matematika 4 theorema.pptx
TRANSCRIPT
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Greens
Theorem
Gauss
Theorem
Stokes
Theorem
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1. Greens Theorem
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(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
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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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Greens Theoremand beyond
Greens Theorem is a crucial component in
the development of many famous works:
James Maxwells EquationsGauss Divergence Theorem
Stokes Integral Theorem
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13.7 Gauss Divergence
Theorem
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(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.
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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.
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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
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13.8 The Integral
Theorem of Stokes
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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.
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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
)(
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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:
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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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