ma215-080-f09 outline 2009-11-30

Last time we looked at the area element dS for a parametric surface S.
Key Point Note what this says: r(u,v) maps a (u,v) pair to a point (x,y,z) in space.
Key Point Thus int_S dS = surface area (an integral on S) = int_D |r_ux r_v| du dv (an integral in the u-v plane).
Note also that r_u and r_v are tangent to S, so their cross-product is perpendicular to it.
Game:
1. Consider the surface r = <u sin(v), u cos(2v), 2v>, 0<=u<=3, 0<=v<=pi. Set up an integral for its area.
2. If its density is d = z, set up an integral for its mass.
If f(x,y,z) is a quantity associated with surface area (e.g., density), then the integral of f(r) dS adds this up over the surface.
Next: Recall that the integral of F.dr says "add up the piece of F that points along C."
For surfaces, a similar quentity is "the piece of F that passes through the surface." This is called the Flux.
Game: For S given by r = <u cos(v), u sin(v), u> (0 <= u <= 3, 0 <= v <= pi) and F = <-y, x, z>:
1. Find a vector perpendicular to S.
2. Find a unit vector normal to S.
3. Find the component of F through S.

2009-11-30: Surface Integrals