ma215-080-f09 outline 2009-12-14

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:
1. The cylinder y²+z²=4 to the left of the plane y=x.
2. The part of the parabaloid y=x²+z² inside (a) the cylinder x² + z²<=4; (b) the cylinder y²+z²<=4.
Key Point
1. The parameterization of a surface just defines the (x,y,z) coordinates of a point on the surface.
2. 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.

2009-12-14: Parameterizing Surfaces