vcprac02s - The University of Sydney School of Mathematics...

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Unformatted text preview: The University of Sydney School of Mathematics and Statistics Solutions to Practice Session 2 MATH2061: Vector Calculus Summer School 2012 1. (a) Find grad f if f ( x, y ) = x + 2 xy 3 y 2 . (b) Find grad g if g ( x, y, z ) = e x cos( yz 2 ). (c) Find if = 1 radicalbig x 2 + y 2 + z 2 . Solution: (a) grad f = (1 + 2 y ) i + (2 x 6 y ) j . (b) grad g = e x cos( yz 2 ) i e x z 2 sin( yz 2 ) j 2 e x yz sin( yz 2 ) k . (c) = x ( x 2 + y 2 + z 2 ) 3 / 2 i + y ( x 2 + y 2 + z 2 ) 3 / 2 j + z ( x 2 + y 2 + z 2 ) 3 / 2 k . 2. (a) Given that F = 2 xy i + ( x 2 + z 2 ) j + (2 yz + 2 z ) k is conservative, find a scalar function such that F = . (b) Find the work done by F along any path C from the point P (1 , 1 , 1) to the point Q (1 , 2 , 3) . Solution: (a) Suppose that is such that F = . Then x = 2 xy, y = x 2 + z 2 , z = 2 yz + 2 z, so that = x 2 y + f ( y, z ) , = x 2 y + z 2 y + g ( x, z ) , = yz 2 +...
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vcprac02s - The University of Sydney School of Mathematics...

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