solutions_vector - PHY2061 R D Field Vector Analysis...

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PHY2061 R. D. Field Solutions Vector Analysis Page 1 of 5 Vector Analysis Solutions Problem 1: Calculate the gradient of the scalar function 3 2 yz x f = . Answer: z yz x y z x x xyz f ˆ 3 ˆ ˆ 2 2 2 3 2 3 + + = ! Solution: The gradient is given by z yz x y z x x xyz z z f y y f x x f f ˆ 3 ˆ ˆ 2 ˆ ˆ ˆ 2 2 3 2 3 + + = + + = ! . Problem 2: Let y x x y F ˆ ˆ + = ! . (a) Calculate r d F ! ! along Path 1 and Path 2 from point P 1 =(0,0) to P 2 =(1,1) as shown in the Figure. (b) Calculate the divergence , F ! ! , and the curl , F ! ! × . Answers: = 1 r d F ! ! 0 0 = × = F F ! ! ! ! Solution: (a) Along Path 1 we have y = x and dy = dx and hence, = = + = + = 1 1 0 1 1 1 2 Path Path y x Path xdx xdy ydx dy F dx F r d F ! ! . Path 2 consists of two parts Path 2a and Path 2b. Along Path 2a we have y = 0 dy = 0 and along Path 2b we have x = 1 dx = 0 . Thus, 1 0 1 0 2 2 2 = + = + = + = dy xdy ydx xdy ydx r d F a Path b Path Path ! ! . (b) The divergence is given by 0 0 0 0 = + + = + + = z F y F x F F z y x ! ! , and the curl is given by y x Path 1 Path 2 (1,1) (0,0)
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