2009-11-25: Surface Integrals and the Area Element dS
Stewart section 17.6
- Last time we introduced parametric surfaces and the area element dS
- Key Point For a surface given by r(u,v), dS = |r_{u}x 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(u^{2}+v^{2})>
- Game:
- Find dS for each of these examples.
- 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_{u}x 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_{u}x r_{v}|.
- Note why dS = |r_{u}x r_{v}|
ma215-080-f09 lecture outline 2009-11-25
Created: Wed Nov 25 12:44:03 EST 2009
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©2009 Gavin LaRose