ma215-080-f09 outline 2009-11-25

Last time we introduced parametric surfaces and the area element dS
Key Point For a surface given by r(u,v), dS = |r_ux r_v| du dv
Example 1: r(u,v) = <2 cos(u), 2 sin(u), v>
Example 2: r(u,v) = <u, v, 4 - 2 sqrt(u²+v²)>
Game:
1. Find dS for each of these examples.
2. Find the surface area of the first.
Note that not every dS is a constant times du dv.
Key Point In the same way that for line integrals, writing ds = |r'(t)| dt writes the integral of pieces of C, ds, as an integral of scaled lengths of pieces of the domain in t, dt, writing dS = |r_ux r_v| du dv write the integral of pieces of the surface S, dS as an integral of uv areas du dv, scaled by |r_ux r_v|.
Note why dS = |r_ux r_v|

2009-11-25: Surface Integrals and the Area Element dS