2. Green’s Theorem
George Green
July 14, 1793 - May 31, 1841
British mathematicianand physicist
First person to try to explain a mathematicaltheory
of electricity and magnetism
!taught-selfAlmost entirely
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 Theory
Consider a simple closed curve C, and let D be
the region enclosed by the curve.
3. :Notes
•in theno holesThe simple, closed curve has
region D
•A direction has been put on the curve with the
convention that the curve C
leftis on theDif the regionpositive orientationhas a
as we traverse the path.
Example
A particle moves once counterclockwise about the circle
of radius 6 about the origin, under the influence of the
force.
dA
y
f
x
g
gdyfdx
C D
jxyixxyeF x ˆ)(ˆ))cosh(( 2/3
5. Green’s Theorem…andbeyond
Green’s Theorem is a crucial component in the
developmentof many famous works:
James Maxwell’sEquations
Gauss’ Divergence Theorem
Stokes’ Integral Theorem
6. Gauss’ Divergence Theorem
Gauss in the House
German mathematician,lived 1777-1855
Born in Braunschweig, Duchy of Braunschweig-
Lüneburg in Northwestern Germany
DisquisitionesPublishedArithmeticaewhen hewas
21(and what haveyoudone 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.
7. 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.
8. Example
Assumethere is a unit circle centered on the
)2
,xz2
origin and a vector field V(x,y,z)=(xyz,y
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
10. •Irish mathematicianand physicist who
attended Pembroke College (Cambridge
University).
•Stokes was the oldest of the trio of natural
philosophers who contributed to the fame of
the Cambridge University school of
th
Mathematical Physics in the middle of the 19
century. The others were:
•James Clark Maxwell - Maxwell’s Equations,
electricity, magnetism and inductance.
•Lord Kelvin - Thermodynamics, absolute
temperature scale.
•Stokes’Theorem
•Interesting 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
11. 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.
So what does it mean?
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.
12. 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.
