2009-12-04: Stokes' Theorem

Stewart section 17.8

• We have for the past few days been considering Surface Integrals, of scalar functions f(x,y,z) and as flux integrals. In section 17.8 we consider the case where the vector field G = curl F and find the flux integral of that.
• Game: Let F = <-y, x, z> and S be the cone with z = r, z <= 3.
1. Find curl F.
2. Parameterize S.
3. Find dS.
4. Evaluate the integral of curl F dot dS on the surface.
• Key Point Stokes' Theorem says the double integral over S of curl F dot dS = the line integral on the boundary of S of F dot dr.
• Note that this is just a non-planar version of Green's theorem.
• Game: Let F = <-y,x,z> and r(t) = <3 cos(t), 3 sin(t), 3>, with 0 <= t <= 2 pi. Find the line integral of F dot dr.
• This just shows that both sides of Stokes' theorem do give the same thing in this case.
• Key Point As with Green's theorem, Stokes' theorem lets us replace a surface integral with a line integral, and vice versa.
ma215-080-f09 lecture outline 2009-12-05
Created: Mon Dec 7 12:20:51 EST 2009