2009-12-14: Parameterizing Surfaces
Stewart section Chapter 17 Review
- How do we parameterize a surface S? Start with r(u,v) = <x,y,z>, where x,y,z are functions of u,v. To find x,y and z, we look at how S is described.
- Game: Parameterize each surface:
- The cylinder y^{2}+z^{2}=4 to the left of the plane y=x.
- The part of the parabaloid y=x^{2}+z^{2} inside (a) the cylinder x^{2} + z^{2}<=4; (b) the cylinder y^{2}+z^{2}<=4.
- Key Point
- The parameterization of a surface just defines the (x,y,z) coordinates of a point on the surface.
- The parameterization must include the domain for u and v.
- How is a surface oriented? If we are given a surface, we must in general specify an orientation (this says which direction through S we call positive). The parameterization of S implicitly defines an orientation: the direction the area vector dS points.
- Game: For (2a) above, find dS to calculate the flux in through S (that is, toward the y-axis.
ma215-080-f09 lecture outline 2009-12-14
Created: Mon Dec 14 13:55:22 EST 2009
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©2009 Gavin LaRose