Exam 13: Boundary-Value Problems in Other Coordinate Systems

In the previous problem, the solution of the eigenvalue problem is

In changing from Cartesian to polar coordinates, $\frac { \partial r } { \partial y }$ is

In the three previous problems, the product solutions are

In the previous four problems, the infinite series solution of the original problem is (for certain values of the constants $a _ { n }$ and $b _ { n }$ )

In the previous problem, separate variables using $u ( r , \theta ) = R ( r ) \Theta ( \theta )$ . The resulting problems are

In changing from Cartesian to polar coordinates, $\frac { \partial r } { \partial x }$ is

In the previous problem, if we also require that $\Theta$ be bounded everywhere, the solution of the eigenvalue problem is

In the previous four problems, the infinite series solution of the original problem is $u = A _ { 0 } + \sum _ { n = 1 } ^ { \infty } r ^ { n } \left( A _ { n } \cos ( n \theta ) + B _ { n } \sin ( n \theta ) \right)$ where Select all that apply.

When changing from Cartesian to spherical coordinates, $\frac { \partial r } { \partial z } =$

Using the separation of the previous problem, the equation for $\Theta$ becomes

In the problem $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 2 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } + \frac { \cot \theta } { r ^ { 2 } } \frac { \partial u } { \partial \theta } = 0$ , separate variables, using $u ( r , \theta ) = R ( r ) \Theta ( \theta )$ . The resulting problems for $R$ and $\Theta$ are

In changing from Cartesian to polar coordinates, $\frac { \partial u } { \partial y }$ is

When changing from Cartesian to spherical coordinates, $\frac { \partial \phi } { \partial x } =$

In the previous three problems, the product solutions are

The equations relating Cartesian and spherical coordinates include Select all that apply.

The solution of $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } + \lambda R = 0 , R ( 1 ) = 0 , R ( 2 ) = 0$ is

Consider the steady-state temperature distribution in a circular disc of radius $C$ centere at the origin, with temperature given as a function, $f ( \theta )$ on the boundary $r = c$ and zero on the boundaries $\theta = 0$ and $\theta = \pi$ The mathematical model of this situation is

In the previous three problems, the solution for $R$ is

In the previous problem, the solution of the eigenvalue problem is

In separating variables, using, $u ( r , \theta ) = R ( r ) \Theta ( \theta )$ , in the equation $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 2 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } + \frac { \cot \theta } { r ^ { 2 } } \frac { \partial u } { \partial \theta } = 0$ , the resulting equation for $R$ and is