Lecture 1Introduction, vector calculus, functions of more variables,

Ing. Jaroslav Jíra, CSc.

Physics for informatics

Introduction

Source of information: http://aldebaran.feld.cvut.cz/ , section Physics for OI

Lecturers: prof. Ing. Stanislav Pekárek, CSc., pekarek@fel.cvut.cz , room 49A

Ing. Jaroslav Jíra, CSc., jira@fel.cvut.cz , room 42

Textbooks: Physics I, Pekárek S., Murla M.

Physics I - seminars, Pekárek S., Murla M.

Internet tests: https://fyzika.feld.cvut.cz/auth/oitest

Conditions for assessment:

- to gain at least 40 points,

- to measure six laboratory experiments and submit reports from them

Scoring system of the Physics for OI

The maximum reachable amount of points from semester is 100. Points from semester go with each student to the exam, where they create a part of the final grade according to the exam rules.

Points can be gained by:

- written tests, max. 50 points. Two tests by 25 points max. (8th and 13th week)

- laboratory reports, max. 30 points. The last five lab reports are marked by up to 6 points each

- tests on the internet, max. 20 points. There are 10 electronical tests available on the internet, each consisting of 8 questions. Correct answering of ALL eight questions results in 2 point gain for the student.

Number of problems to solve Points from the semester

1 90 and more

2 75 – 89

3 65 – 74

4 55 – 64

5 less than 55

Examination – first part:

Every student must solve certain number of problems according to his/her points from the semester.

Examination - second part:

Student answers questions in written form during the written exam. The answers are marked and the total of 30 points can be gained this way.

Then the oral part of the exam follows and each student defends a mark according to the table below. The column resulting in better mark is taken into account.

written exam semester + written exam

A excellent 1 25 120

B very good 1- 23 110

C good 2 20 100

D satisfactory 2- 18 90

E sufficient 3 15 80

Vector calculus - basics

A vector – standard notation for three dimensions kAjAiAAAAA zyxzyx

),,(

Unit vectors i,j,k are vectors of magnitude 1 in directions of the x,y,z axes.

)1,0,0()0,1,0()0,0,1( kji

Magnitude of a vector222zyx AAAAA

Position vector is a vector r from the origin to the current position

kzjyixzyxr

),,(

where x,y,z, are projections of r to the coordinate axes.

Adding and subtracting vectors

),,(

),,(

zyx

zyx

BBBB

AAAA

),,,( zzyyxx BABABABA

),,,( zzyyxx BABABABA

),,,( zyx AkAkAkAk

Multiplying a vector by a scalar

Example of multiplying of a vector by a scalar in a plane

)2,4()1,2(22

)1,2(

uv

u

Multiplication of a vector by a scalar in the Mathematica

Example of addition of three vectors in a plane

The vectors are given: )0,2();3,2();1,2( wvu

Numerical addition gives us

)4,2()031),2(22( wvuz

Graphical solution:

Addition of three vectors in the Mathematica

Example of subtraction of two vectors a plane

The vectors are given: )2,1();3,2( vu

Numerical subtraction gives us

)1,3()23),1(2( vuz

Graphical solution:

Subtraction of two vectors in the Mathematica

Time derivation and time integration of a vector function

ktVjtVitVVVVtV zyxzyx

)()()(),,()(

2

1

2

1

2

1

2

1

)()()()(t

t

z

t

t

y

t

t

x

t

t

dttVkdttVjdttVidttV

kdt

dVj

dt

dVi

dt

dV

dt

dV

dt

dV

dt

dV

dt

tVd zyxzyx

),,(

)(

2

1

2

1

2

1

2

1

)(,)(,)()(t

t

z

t

t

y

t

t

x

t

t

dttVdttVdttVdttV

Determine for any time t: a)

b) the tangential and the radial accelerations

Example of the time derivation of a vector

The motion of a particle is described by the vector equation

ktjtittr 32

3

1)52()(

)(),(),(),(),( tatatvtvtr

][9

1)52()( 642222 mtttzyxtr

]/[22)( 2 smktjtidt

rdtv

]/[22)( smktjdt

vdta

]/[244)( 242222 smtttvvvtv zyx

]/[1244)( 222222 smttaaata zyx

]/[2444]/[2)( 222222 smttaaasmtdt

dvta tnt

Time derivation of a vector in the Mathematica

Time derivation of a vector in the Mathematica -continued

What would happen without Assuming and Refine

What would happen without Simplify

Graphical output of the )(tr

Example of the time integration of a vector

Evaluate the time dependence of the velocity and the position vector for the projectile motion. Initial velocity v0=(10,20) m/s and g=(0,-9.81) m/s2.

]/[),(),0(),()( 00 smvtgvdtgdtdtgdtgdttgv yyxyyx

]/[)2081.9,10()( smttv

2

0 0 0 0( ) ( ) ( , ( ) ) ( , ) [ ]2x y y x y y

tr t v t dt v dt g t v dt v t g v t m

]/[)20905.4,10()( 2 smttttr

Time integration of a vector in the Mathematica

Projectile motion - trajectory:

Study of balistic projectile motion, when components of initial velocity are given

Scalar product (dot product) – is defined as

Where Θ is a smaller angle between vectors

a and b and S is a resulting scalar. Sbaba

Sbaba

i

n

ii

1

cos

Scalar product

cos abbabababaS zzyyxx

For three component vectors we can write

Geometric interpretation – scalar product is equal to the area of rectangle having a and b.cosΘ as sides. Blue and red arrows represent original vectors a and b.

abbaba

baba

abba

0Basic properties of the scalar product

Vector product

nabba

sin

Basic properties of the vector product

0

baba

abbaba

abba

Vector product (cross product) – is defined as

Where Θ is the smaller angle between vectors

a and b and n is unit vector perpendicular to the

plane containing a and b.

Geometric interpretation - the magnitude of the cross product can be interpreted as the positive area A of the parallelogram having a and b as sides

sin abbaA

kbabajbabaibaba

bbb

aaa

kji

bac

xyyxzxxzyzzy

zyx

zyx

)()()(

Component notation

Scalar product and vector product in the Mathematica

Direction of the resulting vector of the vector productcan be determined either by the right hand rule or by the screw rule

Vector triple product

)()()( baccabcba

Scalar triple product

)()()( bacacbcba

V

ccc

bbb

aaa

cba

zyx

zyx

zyx

)(

Geometric interpretation of the scalar triple product is a volume of a paralellepiped V

Scalar field and gradient

),(),( trftrS

Scalar field associates a scalar quantity to every point in a space. This association can be described by a scalar function f and can be also time dependent. (for instance temperature, density or pressure distribution).

The gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is that rate of increase.

z

Sk

y

Sj

x

SiSSgrad

Example: the gradient of the function f(x,y) = −(cos2x + cos2y)2 depicted as a projected vector field on the bottom plane.

Example 2 – finding extremes of the scalar field

Find extremes of the function: )( 22

),( yxexyxh

Extremes can be found by assuming: 0)(

hgrad

In this case : 0),()(

y

h

x

hhgrad

02 )(2)( 2222

yxyx exex

h02 )( 22

yxexyy

h

)(2)( 2222

2 yxyx exe 0y

2

121 2 xx

Answer: there are two extremes )0,2

1();0,

2

1( 21

hh

Extremes of the scalar field in the Mathematica

Vector operators

Gradient

(Nabla operator) z

Sk

y

Sj

x

SiSSgrad

Laplacian2

2

2

2

2

22

z

S

y

S

x

SSgraddivSS

Divergencez

A

y

A

x

AAAdiv zyx

Curl

y

A

x

Ak

x

A

z

Aj

z

A

y

Ai

AAAzyx

kji

AAcurl

xyzx

yz

zyx

Basic mechanical quantities and relations

and their analogies in linear and rotational motion

Linear motion Rotational motion

s, r path, position vector [ m ] φ angle [ rad ]

v velocity [ m*s-1 ] ω anglular velocity [ rad*s-1]

a acceleration [ m*s-2 ] ε angular acceleration [ rad*s-2 ]

F force [ N ] M torque [ N*m]

m mass [ kg ] J moment of inertia [ kg*m2 ]

p linear momentum [ kg*m*s-1] b angular momentum [kg*m2*s-1]

Work W= F s Work W= M φ

Kinetic energy Ek= ½ m v2 Kinetic energy Ek= ½ J ω2

Equation of motion F = m a Equation of motion M = J ε