2009-11-23: Parametrically Defined Surfaces
Stewart section 17.6
- Last time we defined two operations on vector fields: curl F and div F
- Note: curl F = 0 <=> F is conservative
- Today we look at Parametrically Defined Surfaces
Recall how r(t) defines a curve.
- We define surfaces parametrically the same way: Let
- Game:
- Find a parametric representation of this surfaces as r(r,theta). What is the (r,theta) domain?
- If r = <2 cos(theta), 2 sin(theta), t>, pi/2 <= theta <= 2pi, 0 <= t <= 4, sketch the surface.
- For this r, what curve does r(theta,1) trace out?
- Next, we want to be able to integrate over a surface, e.g., to find surface area.
- Key Point dS = |r_{theta} x r_{t}|dtheta dt
- Game: Find dS for r(theta,t) = <2 cos(theta), 2 sin(theta), t>
ma215-080-f09 lecture outline 2009-11-23
Created: Mon Nov 23 13:34:51 EST 2009
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©2009 Gavin LaRose