2220hw2 - Math 2220 Section 2.4 Problem Set 2 Spring 2010...

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Math 2220 Problem Set 2 Spring 2010 Section 2.4 : 10. Determine all the second-order partial derivatives, including the mixed partials, for the function f ( x,y ) = cos( xy ). 14. Do the same thing for f ( x,y ) = e x 2 + y 2 . 18. Consider the function F ( x,y,z ) = 2 x 3 y + xz 2 + y 3 z 5 7 xyz . (a) Find F xx , F yy , and F zz . (b) Calculate the mixed partials F xy , F yx , F xz , F zx , F yz , and F zy and verify Theorem 4.3 (equality of mixed partials). (c) Is F xyx = F xxy ? Could you have known this without resorting to calculation? (d) Is F xyz = F yzx ? 20. The partial differential equation 2 f ∂x 2 + 2 f ∂y 2 + 2 f ∂z 2 = 0 is known as Laplace’s equa- tion (in three variables). Any function f of class C 2 satisfying this equation is called a harmonic function . (a) Is f ( x,y,z ) = x 2 + y 2 2 z 2 harmonic? What about f ( x,y,z ) = x 2 y 2 + z 2 ? (b) We can generalize Laplace’s equation to functions of n variables as 2 f ∂x 2 1 + 2 f ∂x 2 2 + · · · + 2 f ∂x 2 n = 0 Give an example of a harmonic function of n
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