Sections 13.2, 13.7, & 13.8

Monday, April 28, 2008

By: John Helveston, Jim Dorman,

Tyler Aarons, Dale Arbogast

Green’s Theorem

Gauss’ Theorem

Stokes’ Theorem

13.2 Green’s Theorem

(No, he was not French)

George Green

July 14, 1793 - May 31, 1841 British mathematician and physicist First person to try to explain a mathematical theory of

electricity and magnetism Almost entirely self-taught! Published “An Essay on the Application of Mathematical

Analysis to the Theories of Electricity and Magnetism” in 1828.

Entered Cambridge University as an undergraduate in 1833 at age 40.

The TheoryConsider a simple closed curve C, and let D be the region enclosed by the curve. 

Notes:• The simple, closed curve has no holes in the region D• A direction has been put on the curve with the convention that the curve C has a positive orientation if the region D is on the left as we traverse the path.

dAy

f

x

ggdyfdx

C D

ExampleSection 13.2, problem 2

A particle moves once counterclockwise about the circle of radius 6 about

the origin, under the influence of the force:

Calculate the work done.

jxyixxyeF x ˆ)(ˆ))cosh(( 2/3

)sin(6),cos(6)( tttC

)2,0(: tI

6

F

I

dttCtCFW )(')(

dtttttttte t )cos(6),sin(6)cos(6)(sin6)),cos(6cosh()cos(6)sin(62

0

23)cos(6

)sin(6),cos(6)( tttC

jxyixxyeF x ˆ)(ˆ))cosh(( 2/3

Remember:

dttttttet t

2

0

23)cos(6 36)(sin)cos(36))cos(6cosh()cos()sin(36)sin(6

72

Direct computation:

C

sdFWork

dttCsd )('

I

dttCtCFWork )(')(

)sin(6),cos(6)( tttC

jxyixxyeF x ˆ)(ˆ))cosh(( 2/3

Remember:

72

Green’s Theorem:

C

sdFWork

dAy

f

x

ggdyfdx

C D

C

sdFWork

xyg 2/3 )cosh(xxyef x

1x

g1

y

f

)6(22)11( 2 D D

dAdAWork

Green’s Theorem…and beyond

Green’s Theorem is a crucial component in the development of many famous works:James Maxwell’s EquationsGauss’ Divergence TheoremStokes’ Integral Theorem

13.7 Gauss’ Divergence Theorem

(Also not French)

Gauss in the House

German mathematician, lived 1777-1855

Born in Braunschweig, Duchy of Braunschweig-Lüneburg in Northwestern Germany

Published Disquisitiones Arithmeticae when he was 21 (and what have you done today?)

As a workaholic, was once interrupted while working and told his wife was dying. He replied “tell her to wait a moment until I’m finished”.

Gauss’ Divergence Theorem

The integral of a continuously differentiable vector field across a boundary (flux) is equal to the integral of the divergence of that vector field within the region enclosed by the boundary.

Applications The Aerodynamic Continuity Equation

The surface integral of mass flux around a control volume without sources or sinks is equal to the rate of mass storage.

If the flow at a particular point is incompressible, then the net velocity flux around the control volume must be zero.

As net velocity flux at a point requires taking the limit of an integral, one instead merely calculates the divergence.

If the divergence at that point is zero, then it is incompressible. If it is positive, the fluid is expanding, and vice versa

Gauss’s Theorem can be applied to any vector field which obeys an inverse-square law (except at the origin), such as gravity, electrostatic attraction, and even examples in quantum physics such as probability density.

Example Assume there is a unit circle centered on the

origin and a vector field V(x,y,z)= To find the vector flux of the field across the

surface of the sphere, both the unit normal integral and the Gauss’ divergence integral will be computed

),,( 22 xzyxyz

The unit normal of the sphere is defined as

It will be much easier to compute this integral in spherical coordinates, making:

The 3-D surface integral for radius = 1 (plus Jacobian) is equal to:

Unit Normal Integration

),,( zyxn

))cos()sin()cos(,)sin()sin(),cos()sin()cos()sin(( 3322223 V

))cos(),sin()cos(),sin()(sin( n

0)cos()sin()cos()sin()sin()cos()sin()cos()sin( 3243422

00

dd

Now, Gauss’ Divergence Theorem shall be used, and the same result should be obtained

The divergence of the vector V:

The integration results in:

This verifies Gauss’ Theorem Keep in mind however that this is only possible with continuously

differentiable functions, not all functions

Gauss’ Divergence Integration

)sin()sin(2)]cos()sin()sin()cos()sin()cos(2[22 2 xzyyzV

0)(sin)sin(2)]cos()(sin)sin()cos()(sin)cos(2[

)sin()(

232241

0

2

00

21

0

2

00

ddd

dddV

13.8 The Integral Theorem of Stokes

Sir George Gabriel Stokes

(Aug. 13, 1819 – Feb. 1, 1903)

•Irish mathematician and physicist who attended Pembroke College (Cambridge University).

(Again, also not French)

•After graduating as Senior Wrangler (first in class in mathematics) and as a Smith’s Prizemen (award for excellence in research), he was awarded a fellowship and did much of his life’s work at Cambridge.

•Stokes was the oldest of the trio of natural philosophers who contributed to the fame of the Cambridge University school of Mathematical Physics in the middle of the 19th century. The others were:

•James Clark Maxwell - Maxwell’s Equations, electricity, magnetism and inductance.

•Lord Kelvin - Thermodynamics, absolute temperature scale.

•Stokes is remembered for his numerous contributions to science and mathematics which included research in the areas of hydrodynamics, viscosity, elasticity, wave theory of light and optics.

Stokes’ TheoremInteresting Fact : This theorem is also known as the Kelvin – Stokes Theorem because it was actually discovered by Lord Kelvin. Kelvin then presented his discovery in a letter to Stokes. Stokes, who was teaching at Cambridge at the time, made the theory a proof on the Smith’s Prize exam and the name stuck. Additionally, this theorem was used in the derivation of 2 of Maxwell’s Equations!

Given: A three dimensional surface Σ in a vector field F. It’s boundary is denoted by ∂∑ orientation n.

Stokes Theorem:

So what does it mean?

As Greene’s Theorem provides the transformation from a line integral to a surface integral, Stokes’ theorem provides the transformation from a line integral to a surface integral in three-dimensional space.

Simply said, the surface integral of the curl of a vector field over a three dimensional surface is equal

to the line integral of the vector field over the boundary of the surface.

An application from Aerodynamics

The circulation, Γ of a flow is defined as the line integral of the velocity over a closed curve, C:

C

V

ndA

C

sdV

Given: A three dimensional surface in Velocity Field V with boundary C.

Now, by Stokes’ Theorem, we can say that the circulation around the closed contour C is equal to the surface integral of the curl of the velocity field over the surface. Mathematically, this is written as:

dAnVsdVAC

)(

Example from AerodynamicsGiven: An incompressible, steady from where the velocity field is:

jxyy

ixyyxV )3

()( 23

22

vdyudxsdVC

Find: For the plane shown, show that the circulation around the boundary is equal to the surface integral of the curl of the velocity field over the surface (verify Stokes’ Theorem).

Solution:x

y

y=x

(1,1)

1

23

1

0

0

0

23

22 )3

()( dyxyy

dxxyyx1.)

y = 0, x = x

= 0

2.)

y = y, x = 1

1

1

1

0

23

22 )3

()( dyxyy

dxxyyx = -1/4

0

1

0

1

23

22 )3

()( dyxyy

dxxyyx3.)

y = x

= 1/6

12

1TOTAL

Now, evaluating the curl of the velocity vector over the surface.

Example from Aerodynamics (continued)

03

ˆˆˆ

23

22 xyy

xyyx

zyx

kji

V

dAnVA

)(

kxyxy ˆ)2( 22

1

0 0

22 ˆˆ)2()(x

A

dydxkkxyxydAnV

1

0

3

3

1dxx

12

1

12

1)( dAnVsdV

AC

Thus, Stokes’ Theorem is verified:

Summary Green’s Theorem discovered in 1825 Gauss’ Theorem discovered in 1813 Stokes’ Theorem discovered in 1850

Gauss(Germany)

Stokes(Ireland)

Green(England)(ALL NOT FRENCH)