Homework 7 - APPM 2350 HOMEWORK 07 Due Friday, October 15,...

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APPM 2350 HOMEWORK 07 FALL 2010 Due Friday, October 15, 2010 at 4 pm under your TA’s door. 1. Previously in the semester, we derived equations for the distance between a point and a line, and a point and a plane. Your task is to derive the same results, but using the method of Lagrange multipliers. (a) Use Lagrange multipliers to establish the formula D = | ax 0 + by 0 d | a 2 + b 2 for the distance D from the point ( x 0 ,y 0 )tothel ine ax + by = d . (b) Use Lagrange multipliers to establish the formula D = | ax 0 + by 0 + cz 0 d | a 2 + b 2 + c 2 for the distance D from the point ( x 0 ,y 0 ,z 0 )tothel ine ax + by + cz = d . 2. Suppose you have three identical boxes, and some total number of particles, say N . Let n 1 , n 2 , and n 3 denote the number of particles in each box. Of course, we have that 3 ° j =1 n j = N. (1) We can then de±ne the probability of a particle being in the j th box as P j = n j N . Then clearly, 3 ° j =1 P j =1 . (2) Now we de±ne the entropy, S , of our system of boxes and particles as S = 3 ° j =1 P j ln P j
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Homework 7 - APPM 2350 HOMEWORK 07 Due Friday, October 15,...

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