Math 20E First Midterm 48 points April 21, 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. • Hint: In some cases, short cuts can reduce computation. 1. (24 points) Let F = ( x 2 + y 2 ) i + z j + z k , and f = xyz . In each case, either compute the requested item or brie±y explain why it makes no sense. (a) the curl of f (b) the divergence of f (c) ( F × ∇ f ) · F (d) f × F (e) the divergence of F (f) ( F × F ) × F 2. (9 points) For each of the following regions, answer the following questions: (i) Is it open? (ii) Is it connected? (iii) Is it simply connected? You do not need to explain your answers. (a) The set of points ( x, y, z ) with x 2 + y 2 + z 2 < 2 and
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