20e_04s_2e

20e_04s_2e - ze xy i and S is the ellipsoid x 2 + 2 y 2 + 3...

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Math 20E Second Midterm 60 points May 26, 2004 Print Name, ID number and Section on your blue book. BOOKS and CALCULATORS are NOT allowed. One side of one page of NOTES is allowed. You must show your work to receive credit. 1. (18 points) Fill in the blanks with either functions, numbers or words. You need not copy the statements — just write what goes in the blanks. Some examples of words you might or might not use are harmonic irrotational potential scalar solenoidal vector curl divergence gradient laplacian (a) If ∇ × F = ∇ × G for all R , then F G is the of some function. (b) If 2 f ( R ) = 2 g ( R ) for | R | ≤ 1 and f = g for | R | = 1, then f ( R ) g ( R ) = for | R | ≤ 1. (c) If 2 f = 0, we call f a (or an) function. (d) The fundamental theorem of vector analysis states that a nice vector function in a nice domain can be written as the sum of a (or an) and a (or an) . 2. (10 points) Compute ii S ( ∇× F ) · n dS where F ( x, y, z ) =
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Unformatted text preview: ze xy i and S is the ellipsoid x 2 + 2 y 2 + 3 z 2 = 1. 3. (20 points) The function F = 24 xz i 12 z 2 k has zero divergence. (a) Find a vector potential for F ; that is, nd G whose curl is F . (b) Find a vector potential for F that has no k component. (This may or may not be the same as your answer to (a).) 4. (12 points) Let D be the region in the xy-plane where | x | + | y | 1. In other words, it is the region given by 1 x + y 1 and 1 x y 1 . Rewrite ii D ( x y ) 2 e x 2-y 2 dx dy as an integral over u and v by using the substitution u = x + y , v = x y . Remember to describe the domain of integration. Remark: The ( u, v ) integral can be evaluated by integrating over u and then v ; how-ever, you are NOT being asked to evaluate it. END OF EXAM...
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