ma215-080-f09 outline 2009-12-11

Key Point We now have a collection of theorems about vector integrals:
1. The Fundamental Theorem of Calculus for line integrals,
2. Green's Theorem,
3. Stokes' Theorem, and
4. the Divergence Theorem.
It is most often that we use these to replace the integral on the left (a line or surface integral) with the expression on the right (a value, or a surface or volume integral) (though all, especially Stokes' theorem, may be used in either direction).
Game: Let E be the cylindrical volume with radius 3 and height 2, centered on the z-axis and sitting on the xy-plane. Let C be the circle around the base of the volume, and S be the surface contained by C, and let F = <xy, y cos(z), x sin(z)>
1. Find the flux over the surface of E.
2. Find the line integral of F . dr on C using Stokes' theorem.
How easy would it be to work out the line integral in (2) directly?
Game: Let S₁ be the sides of the cylinder, and S₂ the top.
1. Find the flux across S.
2. Use (1) and the previous game to find the flux through S₁+S₂
3. Find the flux through S
Key Point
1. Note that the flux integral of F is different from the flux integral of curlF.
2. We can break up surfaces (and curves) to be able to use a theorem and an easy integral to find flux through (or the line integral on) part of the surface (or curve).

2009-12-11: Theorems of Vector Calculus