MATH209 Study Guide - Regions Of Johannesburg, Dot Product, Parametric Equation

• School: University of Alberta
• Department: Mathematics
• Course: MATH209
• Professor: Dragos Hrimiuc

Document Summary

Example (from class): evaluate the surface area of the upper part of the sphere of radius b which is inside the cylinder x2 + y2 = a2 (0 < a < b). First nd a parametrization for the surface s. to do that, use spherical coordinates: (cid:17) , 0 2 , x( , ) = b sin cos , y( , ) = b sin sin , z( , ) = b cos , or in vector form: Find (cid:126)r and (cid:126)r : (cid:126)r( , ) = b sin cos (cid:126)i + b sin sin (cid:126)j + b cos (cid:126)k, (cid:126)r = Next, compute the cross product: (cid:126)r = (cid:126)i + (cid:126)i + + (b2 sin cos )2 = b2 sin4 cos2 + sin4 sin2 + cos2 sin2 . Next, compute the length of the cross product: (cid:126)r (cid:126)r = (cid:12)(cid:12)(cid:126)ru (cid:126)rv (cid:12)(cid:12) = (cid:126)i cos cos .