m2203sp04finalexamsolutions

m2203sp04finalexamsolutions - S. F. Ellermeyer MATH 2203 -...

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MATH 2203 — Final Exam Solutions May 3, 2004 S. F. Ellermeyer Name Instructions. This exam contains ten problems, but only eight of them will be graded. You may choose any eight to do. Please write DON’T GRADE on the two that you don’t want me to grade. In writing your solution to each problem, include su cient detail and use correct notation. (For instance, don’t forget to write “ = ” when you mean to say that two things are equal.) Your method of solving the problem must be clear to the reader (me). If I have to struggle to understand what you have written, then you might not get full credit or even any credit for your solution even if you get a correct answer. 1. Find the surface area of the surface with parametric equations x = uv y = u + v z = u v u 2 + v 2 1 . Solution: The surface area of this surface, S ,is ZZ S 1 dS = D | r u × r v | dA where D is the unit disk. Since r u × r v = ¯ ¯ ¯ ¯ ¯ ¯ ij k v 11 u 1 1 ¯ ¯ ¯ ¯ ¯ ¯ = 2 i +( u + v ) j v u ) k , we have | r u × r v | = q ( 2) 2 u + v ) 2 v u ) 2 = p 4+2( u 2 + v 2 ) = 2 2+ u 2 + v 2 so the surface area is D | r u × r v | dA = 2 D u 2 + v 2 dA = 2 Z 2 π 0 Z 1 0 r 2 rdrd θ = μ 2 6 8 3 π . 1
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2. Draw a sketch of the vector f eld F ( x, y )= x i y j . Make sure to include all four quadrants in the xy plane in your picture so that the overall nature of the vector f eld is evident.
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m2203sp04finalexamsolutions - S. F. Ellermeyer MATH 2203 -...

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