Midterm09 - Optics 287, Math 287 March 19, 2009 11:05 to...

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Optics 287, Math 287 March 19, 2009 11:05 to 12:20 Midterm Exam 1. a) Calculate the gradient, divergence or curl (whichever applies) of A ( r ) = r n r , where n is an integer. (15 pts.) b) For what values of n is the divergence constant? (5 pts.) c) For what values of n is the divergence equal to zero? (5 pts.) d) For what values of n is the curl zero? (5 pts.) e) Consider the integral of the divergence of A over the volume contained in a unit sphere centered at the origin. What is the value of this integral for any n ? (10 pts.) 2. (15 pts) Consider the function f ( x ) = 1 x , 0 x < 1, 1 x ,
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