Bridging the Gap between Mathematics and the Physical...

LanguageVector Calculus

Differential Calculus

Bridging the Gap between Mathematics

and the Physical Sciences

Tevian Dray

Department of Mathematics

Oregon State University

http://www.math.oregonstate.edu/~tevian

Tevian Dray Bridging the Gap between Mathematics and the Physical Sciences

Math vs. PhysicsFunctionsDot Product

Mathematics vs. Physics

What are Functions?

Suppose the temperature on a rectangular slab of metal is given by

T (x , y) = k(x2 + y2)

where k is a constant. What is T (r , θ)?

What are Functions?

T (x , y) = k(x2 + y2)

A: T (r , θ) = kr2

What are Functions?

T (x , y) = k(x2 + y2)

A: T (r , θ) = kr2

B: T (r , θ) = k(r2 + θ2)

What are Functions?

T (x , y) = k(x2 + y2)

A: T (r , θ) = kr2

B: T (r , θ) = k(r2 + θ2)

What are Functions?

T = f (x , y) = k(x2 + y2)

T = g(r , θ) = kr2

What are Functions?

T = f (x , y) = k(x2 + y2)

T = g(r , θ) = kr2

PHYSICS

T = T (x , y) = k(x2 + y2)

T = T (r , θ) = kr2

What are Functions?

T = f (x , y) = k(x2 + y2)

T = g(r , θ) = kr2

PHYSICS

T = T (x , y) = k(x2 + y2)

T = T (r , θ) = kr2

Two disciplines separated by a common language...

Physics is about things.

Physicists can’t change the problem.

Physics is about things.

Physicists can’t change the problem.

Mathematicians do algebra.

Physicists do geometry.

Projection:

~u ·~v = |~u||~v| cos θ

~u ·~v = uxvx + uyvy

Projection:

~u ·~v = |~u||~v| cos θ

Law of Cosines:

(~u −~v) · (~u −~v) = ~u · ~u +~v ·~v − 2~u ·~v|~u −~v|2 = |~u|2 + |~v|2 − 2|~u||~v| cos θ

Projection:

~u ·~v = |~u||~v| cos θ

Law of Cosines:

(~u −~v) · (~u −~v) = ~u · ~u +~v ·~v − 2~u ·~v|~u −~v|2 = |~u|2 + |~v|2 − 2|~u||~v| cos θ

Addition Formulas:

~u = cos α ı̂ + sinα ̂

~v = cos β ı̂ + sinβ ̂

~u ·~v = cos(α − β)

Find the angle between the diagonal of a cubeand one of its edges.

Algebra:

~u = ı̂ + ̂ + k̂

~v = ı̂

=⇒ ~u ·~v = 1

Geometry:~u ·~v = |~u||~v| cos θ =

√3 cos θ

Algebra:

~u = ı̂ + ̂ + k̂

~v = ı̂

=⇒ ~u ·~v = 1

Geometry:~u ·~v = |~u||~v| cos θ =

√3 cos θ

Need both!

Bridge ProjectVector DifferentialsGradient

The Vector Calculus Bridge Project

http://www.math.oregonstate.edu/bridge

Differentials (Use what you know!)

Multiple representations

Symmetry (adapted bases, coordinates)

Geometry (vectors, div, grad, curl)

Differentials (Use what you know!)

Multiple representations

Symmetry (adapted bases, coordinates)

Geometry (vectors, div, grad, curl)

Small group activities

Instructor’s guide

Online text (http://www.math.oregonstate.edu/BridgeBook)

Vector Differentials

dy ̂d~r

dx ı̂

d~r = dx ı̂ + dy ̂

dy ̂d~r

dx ı̂

d~r = dx ı̂ + dy ̂

r dφ φ̂ dr r̂

d~r = dr r̂ + r dφ φ̂

dy ̂d~r

dx ı̂

d~r = dx ı̂ + dy ̂

r dφ φ̂ dr r̂

d~r = dr r̂ + r dφ φ̂

ds = |d~r|d~A = d~r1 × d~r2

dA = |d~r1 × d~r2|dV = (d~r1 × d~r2) · d~r3

Gradient

Master Formula:df = ~∇f · d~r

Gradient

Master Formula:df = ~∇f · d~r

f = const =⇒ df = 0 =⇒ ~∇f ⊥ d~r

ds= ~∇f · d~r

The Hill

Suppose you are standing on ahill. You have a topographic map,which uses rectangular coordinates(x , y) measured in miles. Yourglobal positioning system says yourpresent location is at one of thepoints shown. Your guidebook tellsyou that the height h of the hill infeet above sea level is given by

h = a − bx2 − cy2

where a = 5000 ft, b = 30 ftmi2

and c = 10 ftmi2

-6 -4 -2 0 2 4 6x

DefinitionDifferentials

Coherent Calculus

co-he-rent:

logically or aesthetically ordered

Coherent Calculus

coherent:

cal-cu-lus:

a method of computation in a special notation

Coherent Calculus

coherent:

calculus:

a method of computation in a special notation

differential calculus:

a branch of mathematics concerned chiefly with the study of therate of change of functions with respect to their variablesespecially through the use of derivatives and differentials

Differentials

d(u + cv) = du + c dv

d(uv) = u dv + v du

Differentials

d(u + cv) = du + c dv

d(uv) = u dv + v du

d (un) = nun−1 du

d (eu) = eu du

d(sin u) = cos u du

d(cos u) = − sin u du

d(ln u) =1

Derivatives

Derivatives:

f (x) = sin(x) =⇒ f ′(x) = cos(x)

Derivatives

Derivatives:d

dusin u =

d sin u

du= cos u

Derivatives

Derivatives:d

dusin u =

d sin u

du= cos u

Chain rule:

h(x) = f(

=⇒ h′(x) = f ′(

g ′(x)

Derivatives

Derivatives:d

dusin u =

d sin u

du= cos u

Chain rule:

dxsin u =

d sin u

dx= cos u

Derivatives

Derivatives:d

dusin u =

d sin u

du= cos u

Chain rule:

dxsin u =

d sin u

dx= cos u

Inverse functions:

g(x) = f −1(x) =⇒ g ′(x) =1

f ′(

Derivatives

Derivatives:d

dusin u =

d sin u

du= cos u

Chain rule:

dxsin u =

d sin u

dx= cos u

Inverse functions:

duln u =

du/dq=

deq/dq=

Derivatives

Instead of:

chain rule

related rates

implicit differentiation

derivatives of inverse functions

difficulties of interpretation (units!)

Derivatives

Instead of:

chain rule

related rates

One coherent idea:

Derivatives

Instead of:

chain rule

related rates

One coherent idea:

“Zap equations with d”

Derivatives

Instead of:

chain rule

related rates

One coherent idea:

“Zap equations with d”

Tevian Dray & Corinne A. Manogue,Putting Differentials Back into Calculus,

College Math. J., (to apppear).

LanguageFunctions

Vector CalculusDifferential Calculus

A Radical View of Calculus

The central idea in calculusis not the limit.

The central idea ofderivatives is not slope.

The central idea of integralsis not area.

The central idea of curvesand surfaces is notparameterization.

The central representation ofa function is not its graph.

LanguageFunctions

A Radical View of Calculus

The central idea in calculusis not the limit.

The central idea ofderivatives is not slope.

The central idea of integralsis not area.

The central idea of curvesand surfaces is notparameterization.

The central representation ofa function is not its graph.

The central idea in calculusis the differential.

The central idea ofderivatives is rate of change.

The central idea of integralsis total amount.

The central idea of curvesand surfaces is “use whatyou know”.

The central representationof a function is dataattached to the domain.

LanguageFunctions

SUMMARY

LanguageFunctions

SUMMARY

I took this class a year ago, and I still remember all of it...

LanguageFunctions

Bridging the Gap between Mathematics and the Physical...

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