Special Integrations

Green’s Theorem

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 Theory

Consider 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.

Example

A particle moves once counterclockwise about the circle of radius 6 about the origin, under the influence of the force.

∫Cfdx+gdy=∬

D(∂ g∂ x

− ∂ f∂ y )dA

F=( ex− y+x cosh ( x )) i+( y3/2+x ) j

Calculate the work done.

C ( t )=(6cos ( t ) ,6sin ( t ))I : t=(0,2π )

Green’s Theorem…and beyond

Green’s Theorem is a crucial component in the development of many famous works:

James Maxwell’s Equations

Gauss’ Divergence Theorem

Stokes’ Integral Theorem

Gauss’ Divergence Theorem

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)=(xyz,y2 ,xz2)

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

The Integral Theorem of Stokes

•Irish mathematician and 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 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’ 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

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 .

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.