/wiki/Union_%28set_theory%29

http://en.wikipedia.org/wiki/Join_%28mathematics%29

17

if it exists and say that the family a is unconditionally summable. Saying that the sum S is the

limit of finite partial sums means that for every neighborhood V of 0 in X, there is a finite subset

A0 of I such that

Because F is not totally ordered, this is not a limit of a sequence of partial sums, but rather of a

net.

For every W, neighborhood of 0 in X, there is a smaller neighborhood V such that V − V ⊂ W. It

follows that the finite partial sums of an unconditionally summable family ai, i ∈ I, form a

Cauchy net, that is: for every W, neighborhood of 0 in X, there is a finite subset A0 of I such that

When X is complete, a family a is unconditionally summable in X if and only if the finite sums

satisfy the latter Cauchy net condition. When X is complete and ai, i ∈ I, is unconditionally

summable in X, then for every subset J ⊂ I, the corresponding subfamily aj, j ∈ J, is also

unconditionally summable in X.

When the sum of a family of non-negative numbers, in the extended sense defined before, is

finite, then it coincides with the sum in the topological group X = R.

If a family a in X is unconditionally summable, then for every W, neighborhood of 0 in X, there

is a finite subset A0 of I such that ai ∈ W for every i not in A0. If X is first-countable, it follows

that the set of i ∈ I such that ai ≠ 0 is countable. This need not be true in a general abelian

topological group (see examples below).

[] Unconditionally convergent series

Suppose that I = N. If a family an, n ∈ N, is unconditionally summable in an abelian Hausdorff

topological group X, then the series in the usual sense converges and has the same sum,

By nature, the definition of unconditional summability is insensitive to the order of the

summation. When ∑an is unconditionally summable, then the series remains convergent after

any permutation σ of the set N of indices, with the same sum,

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It can be proved that the converse holds: is a series ∑an converges after any permutation, then it

is unconditionally convergent. When X is complete, then unconditional convergence is also

equivalent to the fact that all subseries are convergent; if X is a Banach space, this is equivalent

to say that for every sequence of signs εn = 1 or −1, the series

converges in X. If X is a Banach space, then one may define the notion of absolute convergence.

A series ∑an of vectors in X converges absolutely if

If a series of vectors in a Banach space converges absolutely then it converges unconditionally,

but the converse only holds in finite dimensional Banach spaces (theorem of Dvoretzky &

Rogers (1950)).

[] Well-ordered sums

Conditionally convergent series can be considered if I is a well-ordered set, for example an

ordinal α0. One may define by transfinite recursion:

and for a limit ordinal α,

if this limit exists. If all limits exist up to α0, then the series converges.

[] Examples

1. Given a function f : X→Y, with Y an abelian topological group, define for every a ∈ X

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19

the function whose support is a singleton {a}. Then

in the topology of pointwise convergence (that is, the sum is taken in the infinite product

group YX

).

2. In the definition of partitions of unity, one constructs sums of functions over arbitrary

index set I,

While, formally, this requires a notion of sums of uncountable series, by construction

there are, for every given x, only finitely many nonzero terms in the sum, so issues

regarding convergence of such sums do not arise. Actually, one usually assumes more:

the family of functions is locally finite, i.e., for every x there is a neighborhood of x in

which all but a finite number of functions vanish. Any regularity property of the φi, such

as continuity, differentiability, that is preserved under finite sums will be preserved for

the sum of any subcollection of this family of functions.

3. On the first uncountable ordinal ω1 viewed as a topological space in the order topology,

the constant function f: [0,ω1) → [0,ω1] given by f(α) = 1 satisfies

(in other words, ω1 copies of 1 is ω1) only if one takes a limit over all countable partial

sums, rather than finite partial sums. This space is not separable.

Taylor series

"Series expansion" redirects here. For other notions of the term, see series (mathematics).

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20

As the degree of the Taylor polynomial rises, it approaches the correct function. This image

shows sinx (in black) and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.

The exponential function (in blue), and the sum of the first n+1 terms of its Taylor series at 0 (in

red).

In mathematics, the Taylor series is a representation of a function as an infinite sum of terms

calculated from the values of its derivatives at a single point. It is named after the English

mathematician Brook Taylor. If the series is centered at zero, the series is also called a

Maclaurin series, named after the Scottish mathematician Colin Maclaurin. It is common

practice to use a finite number of terms of the series to approximate a function. The Taylor series

may be regarded as the limit of the Taylor polynomials.

[] Definition

The Taylor series of a real or complex function ƒ(x) that is infinitely differentiable in a

neighborhood of a real or complex number a is the power series

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21

which can be written in the more compact sigma notation as

where n! denotes the factorial of n and ƒ (n)

(a) denotes the nth derivative of ƒ evaluated at the

point a. The zeroth derivative of ƒ is defined to be ƒ itself and (x − a)0 and 0! are both defined to

be 1.

[] Maclaurin series

In the particular case where a = 0, the series is also called a Maclaurin series:

[] Derivation

The Maclaurin / Taylor series can be derived in the following manner.

An arbitrary function may be defined by a power series:

Evaluating at x = 0, we have:

f(0) = a0

Differentiating the function,

Evaluating at x = 0,

f'(0) = a1

Differentiating the function again,

Evaluating at x = 0,

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22

Generalizing,

Where fn(0) is the n

th derivative of f(0).

Substituting the respective values of an in the power expansion,

Which is a particular case of the Taylor series (also known as Maclaurin series).

Generalizing further, we have

Which is the Taylor series.

[1]

[] Examples

The Maclaurin series for any polynomial is the polynomial itself.

The Maclaurin series for (1 − x)−1

is the geometric series

so the Taylor series for x−1

at a = 1 is

By integrating the above Maclaurin series we find the Maclaurin series for −ln(1 − x), where ln

denotes the natural logarithm:

and the corresponding Taylor series for ln(x) at a = 1 is

The Taylor series for the exponential function ex at a = 0 is

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23

The above expansion holds because the derivative of ex with respect to x is also e

x and e

0

equals 1. This leaves the terms (x − 0)n in the numerator and n! in the denominator for each term

in the infinite sum.

[] Convergence

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for

a full period centered at the origin.

The Taylor polynomials for log(1+x) only provide accurate approximations in the range −1 < x ≤

1. Note that, for x > 1, the Taylor polynomials of higher degree are worse approximations.

Taylor series need not in general be convergent. More precisely, the set of functions with a

convergent Taylor series is a meager set in the Frechet space of smooth functions. In spite of

this, for many functions that arise in practice, the Taylor series does converge.

The limit of a convergent Taylor series of a function f need not in general be equal to the

function value f(x), but in practice often it is. For example, the function

is infinitely differentiable at x = 0, and has all derivatives zero there. Consequently, the Taylor

series of f(x) is zero. However, f(x) is not equal to the zero function, and so it is not equal to its

Taylor series.


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If f(x) is equal to its Taylor series in a neighborhood of a, it is said to be analytic in this

neighborhood. If f(x) is equal to its Taylor series everywhere it is called entire. The exponential

function ex and the trigonometric functions sine and cosine are examples of entire functions.

Examples of functions that are not entire include the logarithm, the trigonometric function

tangent, and its inverse arctan. For these functions the Taylor series do not converge if x is far

from a.

Taylor series can be used to calculate the value of an entire function in every point, if the value

of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series

for entire functions include:

1. The partial sums (the Taylor polynomials) of the series can be used as approximations of

the entire function. These approximations are good if sufficiently many terms are

included.

2. The series representation simplifies many mathematical proofs.

Pictured on the right is an accurate approximation of sin(x) around the point a = 0. The pink

curve is a polynomial of degree seven:

The error in this approximation is no more than |x|9/9!. In particular, for −1 < x < 1, the error is

less than 0.000003.

In contrast, also shown is a picture of the natural logarithm function log(1 + x) and some of its

Taylor polynomials around a = 0. These approximations converge to the function only in the

region −1 < x ≤ 1; outside of this region the higher-degree Taylor polynomials are worse

approximations for the function. This is similar to Runge's phenomenon.

The error incurred in approximating a function by its nth-degree Taylor polynomial, is called

the remainder or residual and is denoted by the function Rn(x). Taylor's theorem can be used to

obtain a bound on the size of the remainder.

[] History

The Greek philosopher Zeno considered the problem of summing an infinite series to achieve a

finite result, but rejected it as an impossibility: the result was Zeno's paradox. Later, Aristotle

proposed a philosophical resolution of the paradox, but the mathematical content was apparently

unresolved until taken up by Democritus and then Archimedes. It was through Archimedes's

method of exhaustion that an infinite number of progressive subdivisions could be performed to

achieve a finite result.[2]

Liu Hui independently employed a similar method a few centuries

later.[3]

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25

In the 14th century, the earliest examples of the use of Taylor series and closely-related methods

were given by Madhava of Sangamagrama.[4]

Though no record of his work survives, writings of

later Indian mathematicians suggest that he found a number of special cases of the Taylor series,

including those for the trigonometric functions of sine, cosine, tangent, and arctangent. The

Kerala school of astronomy and mathematics further expanded his works with various series

expansions and rational approximations until the 16th century.

In the 17th century, James Gregory also worked in this area and published several Maclaurin

series. It was not until 1715 however that a general method for constructing these series for all

functions for which they exist was finally provided by Brook Taylor,[5]

after whom the series are

now named.

The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published

the special case of the Taylor result in the 18th century.

[] Properties

The function e

−1/x² is not analytic at x = 0: the Taylor series is identically 0, although the function

is not.

If this series converges for every x in the interval (a − r, a + r) and the sum is equal to f(x), then

the function f(x) is said to be analytic in the interval (a − r, a + r). If this is true for any r then

the function is said to be an entire function. To check whether the series converges towards f(x),

one normally uses estimates for the remainder term of Taylor's theorem. A function is analytic if

and only if it can be represented as a power series; the coefficients in that power series are then

necessarily the ones given in the above Taylor series formula.

The importance of such a power series representation is at least fourfold. First, differentiation

and integration of power series can be performed term by term and is hence particularly easy.

Second, an analytic function can be uniquely extended to a holomorphic function defined on an

open disk in the complex plane, which makes the whole machinery of complex analysis

available. Third, the (truncated) series can be used to compute function values approximately

(often by recasting the polynomial into the Chebyshev form and evaluating it with the Clenshaw

algorithm).


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26

Fourth, algebraic operations can often be done much more readily on the power series

representation; for instance the simplest proof of Euler's formula uses the Taylor series

expansions for sine, cosine, and exponential functions. This result is of fundamental importance

in such fields as harmonic analysis.

Another reason why the Taylor series is the natural power series for studying a function f is that,

given the value of f and its derivatives at a point a, the Taylor series is in some sense the most

likely function that fits the given data.[6]

Note that there are examples of infinitely differentiable functions f(x) whose Taylor series

converge, but are not equal to f(x). For instance, the function defined pointwise by f(x) = e−1/x²

if

x ≠ 0 and f(0) = 0 is an example of a non-analytic smooth function. All its derivatives at x = 0 are

zero, so the Taylor series of f(x) at 0 is zero everywhere, even though the function is nonzero for

every x ≠ 0. This particular pathology does not afflict Taylor series in complex analysis. There,

the area of convergence of a Taylor series is always a disk in the complex plane (possibly with

radius 0), and where the Taylor series converges, it converges to the function value. Notice that

e−1/z²

does not approach 0 as z approaches 0 along the imaginary axis, hence this function is not

continuous as a function on the complex plane.

Since every sequence of real or complex numbers can appear as coefficients in the Taylor series

of an infinitely differentiable function defined on the real line,[Proof]

the radius of convergence of

a Taylor series can be zero. There are even infinitely differentiable functions defined on the real

line whose Taylor series have a radius of convergence 0 everywhere.[7]

Some functions cannot be written as Taylor series because they have a singularity; in these cases,

one can often still achieve a series expansion if one allows also negative powers of the variable

x; see Laurent series. For example, f(x) = e−1/x²

can be written as a Laurent series.

[] List of Maclaurin series of some common functions

See also List of mathematical series

The real part of the cosine function in the complex plane.

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27

An 8th degree approximation of the cosine function in the complex plane.

The two above curves put together.

Several important Maclaurin series expansions follow.[8]

All these expansions are valid for

complex arguments x.

Exponential function:

Natural logarithm:

Finite geometric series:

Infinite geometric series:

Variants of the infinite geometric series:






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28

Square root:

Binomial series (includes the square root for α = 1/2 and the infinite geometric series for α = −1):

with generalized binomial coefficients

Trigonometric functions:

where the Bs are Bernoulli numbers.

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29

Hyperbolic functions:

Lambert's W function:

The numbers Bk appearing in the summation expansions of tan(x) and tanh(x) are the Bernoulli

numbers. The Ek in the expansion of sec(x) are Euler numbers.

[] Calculation of Taylor series

Several methods exist for the calculation of Taylor series of a large number of functions. One

can attempt to use the Taylor series as-is and generalize the form of the coefficients, or one can

use manipulations such as substitution, multiplication or division, addition or subtraction of

standard Taylor series to construct the Taylor series of a function, by virtue of Taylor series

being power series. In some cases, one can also derive the Taylor series by repeatedly applying

integration by parts. Particularly convenient is the use of computer algebra systems to calculate

Taylor series.

[] First example

Compute the 7th

degree Maclaurin polynomial for the function

.

First, rewrite the function as

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30

.

We have for the natural logarithm (by using the big O notation)

and for the cosine function

The latter series expansion has a zero constant term, which enables us to substitute the second

series into the first one and to easily omit terms of higher order than the 7th

degree by using the

big O notation:

Since the cosine is an even function, the coefficients for all the odd powers x, x3, x

5, x

7, ... have to

be zero.

[] Second example

Suppose we want the Taylor series at 0 of the function

.

We have for the exponential function

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31

and, as in the first example,

Assume the power series is

Then multiplication with the denominator and substitution of the series of the cosine yields

Collecting the terms up to fourth order yields

Comparing coefficients with the above series of the exponential function yields the desired

Taylor series

[] Taylor series as definitions

Classically, algebraic functions are defined by an algebraic equation, and transcendental

functions (including those discussed above) are defined by some property that holds for them,

such as a differential equation. For example the exponential function is the function which is

equal to its own derivative everywhere, and assumes the value 1 at the origin. However, one may

equally well define an analytic function by its Taylor series.

Taylor series are used to define functions and "operators" in diverse areas of mathematics. In

particular, this is true in areas where the classical definitions of functions break down. For

example, using Taylor series, one may define analytical functions of matrices and operators, such

as the matrix exponential or matrix logarithm.

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32

In other areas, such as formal analysis, it is more convenient to work directly with the power

series themselves. Thus one may define a solution of a differential equation as a power series

which, one hopes to prove, is the Taylor series of the desired solution.

[] Taylor series in several variables

The Taylor series may also be generalized to functions of more than one variable with

For example, for a function that depends on two variables, x and y, the Taylor series to second

order about the point (a, b) is:

where the subscripts denote the respective partial derivatives.

A second-order Taylor series expansion of a scalar-valued function of more than one variable

can be written compactly as

where is the gradient of evaluated at and is the Hessian matrix.

Applying the multi-index notation the Taylor series for several variables becomes

which is to be understood as a still more abbreviated multi-index version of the first equation of

this paragraph, again in full analogy to the single variable case.

Matrix calculus



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33

In mathematics, matrix calculus is a specialized notation for doing multivariable calculus,

especially over spaces of matrices, where it defines the matrix derivative. This notation is well-

suited to describing systems of differential equations, and taking derivatives of matrix-valued

functions with respect to matrix variables. This notation is commonly used in statistics and

engineering, while the tensor index notation is preferred in physics.

[] Notice

This article uses another definition for vector and matrix calculus than the form often

encountered within the field of estimation theory and pattern recognition. The resulting equations

will therefore appear to be transposed when compared to the equations used in textbooks within

these fields.

[] Notation

Let M(n,m) denote the space of real n×m matrices with n rows and m columns, such matrices

will be denoted using bold capital letters: A, X, Y, etc. An element of M(n,1), that is, a column

vector, is denoted with a boldface lowercase letter: a, x, y, etc. An element of M(1,1) is a scalar,

denoted with lowercase italic typeface: a, t, x, etc. XT denotes matrix transpose, tr(X) is trace,

and det(X) is the determinant. All functions are assumed to be of differentiability class C1 unless

otherwise noted. Generally letters from first half of the alphabet (a, b, c, …) will be used to

denote constants, and from the second half (t, x, y, …) to denote variables.

[] Vector calculus

Main article: Vector calculus

Because the space M(n,1) is identified with the Euclidean space Rn and M(1,1) is identified with

R, the notations developed here can accommodate the usual operations of vector calculus.

The tangent vector to a curve x : R → Rn is

The gradient of a scalar function f : Rn → R

The directional derivative of f in the direction of v is then


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34

The pushforward or differential of a function f : Rm → R

n is described by the Jacobian

matrix

The pushforward along f of a vector v in Rm is

[] Matrix calculus

For the purposes of defining derivatives of simple functions, not much changes with matrix

spaces; the space of n×m matrices is isomorphic to the vector space Rnm

.[dubious – discuss]

The three

derivatives familiar from vector calculus have close analogues here, though beware the

complications that arise in the identities below.

The tangent vector of a curve F : R → M(n,m)

The gradient of a scalar function f : M(n,m) → R

Notice that the indexing of the gradient with respect to X is transposed as compared with

the indexing of X. The directional derivative of f in the direction of matrix Y is given by

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35

The differential or the matrix derivative of a function F : M(n,m) → M(p,q) is an element

of M(p,q) ⊗ M(m,n), a fourth-rank tensor (the reversal of m and n here indicates the dual

space of M(n,m)). In short it is an m×n matrix each of whose entries is a p×q matrix.[citation

needed]

and note that each ∂F/∂Xi,j is a p×q matrix defined as above. Note also that this matrix has

its indexing transposed; m rows and n columns. The pushforward along F of an n×m

matrix Y in M(n,m) is then

as formal block matrices.

Note that this definition encompasses all of the preceding definitions as special cases.

According to Jan R. Magnus and Heinz Neudecker, the following notations are both unsuitable,

as the determinants of the resulting matrices would have "no interpretation" and "a useful chain

rule does not exist" if these notations are being used:[1]

1.

2.

The Jacobian matrix, according to Magnus and Neudecker,[1]

is

[contradiction]

[] Identities

This section's factual accuracy is disputed. Please see the relevant discussion on the talk

page. (July 2009)

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36

Note that matrix multiplication is not commutative, so in these identities, the order must not be

changed.

Chain rule: If Z is a function of Y which in turn is a function of X, and these are all

column vectors, then[citation needed]

Product rule:In all cases where the derivatives do not involve tensor products (for

example, Y has more than one row and X has more than one column),[citation needed]

[] Examples

[] Derivative of linear functions

This section lists some commonly used vector derivative formulas for linear equations evaluating

to a vector.

[] Derivative of quadratic functions

This section lists some commonly used vector derivative formulas for quadratic matrix equations

evaluating to a scalar.

Related to this is the derivative of the Euclidean norm:

http://en.wikipedia.org/wiki/Commutative

http://en.wikipedia.org/wiki/Chain_rule


http://en.wikipedia.org/wiki/Product_rule


http://en.wikipedia.org/wiki/Euclidean_norm

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[] Derivative of matrix traces

This section shows examples of matrix differentiation of common trace equations.

[] Derivative of matrix determinant

[] Relation to other derivatives

The matrix derivative is a convenient notation for keeping track of partial derivatives for doing

calculations. The Fréchet derivative is the standard way in the setting of functional analysis to

take derivatives with respect to vectors. In the case that a matrix function of a matrix is Fréchet

differentiable, the two derivatives will agree up to translation of notations. As is the case in

general for partial derivatives, some formulae may extend under weaker analytic conditions than

the existence of the derivative as approximating linear mapping.

Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with

respect to one of those variables, with the others held constant (as opposed to the total derivative,

in which all variables are allowed to vary). Partial derivatives are used in vector calculus and

differential geometry.

The partial derivative of a function f with respect to the variable x is variously denoted by

The partial-derivative symbol ∂ is a rounded letter, derived but distinct from the Greek letter

delta. The notation was introduced by Adrien-Marie Legendre and gained general acceptance

after its reintroduction by Carl Gustav Jacob Jacobi.[1]

[] Introduction

Suppose that ƒ is a function of more than one variable. For instance,

http://en.wikipedia.org/wiki/Trace_%28matrix%29

http://en.wikipedia.org/wiki/Fr%C3%A9chet_derivative

http://en.wikipedia.org/wiki/Functional_analysis

http://en.wikipedia.org/wiki/Partial_derivative




http://en.wikipedia.org/wiki/Total_derivative

http://en.wikipedia.org/wiki/Vector_calculus

http://en.wikipedia.org/wiki/Differential_geometry

http://en.wikipedia.org/wiki/%E2%88%82

http://en.wikipedia.org/wiki/Delta_%28letter%29

http://en.wikipedia.org/wiki/Adrien-Marie_Legendre

http://en.wikipedia.org/wiki/Carl_Gustav_Jacob_Jacobi

http://en.wikipedia.org/wiki/Partial_derivative#cite_note-0

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A graph of z = x

2 + xy + y

2. For the partial derivative at (1, 1, 3) that leaves y constant, the

corresponding tangent line is parallel to the xz-plane.

A slice of the graph above at y= 1

It is difficult to describe the derivative of such a function, as there are an infinite number of

tangent lines to every point on this surface. Partial differentiation is the act of choosing one of

these lines and finding its slope. Usually, the lines of most interest are those that are parallel to

the xz-plane, and those that are parallel to the yz-plane.

A good way to find these parallel lines is to treat the other variable as a constant. For example, to

find the slope of the line tangent to the function at (1, 1, 3) that is parallel to the xz-plane, we

treat the y variable as constant. The graph and this plane are shown on the right. On the left, we

see the way the function looks on the plane y = 1. By finding the derivative of the equation while

assuming that y is a constant, we discover that the slope of ƒ at the point (x, y, z) is:

So at (1, 1, 3), by substitution, the slope is 3. Therefore

at the point (1, 1, 3). That is, the partial derivative of z with respect to x at (1, 1, 3) is 3.

http://en.wikipedia.org/wiki/Tangent




http://en.wikipedia.org/wiki/File:Grafico_3d_x2+xy+y2.png

http://en.wikipedia.org/wiki/File:X2+x+1.png

39

[] Definition

[] Basic definition

The function f can be reinterpreted as a family of functions of one variable indexed by the other

variables:

In other words, every value of x defines a function, denoted fx, which is a function of one real

number.[2]

That is,

Once a value of x is chosen, say a, then f(x,y) determines a function fa which sends y to a2 + ay +

y2:

In this expression, a is a constant, not a variable, so fa is a function of only one real variable, that

being y. Consequently, the definition of the derivative for a function of one variable applies:

The above procedure can be performed for any choice of a. Assembling the derivatives together

into a function gives a function which describes the variation of f in the y direction:

This is the partial derivative of f with respect to y. Here ∂ is a rounded d called the partial

derivative symbol. To distinguish it from the letter d, ∂ is sometimes pronounced "del" or

"partial" instead of "dee".

In general, the partial derivative of a function f(x1,...,xn) in the direction xi at the point (a1,...,an)

is defined to be:

http://en.wikipedia.org/wiki/Partial_derivative#cite_note-1

40

In the above difference quotient, all the variables except xi are held fixed. That choice of fixed

values determines a function of one variable

, and by definition,

In other words, the different choices of a index a family of one-variable functions just as in the

example above. This expression also shows that the computation of partial derivatives reduces to

the computation of one-variable derivatives.

An important example of a function of several variables is the case of a scalar-valued function

f(x1,...xn) on a domain in Euclidean space Rn (e.g., on R

2 or R

3). In this case f has a partial

derivative ∂f/∂xj with respect to each variable xj. At the point a, these partial derivatives define

the vector

This vector is called the gradient of f at a. If f is differentiable at every point in some domain,

then the gradient is a vector-valued function ∇f which takes the point a to the vector ∇f(a).

Consequently, the gradient produces a vector field.

A common abuse of notation is to define the del operator (∇) as follows in three-dimensional

Euclidean space R3 with unit vectors :

Or, more generally, for n-dimensional Euclidean space Rn with coordinates (x1, x2, x3,...,xn) and

unit vectors ( ):

[] Formal definition

Like ordinary derivatives, the partial derivative is defined as a limit. Let U be an open subset of

Rn and f : U → R a function. The partial derivative of f at the point a = (a1, ..., an) ∈ U with

respect to the i-th variable ai is defined as

http://en.wikipedia.org/wiki/Scalar-valued_function


http://en.wikipedia.org/wiki/Vector_field

http://en.wikipedia.org/wiki/Abuse_of_notation

http://en.wikipedia.org/wiki/Del_operator

http://en.wikipedia.org/wiki/Euclidean_space

http://en.wikipedia.org/wiki/Unit_vectors

http://en.wikipedia.org/wiki/Limit_of_a_function

http://en.wikipedia.org/wiki/Open_set

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Even if all partial derivatives ∂f/∂ai(a) exist at a given point a, the function need not be

continuous there. However, if all partial derivatives exist in a neighborhood of a and are

continuous there, then f is totally differentiable in that neighborhood and the total derivative is

continuous. In this case, it is said that f is a C1 function. This can be used to generalize for vector

valued functions (f : U → R'm) by carefully using a componentwise argument.

The partial derivative can be seen as another function defined on U and can again be partially

differentiated. If all mixed second order partial derivatives are continuous at a point (or on a set),

f is termed a C2 function at that point (or on that set); in this case, the partial derivatives can be

exchanged by Clairaut's theorem:

[] Examples

The volume of a cone depends on height and radius

The volume V of a cone depends on the cone's height h and its radius r according to the formula

The partial derivative of V with respect to r is

http://en.wikipedia.org/wiki/Continuous_function

http://en.wikipedia.org/wiki/Neighborhood_%28topology%29


http://en.wikipedia.org/wiki/Symmetry_of_second_derivatives#Clairaut.27s_theorem

http://en.wikipedia.org/wiki/Volume

http://en.wikipedia.org/wiki/Cone_%28geometry%29

http://en.wikipedia.org/wiki/Height

http://en.wikipedia.org/wiki/Radius

http://en.wikipedia.org/wiki/File:Cone_3d.png

http://en.wikipedia.org/wiki/File:Cone_3d.png

42

which represents the rate with which a cone's volume changes if its radius is varied and its height

is kept constant. The partial derivative with respect to h is

which represents the rate with which the volume changes if its height is varied and its radius is

kept constant.

By contrast, the total derivative of V with respect to r and h are respectively

and

The difference between the total and partial derivative is the elimination of indirect dependencies

between variables in the latter.

If (for some arbitrary reason) the cone's proportions have to stay the same, and the height and

radius are in a fixed ratio k,

This gives the total derivative with respect to r:

Equations involving an unknown function's partial derivatives are called partial differential

equations and are common in physics, engineering, and other sciences and applied disciplines.

[] Notation

For the following examples, let f be a function in x, y and z.


http://en.wikipedia.org/wiki/Partial_differential_equation

http://en.wikipedia.org/wiki/Partial_differential_equation

http://en.wikipedia.org/wiki/Physics

http://en.wikipedia.org/wiki/Engineering

http://en.wikipedia.org/wiki/Science

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First-order partial derivatives:

Second-order partial derivatives:

Second-order mixed derivatives:

Higher-order partial and mixed derivatives:

When dealing with functions of multiple variables, some of these variables may be related to

each other, and it may be necessary to specify explicitly which variables are being held constant.

In fields such as statistical mechanics, the partial derivative of f with respect to x, holding y and z

constant, is often expressed as

[] Antiderivative analogue

There is a concept for partial derivatives that is analogous to antiderivatives for regular

derivatives. Given a partial derivative, it allows for the partial recovery of the original function.

Consider the example of . The "partial" integral can be taken with respect to x

(treating y as constant, in a similar manner to partial derivation):

Here, the "constant" of integration is no longer a constant, but instead a function of all the

variables of the original function except x. The reason for this is that all the other variables are

http://en.wikipedia.org/wiki/Mixed_derivatives

http://en.wikipedia.org/wiki/Statistical_mechanics

http://en.wikipedia.org/wiki/Antiderivative

http://en.wikipedia.org/wiki/Constant_of_integration

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treated as constant when taking the partial derivative, so any function which does not involve x

will disappear when taking the partial derivative, and we have to account for this when we take

the antiderivative. The most general way to represent this is to have the "constant" represent an

unknown function of all the other variables.

Thus the set of functions x2 + xy + g(y), where g is any one-argument function, represents the

entire set of functions in variables x,y that could have produced the x-partial derivative 2x+y.

If all the partial derivatives of a function are known (for example, with the gradient), then the

antiderivatives can be matched via the above process to reconstruct the original function up to a

constant.


Mathematical Physics 1

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