Let f(x, y) be a function of two variables x and y. We define the laplacian of f

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Let f(x, y) be a function of two variables x and y. We define the laplacian of f to be the combination of derivatives 2f = 2f/ x2 + 2f/ y2 This operator is extremely useful in modeling the physical nature of diffusion. Suppose we use the polar coordinate substitution x = r cos( theta ) and y = r sin( theta ). The goal of these exercises is to compute 2f To begin, consider the expressions x = rcos( theta ) and y = r sin(theta). Using implicit differentiation, show: r/ x cos( theta ) - r sin( theta ) theta/ x = 1 r/ x sin(theta) - r cos( theta ) theta / x = 0 Solve these equations for r/ x and theta/ x: r/ x= theta / = The chain rule says that f/ x = f/ x r/ x + f/ theta r/ theta Moreover, 2f/ x2 = / x( f/ x)= / r( f/ x) r/ x+ / theta ( f/ x) theta / x Substitute your results from (b) to find an expression for 2f/ x2 in terms of partial derivatives of f with respect to r and theta: 2f/ x2= 2f/ r2+ f/ r+ 2f/ theta r+ f/ theta Repeat parts (a), (b). and (c) to find 2f/ y2 in terms of partial derivatives of f with respect to r and theta. Combine the two equations to get the polar form for the laplacian: 2f = 2f/ r2+ 2f/ theta 2+ 2f/ theta r+ f/ r+ f/ theta

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