This preview shows page 1. Sign up to view the full content.
Unformatted text preview: ∂β has a nonzero vertical component while ∂ −→ p ∂α does not. If cos β = 0, we simply note that the two partial derivative vectors are perpendicular to each other (in fact, in retrospect, this is true whatever value β has). Thus, every point is a regular point. Of course, increasing either α or β by 2 π will put us at the same position, so to get a coordinate patch we need to restrict each of our parameters to intervals of length < 2 π . To de±ne the tangent plane to a regularly parametrized surface, we can think, as we did for the graph of a function, in terms of slicing the surface...
View Full Document
This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
- Fall '08