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# a5f08 - ASSIGNMENT#5 1 Let F be a C2 vector field on R 3...

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ASSIGNMENT #5 1. Let F be a C 2 vector field on 3 R . Prove that . F F F 2 div grad curl curl = 2. Let and 0 k j i r + + = z y x r = r . In each case find a constant k such that the expression is turned into an identity. Using some basic identities might help. Calculations in fact are no longer then one line. (a) (b) kr r = 3 2 3 )] ( [ r k r r = (c) 0 ) ( = k r r (d) 4 2 2 ) ( r k r = r 3. The height of a circular fence (radius 10m, centered at origin) at position is ) , ( y x given by the function , so the height varies from 3m to 5m. ) ( 01 . 0 4 ) , ( 2 2 y x y x h + = Suppose that 1L of paint covers 100m 2 . Sketch the fence and determine how much paint you will need if you paint both sides of the fence. 4. It is often very useful to parametrize a curve C : b t t = 0 ), ( g x with respect to arc length du u t s s = = 0 ) ( ) ( g t to get C : ) ( 0 ), ( b s s s = f x . Consider the parametrization of C : 4 0 ), 2 sin , 2 cos 2 , 2 sin ( ) ( π = t t t t t x .

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a5f08 - ASSIGNMENT#5 1 Let F be a C2 vector field on R 3...

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