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# hwk3 - Math 31BH Homework 3 Due Wednesday February 2 Part I...

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Math 31BH - Homework 3. Due Wednesday, February 2. Part I. 1. ( Wednesday, January 26. ) Laplacian and harmonic functions. (i) The temperature T ( x, y ) in a long thin plane at the point ( x, y ) satisfies Laplace’s equation T xx + T yy = 0 . Does the function T ( x, y ) = ln( x 2 + y 2 ) satisfy Laplace’s equation? (ii) Functions which satisfy the Laplace equation are called harmonic. Show that T ( x, y ) = e x cos y is harmonic. See also Problem 5 (ii). (iii) * The function in (i) has the special form T ( x, y ) = h ( x 2 + y 2 ) where h ( z ) = ln z . Find all harmonic functions of the form T ( x, y ) = h ( x 2 + y 2 ) . (iv) The Laplacian of a function f ( x 1 , . . . , x n ) of n variables is defined to be Δ f = 2 f ∂x 2 1 + . . . + 2 f ∂x 2 n . How would you define a harmonic function of n variables? 2. ( Wednesday, January 26. ) d’Alambertian and waves. The d’Alambertian of the function u ( x 1 , . . . , x n , t ) of ( n + 1) variables is defined to be u = 2 u ∂x 2 1 + . . . + 2 u ∂x 2 n - 1 c 2 2 u ∂t 2 . This differs from the Laplacian defined in the previous problem by the choice of signs (+ , + , . . . , + , - ) in front of the derivatives. This choice of signs is dictated by special relativity. The constant c has the physical interpretation as the speed.

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hwk3 - Math 31BH Homework 3 Due Wednesday February 2 Part I...

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