2. The Laplacian operator takes different forms in different coordinate systems. Con- sider the Laplace equation...

 

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2. The Laplacian operator takes different forms in different coordinate systems. Con- sider the Laplace equation in two dimensions, with the Laplacian operator expressed in polar coordinates, (r,0): D² = 10 (2) rər 18 rər + When the Laplace equation V2 = 0 is separated in polar coordinates (r, 0), the 0- equation has the form f"(0) = Xƒ(0). Because is an angle here, going around by 27 should get us back to the same point; this means that ƒ(0+2) = f(0). This is a type of boundary condition, called the periodic boundary condition. 1 8² r² 20² Solve the (eigenvalue) problem f"(0) = Xƒ(0) with periodic boundary condition f(0 + 2π) = f(0). Use complex exponentials rather than trigonometric functions to get a simpler answer. You should be able to show that there are no solutions when A> 0 and that the allowed values of A are A = -n² for integer n (including 0). Hint: Look at the solution to Worksheet 3, Q 2, including the complex exponential part. 3. Consider Laplace equation V2f = 0 in two dimensions, with the Laplacian opera- tor expressed in polar coordinates, (r, 0): (u(²,0)). Using your answer to the previous question, write down the general solution to this equation that is smooth (and finite) at r = 0. + 18² r² 002¹(r, 0) = 0. 2. The Laplacian operator takes different forms in different coordinate systems. Con- sider the Laplace equation in two dimensions, with the Laplacian operator expressed in polar coordinates, (r,0): D² = 10 (2) rər 18 rər + When the Laplace equation V2 = 0 is separated in polar coordinates (r, 0), the 0- equation has the form f"(0) = Xƒ(0). Because is an angle here, going around by 27 should get us back to the same point; this means that ƒ(0+2) = f(0). This is a type of boundary condition, called the periodic boundary condition. 1 8² r² 20² Solve the (eigenvalue) problem f"(0) = Xƒ(0) with periodic boundary condition f(0 + 2π) = f(0). Use complex exponentials rather than trigonometric functions to get a simpler answer. You should be able to show that there are no solutions when A> 0 and that the allowed values of A are A = -n² for integer n (including 0). Hint: Look at the solution to Worksheet 3, Q 2, including the complex exponential part. 3. Consider Laplace equation V2f = 0 in two dimensions, with the Laplacian opera- tor expressed in polar coordinates, (r, 0): (u(²,0)). Using your answer to the previous question, write down the general solution to this equation that is smooth (and finite) at r = 0. + 18² r² 002¹(r, 0) = 0.

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