matematika 4 theorema.pptx

8/13/2019 MATEMATIKA 4 Theorema.pptx

1/23

Greens

Theorem

Gauss

Theorem

Stokes

Theorem


2/23

1. Greens Theorem


3/23

(No, he was not French)

George Green

Jul

y 14, 1793 - May 31, 1841

British mathematician and physicist

First person to try to explain a mathematical theory ofelectricity and magnetism

Almost entirely self-taught!

Published An Essay on the Application of MathematicalAnalysis to the Theories of Electricity and Magnetismin

1828. Entered Cambridge University as an undergraduate in

1833 at age 40.


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The Theory

Consider a simple closed curve C,and let Dbe the regionenclosed by the curve.

Notes:

The simple, closed curve has no holesin the region D

A direction has been put on the curve with the convention that the curve C

has a positive orientationif the region Dis on the leftas we traverse the path.

dAy

f

x

ggdyfdx

C D


5/23

Example

Section 13.2, problem 2A 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 )())cosh(( 2/3

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

)2,0(: tI

6

F


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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 )())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 )(')(


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8/23

Greens Theoremand beyond

Greens Theorem is a crucial component in

the development of many famous works:

James Maxwells EquationsGauss Divergence Theorem

Stokes Integral Theorem


9/23

13.7 Gauss Divergence

Theorem


10/23

(Also not French)

Gauss in the House

German mathematician, lived1777-1855

Born in Braunschweig, Duchy ofBraunschweig-Lneburg inNorthwestern Germany

Published DisquisitionesArithmeticaewhen he was 21 (andwhat have youdone today?)

As a workaholic, was onceinterrupted while working and toldhis wife was dying. He repliedtell her to wait a moment untilIm finished.


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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 vectorfield within the region enclosed by the

boundary.


12/23

Applications The Aerodynamic Continuity Equation

The surface integral of mass flux around a control volumewithout sources or sinks is equal to the rate of mass storage.

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

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

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

Gausss 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 suchas probability density.


13/23


14/23

The unit normal of the sphere is defined as

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

The 3-D surface integral for radius = 1 (plus Jacobian) isequal 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( 324342

2

00

dd


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Now, Gauss Divergence Theorem shall be used, and the sameresult 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 continuouslydifferentiable 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()(

23224

1

0

2

00

2

1

0

2

00

ddd

dddV


16/23

13.8 The Integral

Theorem of Stokes


17/23

Sir George Gabriel Stokes

(Aug. 13, 1819Feb. 1, 1903)

Irish mathematician and physicist who attended

Pembroke College (Cambridge University).

(Again, also not French)

After graduating as Senior Wrangler (first in class inmathematics) and as a Smiths Prizemen (award for

excellence in research), he was awarded a fellowship

and did much of his lifes work at Cambridge.

Stokes was the oldest of the trio of natural

philosophers who contributed to the fame of theCambridge University school of Mathematical Physics

in the middle of the 19thcentury. The others were:

James Clark Maxwell - Maxwells Equations,

electricity, magnetism and inductance.

Lord Kelvin - Thermodynamics, absolutetemperature 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.


18/23

Stokes TheoremInteresting Fact :This theorem is also known as the KelvinStokes 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 Smiths Prize exam and the namestuck. Additionally, this theorem was used in the derivation of 2 of Maxwells Equations!

Given:A three dimensional surfacein a vector field F. Its

boundary is denoted by orientation n.

Stokes Theorem:

So what does it mean?

As Greenes 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-dimensionalspace.

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.


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An application from Aerodynamics

The circulation, 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:

dAnVsdV

AC

)(


20/23

Example from Aerodynamics

Given:An incompressible, steady from where the velocity field is:

jxy

y

ixyyxV )3()( 2

322

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

1

TOTAL


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

Example from Aerodynamics (continued)

03

23

22 xyy

xyyx

zyx

kji

V

dAnVA

)(

kxyxy )2( 22

1

0 0

22 )2()(

x

A

dydxkkxyxydAnV

1

0

3

3

1dxx

12

1

12

1)( dAnVsdV

AC

Thus, Stokes Theorem is verified:


22/23

Summary

Greens Theorem discovered in 1825 Gauss Theorem discovered in 1813

Stokes Theorem discovered in 1850

Gauss

(Germany)

Stokes

(Ireland)Green

(England)


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