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8.022pset0solution

# 8.022pset0solution - Massachusetts Institute of Technology...

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Massachusetts Institute of Technology Department of Physics 8.022 Spring 2004 Solution to Assignment 0: Math review/practice 1. Know your recitation ... hopefully, you don’t need to see a solution set for this one ... 2. Partial derivatives and coordinate conversions. (a) ∂f ∂x = - 1 ( x 2 + y 2 + z 2 ) 2 ∂x q x 2 + y 2 + z 2 = - x ( x 2 + y 2 + z 2 ) 3 / 2 . By symmetry, it should be obvious that ∂f ∂y = - y ( x 2 + y 2 + z 2 ) 3 / 2 , ∂f ∂z = - z ( x 2 + y 2 + z 2 ) 3 / 2 . (b) Note that the magnitude of the radial displacement vector | ~ r | = x 2 + y 2 + z 2 . For simplicity, let’s denote this r . Then, ~ f = - x ˆ x + y ˆ y + z ˆ z ( x 2 + y 2 + z 2 ) 3 / 2 , = - ~ r r 3 , = - ˆ r r 2 . On the last line, we’ve defined the unit vector ˆ r = ~ r/r . It is the unit vector that points in the radial direction (with respect to a specified origin). You should be aware that flipping back and forth between the forms on the 2nd and 3rd lines is something we do a lot, depending on which is more convenient. (c) When we convert to cylindrical coordinates, x 2 + y 2 + z 2 becomes ( r c ) 2 cos 2 φ + ( r c ) 2 sin 2 φ + z 2 = r 2 c + z 2 . The function f thus becomes f = 1 / q r 2 c + z 2 . The partial derivatives we want are therefore given by ∂f ∂r c = - r c ( r 2 c + z 2 ) 3 / 2 , ∂f ∂φ = 0 , ∂f ∂z = - z ( r 2 c + z 2 ) 3 / 2 .

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(d) In these coordinates, ~ f = - r c ˆ r c + z ˆ z ( r 2 c + z 2 ) 3 / 2 . This is equivalent to the answer we worked out in part (b) provided that the radial displacement vector is ~ r = r c ˆ r c + z ˆ z . (e) When we convert to spherical coordinates, we have x 2 + y 2 + z 2 = r 2 sin 2 θ cos 2 φ + r 2 sin 2 θ sin 2 φ + r 2 cos 2 θ = r 2 sin 2 θ + r 2 cos 2 θ = r 2 . So, f = 1 /r . [ CAUTION : For some reason, many math classes and textbooks define spherical coordinates in such a way that θ and φ are reversed relative to what we use. I have no clue why that is; all I know is that every physics textbook, physics research paper, and physics brain I’ve ever encountered uses them in the way that I’ve defined here. If you learned about spherical coordinates (or “spherical polar coordinates”) in
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8.022pset0solution - Massachusetts Institute of Technology...

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