20e_04s_fs

20e_04s_fs - Math 20E Final Exam Solutions June 2004 1 Let...

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Unformatted text preview: Math 20E Final Exam Solutions June 2004 1. Let C = A × B . By the definition of the cross product, C is perpendicular to A and C × A is perpendicular to C and A . Since the length of a cross product of two perpendicular vectors is the product of their lengths, | C × A | = | C || A | . 2. Since ∂ R /∂u = cos v i − sin v j − 2 u k and ∂ R /∂v = − u sin v i − u cos j , we have the normal N = ∂ R ∂u × ∂ R ∂v = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle i j k cos v − sin v − 2 u − u sin v − u cos v vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = − 2 u 2 cos v i + 2 u 2 sin v j − u k . Since | N | = radicalbig 4 u 4 cos 2 v + 4 u 4 sin 2 v + u 2 = | u | √ 4 u 2 + 1 and u ≥ 0, the desired answer is − N | N | = 2 u cos v i √ 4 u 2 + 1 − 2 u sin v j √ 4 u 2 + 1 + k √ 4 u 2 + 1 ....
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20e_04s_fs - Math 20E Final Exam Solutions June 2004 1 Let...

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