ma215-080-f09 outline 2009-12-07

Last time we introduced Stokes' theorem, and noted that it allows us to replace a surface integral with a line integral, or vice versa. Our goal is to evaluate whichever of these is the easiest.
Note that we said that curl F is the tendency of the vector field F to rotate things. So Green's and Stokes' theorems both say that this tendency on a surface is captured by the behavior on the boundary!
Example: Consider a surface S and F = <2x, x²y, -xz>. Find the double integral on S of curl F dot dS.
Game:
1. The boundary of S consists of two parts. Find r(t) on each.
2. Find the integral of F dot dr on the boundary.
Note that the surface T [figure] has the same boundary as S.
Game: Find the integral of curl F dot dS on T.
1. Find dS (hint: note that dS = dx dz).
2. Find curl F.
3. Find the integral.
Key Point
1. dS = r_u x r_v du dv = N dS. Thus if N = i (or j or k), we can write down dS without parameterizing the surface!
2. Stokes' theorem lets us evaluate either a surface integral or a line integral.
3. It also lets us evaluate a surface integral on a different surface, if the second surface has the same boundary.

2009-12-07: More Stokes' Theorem