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4321_HW2FA08

# 4321_HW2FA08 - A = x x ^ 2 y y ^ 3 z z ^ and B = 3y x ^-2x...

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Physics 4321 Homework Set 2 September 3, 2008 Due: September 8, 2008 Recall that the units vectors parallel to x, y, and z are given by x ^ , y ^ , and z ^ , respectively. 1. Calculate the gradients of each of the following functions. (a) f ( x , y , z ) = x 2 + y 3 + z 4 ; (b) f ( x , y , z ) = x 2 y 3 z 4 ; (c) f ( x , y , z ) = e x sin( y ) ln( z ). 2. Let r be the separation vector from a point ( x ’, y ’, z ’) to the point ( x , y , z ), and let r be its length. Show that the gradient of (1/ r ) is given by grad (1/ r ) = - r / r 3 . 3. Calculate the divergence and curl of each of the vector functions (a) v = x 2 x ^ + 3 xz 2 y ^ - 2 xz z ^ and (b) v = y 2 x ^ + (2 xy + z 2 ) y ^ + 2 yz z ^ . 4. Write out the components of the vector ( grad )B 5. Check product rule (iv) (page 21) by evaluating each term separately for
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Unformatted text preview: A = x x ^ + 2 y y ^ + 3 z z ^ and B = 3y x ^-2x y ^ . 6. Calculate the Laplacian of (a) T = x 2 + 2 xy + 3 z +4 and (b) v = x 2 x ^ + 3 xz 2 y ^- 2 xz z ^ . 7. Calculate the line integral of the function v = x 2 x ^ + 2 yz y ^ + y 2 z ^ along the paths (a) (0,0,0) → (1,0,0) → (1,1,0) → (1,1,1) and (b) the direct straight line. 8. If v = 4 xz x ^- y 2 y ^ + yz z ^ , calculate its surface integral over the surface bounded by the cube x = 0, x = 1, y = 0, y = 1, z = 0, z = 1. Notice that you will have to set up an integral over each of the six surfaces....
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