Deck 13: Boundary-Value Problems in Other Coordinate Systems

Question 1

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

A) $\lambda = n \pi , T = a _ { n } \cos ( n \pi i ) + b _ { n } \sin ( n \pi t )$
B) $\lambda = n ^ { 2 } \pi ^ { 2 } , T = a _ { n } \cos ( n \pi t ) + b _ { n } \sin ( n \pi t )$
C) $\lambda = z _ { n } ^ { 2 } / 4$ , where $J _ { 0 } \left( z _ { n } \right) = 0 , R = J _ { 0 } \left( z _ { n } r / 2 \right)$
D) $\lambda = z _ { n } / 2$ , where $J _ { 0 } \left( z _ { n } \right) = 0 , R = J _ { 0 } \left( z _ { n } r / 2 \right)$
E) $\lambda = z _ { n } ^ { 2 }$ , where $J _ { 0 } \left( z _ { n } \right) = 0 , R = J _ { 0 } \left( z _ { n } r \right)$

Answer 1

$\lambda = z _ { n } ^ { 2 } / 4$ , where $J _ { 0 } \left( z _ { n } \right) = 0 , R = J _ { 0 } \left( z _ { n } r / 2 \right)$

Question 2

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

A) $\Theta ^ { \prime \prime } - \lambda \Theta = 0$
B) $\Theta ^ { \prime \prime } + \lambda \Theta = 0$
C) $\sin ( \theta ) \Theta ^ { \prime \prime } + \cos ( \theta ) \Theta ^ { \prime } + \lambda \sin ( \theta ) \Theta = 0$
D) $\cos ( \theta ) \Theta ^ { \prime \prime } - \sin ( \theta ) \Theta ^ { \prime } + \lambda \sin ( \theta ) \Theta = 0$
E) $\sin ( \theta ) \Theta ^ { \prime \prime } - \cos ( \theta ) \Theta ^ { \prime } + \lambda \sin ( \theta ) \Theta = 0$

Answer 2

$\sin ( \theta ) \Theta ^ { \prime \prime } + \cos ( \theta ) \Theta ^ { \prime } + \lambda \sin ( \theta ) \Theta = 0$

Question 3

The two dimensional wave equation in polar coordinates is

A) $a ^ { 2 } \left( \frac { \partial ^ { 2 } y } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } \right) = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }$
B) $a ^ { 2 } \left( \frac { \partial ^ { 2 } y } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } \right) = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }$
C) $a ^ { 2 } \left( \frac { \partial ^ { 2 } y } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } \right) = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }$
D) $a ^ { 2 } \left( \frac { \partial ^ { 2 } y } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { 1 } { r } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } \right) = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }$
E) $a ^ { 2 } \left( \frac { \partial ^ { 2 } y } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } - \frac { 1 } { r } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } \right) = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }$

Answer 3

$a ^ { 2 } \left( \frac { \partial ^ { 2 } y } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } \right) = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } }$

Question 4

Consider the vibrations of a circular membrane of radius 2 clamped along the circumference, with an initial displacement of $1 - \sin ( \pi r / 2 )$ and an initial velocity of zero. The mathematical model for this situation is

A) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi r / 2 ) , u , ( r , 0 ) = 0$
B) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi r / 2 ) , u , ( r , 0 ) = 0$
C) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } = - \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi / 2 ) , u _ { t } ( r , 0 ) = 0$
D) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } = - \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi r / 2 ) , u _ { t } ( r , 0 ) = 0$
E) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } = \frac { \partial ^ { 2 } u } { \partial t ^ { 2 } } = 0 , u ( 2 , t ) = 0 , u ( r , 0 ) = 1 - \sin ( \pi r / 2 ) , u , ( r , 0 ) = 0$

Question 5

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

A) $u = \sum _ { n = 1 } ^ { \infty } a _ { n } J _ { 0 } \left( z _ { n } r / 2 \right) \cos \left( z _ { n } t / 2 \right)$
B) $u = \sum _ { n = 1 } ^ { \infty } b _ { n } J _ { 0 } \left( z _ { n } r / 2 \right) \sin \left( z _ { n } t / 2 \right)$
C) $u = \sum _ { n = 1 } ^ { \infty } b _ { n } J _ { 0 } ( n \pi r ) \sin ( n \pi t )$
D) $u = \sum _ { n = 1 } ^ { \infty } a _ { n } J _ { 0 } ( n \pi r ) \cos ( n \pi t )$
E) $u = \sum _ { n - 1 } ^ { \infty } J _ { 0 } \left( z _ { n } r / 2 \right) \left( a _ { n } \cos ( n \pi t ) + b _ { n } \sin ( n \pi t ) \right)$

Question 6

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

A) $y = r \cos \phi \cos \theta$
B) $x = r \cos \phi \sin \theta$
C) $r ^ { 2 } = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$
D) $z = r \cos \theta$
E) $\tan \phi = y / x$

Question 7

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

A) $R ^ { \prime \prime } + r R ^ { \prime } - r ^ { 2 } \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$
B) $R ^ { \prime \prime } + r R ^ { \prime } + r ^ { 2 } \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$
C) $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } - \lambda R = 0 , \Theta ^ { \prime \prime } - \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$
D) $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } + \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$
E) $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } - \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( 0 ) = 0 , \Theta ( \pi ) = 0$

Question 8

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

A) $\lambda = n \pi / \ln 2 , R = \sin ( n \pi \ln r / \ln 2 )$
B) $\lambda = ( n \pi / \ln 2 ) ^ { 2 } , R = \sin ( n \pi \ln r / \ln 2 )$
C) $\lambda = ( n \pi / \ln 2 ) ^ { 2 } , R = \cos ( n \pi \ln r / \ln 2 )$
D) $\lambda = n \pi / \ln 2 , R = \cos ( n \pi \ln r / \ln 2 )$
E) none of the above

Question 9

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

A) $A _ { n } = \int _ { 0 } ^ { 2 \pi } f ( \theta ) \cos ( n \theta ) d \theta / \left( c ^ { n } \pi \right)$
B) $B _ { n } = \int _ { 0 } ^ { 2 \pi } f ( \theta ) \sin ( n \theta ) d \theta / \left( c ^ { n } \pi \right)$
C) $B _ { n } = \int _ { 0 } ^ { 2 \pi } f ( \theta ) \cos ( n \theta ) d \theta / \left( c ^ { n } \pi \right)$
D) $A _ { n } = 0$
E) $B _ { n } = 0$

Question 10

In changing from Cartesian to spherical coordinates, $\frac { \partial r } { \partial x }$ becomes

A) $\sin \phi \sin \theta$
B) $\sin \phi \cos \theta$
C) $\cos \phi \sin \theta$
D) $\cos \phi \cos \theta$
E) $\sin \phi$

Question 11

In the three previous problems, the product solutions are

A) $u _ { n } = r ^ { n } \left( c _ { n } e ^ { n \theta } + d _ { n } e ^ { - n \theta } \right) , n = 0,1,2 , \ldots$
B) $u _ { n } = r ^ { n } \left( c _ { n } e ^ { n \theta } + d _ { n } e ^ { - n \theta } \right) , n = 1,2,3 , \ldots$
C) $u _ { n } = r ^ { n } \left( c _ { n } \cos ( n \theta ) + d _ { n } \sin ( n \theta ) \right) , n = 0,1,2 , \ldots$
D) $u _ { n } = r ^ { n } \cos ( n \theta ) , n = 0,1,2 , \ldots$
E) $u _ { n } = r ^ { n } \sin ( n \theta ) , n = 1,2,3 , \ldots$

Question 12

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

A) $R = r ^ { n }$
B) $R = r ^ { - n - 1 }$
C) $R = c _ { 1 } r ^ { n } + c _ { 2 } r ^ { - n - 1 }$
D) $R = c _ { 1 } r ^ { n } + c _ { 2 } r ^ { n + 1 }$
E) $R = c _ { 1 } r ^ { n } + c _ { 2 } r ^ { n - 1 }$

Question 13

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

A) $\lambda = n ^ { 2 } , R = r ^ { n } , n = 0,1,2 , \ldots$
B) $\lambda = n , R = r ^ { n } , n = 0,1,2 , \ldots$
C) $\lambda = n , \Theta = c _ { n } \cos ( n \theta ) + d _ { n } \sin ( n \theta ) , n = 0,1,2 , .$
D) $\lambda = n ^ { 2 } , \Theta = d _ { n } \sin ( n \theta ) , n = 1,2,3 , .$
E) $\lambda = n ^ { 2 } , \Theta = c _ { n } \cos ( n \theta ) , n = 0,1,2 , \ldots$

Question 14

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

A) $r ^ { 2 } R ^ { \prime \prime } + 2 r R ^ { \prime } - \lambda R = 0$
B) $r ^ { 2 } R ^ { \prime \prime } - 2 r R ^ { \prime } - \lambda R = 0$
C) $r R ^ { \prime \prime } + 2 r R ^ { \prime } - \lambda r R = 0$
D) $r R ^ { \prime \prime } + 2 R ^ { \prime } - \lambda r R = 0$
E) $r ^ { 2 } R ^ { \prime \prime } - 2 r R ^ { \prime } - \lambda r R = 0$

Question 15

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

A) $\lambda = n ( n + 1 ) , \Theta = P _ { n } ( \sin \theta ) , n = 1,2,3 , \ldots$
B) $\lambda = n ( n - 1 ) , \Theta = \sin ( n \theta ) , n = 1,2,3 , \ldots$
C) $\lambda = n ( n - 1 ) , \Theta = \cos ( n \theta ) , n = 1,2,3 , \ldots$
D) $\lambda = n ( n + 1 ) , \Theta = P _ { n } ( \cos \theta ) , n = 1,2,3 , \ldots$
E) $\lambda = n ( n - 1 ) , \Theta = P _ { n } ( \sin \theta ) , n = 1,2,3 , \ldots$

Question 16

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

A) $\sin \theta$
B) $\cos \theta$
C) $\sin \theta / r$
D) $\cos \theta / r$
E) $- \sin \theta / r$

Question 17

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

A) $\cos \theta \frac { \partial u } { \partial r } - \sin \theta \frac { \partial u } { \partial \theta } / r$
B) $\cos \theta \frac { \partial u } { \partial r } + \cos \theta \frac { \partial u } { \partial \theta } / r$
C) $\sin \theta \frac { \partial u } { \partial r } + \cos \theta \frac { \partial u } { \partial \theta }$
D) $\sin \theta \frac { \partial u } { \partial r } + \cos \theta \frac { \partial u } { \partial \theta } / r$
E) $\sin \theta \frac { \partial u } { \partial r } - \cos \theta \frac { \partial u } { \partial \theta } / r$

Question 18

In the previous problem, after separating variables using $u ( r , t ) = R ( r ) T ( t )$ . the resulting problems are

A) $r R ^ { \prime \prime } + R ^ { \prime } + r \lambda R = 0 , R ( 0 )$ is bounded $R ( 2 ) = 0 , T ^ { \prime \prime } - \lambda T = 0 , T ^ { \prime } ( 0 ) = 0$
B) $r R ^ { \prime \prime } + R ^ { \prime } + r \lambda R = 0 , R ( 0 )$ is bounded $R ( 2 ) = 0 , T ^ { \prime \prime } + \lambda T = 0 , T ^ { \prime } ( 0 ) = 0$
C) $r R ^ { \prime \prime } + R ^ { \prime } - r \lambda R = 0 , R ( 0 )$ is bounded $R ( 2 ) = 0 , T ^ { \prime \prime } + \lambda T = 0 , T ^ { \prime } ( 0 ) = 0$
D) $r R ^ { \prime \prime } + R ^ { \prime } + \lambda R = 0 , R ( 0 )$ is bounded $R ( 2 ) = 0 , T ^ { \prime \prime } + \lambda T = 0 , T ^ { \prime } ( 0 ) = 0$
E) $r R ^ { \prime \prime } + R ^ { \prime } + \lambda R = 0 , R ( 0 )$ is bounded $R ( 2 ) = 0 , T ^ { \prime \prime } - \lambda T = 0 , T ^ { \prime } ( 0 ) = 0$

Question 19

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

A) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta ) , u ( r , 0 ) = 0 , u ( r , \pi ) = 0$
B) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta ) , u ( r , 0 ) = 0 , u ( r , \pi ) = 0$
C) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta ) , u ( r , 0 ) = 0 , u ( r , \pi ) = 0$
D) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { 1 } { r } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta ) , u ( r , 0 ) = 0 , u ( r , \pi ) = 0$
E) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta ) , u ( r , 0 ) = 0 , u ( r , \pi ) = 0$

Question 20

In the previous three problems, the product solutions are

A) $u = J _ { 0 } \left( z _ { n } r / 2 \right) \left( a _ { n } \cos \left( z _ { n } t / 2 \right) + b _ { n } \sin \left( z _ { n } t / 2 \right) \right)$
B) $u = J _ { 0 } \left( z _ { n } r / 2 \right) a _ { n } \cos \left( z _ { n } t / 2 \right)$
C) $u = J _ { 0 } \left( z _ { n } r / 2 \right) b _ { n } \sin \left( z _ { n } t / 2 \right)$
D) $u = J _ { 0 } ( n \pi r ) b _ { n } \sin ( n \pi i )$
E) $u = J _ { 0 } ( n \pi r ) a _ { n } \cos ( n \pi i )$

Question 21

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

A) $\lambda = n , \Theta = c _ { n } \cos ( n \theta ) + d _ { n } \sin ( n \theta ) , n = 0,1,2 , \ldots$
B) $\lambda = n ^ { 2 } , \ominus = c _ { n } \cos ( n \theta ) + d _ { n } \sin ( n \theta ) , n = 0,1,2 , \ldots$
C) $\lambda = n ^ { 2 } , \ominus = c _ { n } \cos ( n \theta ) + d _ { n } \sin ( n \theta ) , n = 1,2,3 , \ldots$
D) $\lambda = n ^ { 2 } , R = r ^ { n } , n = 0,1,2 , \ldots$
E) $\lambda = n , R = r ^ { n } , n = 0,1,2 , \ldots$

Question 22

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

A) $\sin \theta$
B) $\cos \theta$
C) $\sin \theta / r$
D) $\cos \theta / r$
E) $- \sin \theta / r$

Question 23

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

A) $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } - \lambda R = 0 , \Theta ^ { \prime \prime } - \lambda \Theta = 0 , \Theta ( \theta ) = \Theta ( \theta + 2 \pi )$
B) $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } + \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( \theta ) = \Theta ( \theta + 2 \pi )$
C) $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } - \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( \theta ) = \Theta ( \theta + 2 \pi )$
D) $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } - \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( \theta ) = \Theta ( \theta + \pi )$
E) $r ^ { 2 } R ^ { \prime \prime } + r R ^ { \prime } + \lambda R = 0 , \Theta ^ { \prime \prime } + \lambda \Theta = 0 , \Theta ( \theta ) = \Theta ( \theta + \pi )$

Question 24

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

A) $\sin \theta \frac { \partial u } { \partial r } + \cos \theta \frac { \partial u } { \partial \theta } / r$
B) $\sin \theta \frac { \partial u } { \partial r } - \cos \theta \frac { \partial u } { \partial \theta } / r$
C) $\cos \theta \frac { \partial u } { \partial r } - \sin \theta \frac { \partial u } { \partial \theta } / r$
D) $\cos \theta \frac { \partial u } { \partial r } + \cos \theta \frac { \partial u } { \partial \theta } / r$
E) $\sin \theta \frac { \partial u } { \partial r } + \cos \theta \frac { \partial u } { \partial \theta }$

Question 25

The relationships between Cartesian and spherical coordinates are Select all that apply.

A) $r ^ { 2 } = x ^ { 2 } + y ^ { 2 }$
B) $\tan \phi = y / x$
C) $x = r \cos \phi \sin \theta$
D) $y = r \cos \phi \cos \theta$
E) $z = r \cos \theta$

Question 26

In the previous three problems, the product solutions are

A) $u = J _ { 0 } ( n \pi r / 3 ) \cos ( n \pi z / 3 )$
B) $u = J _ { 0 } \left( n ^ { 2 } \pi ^ { 2 } r / 9 \right) \cos ( n \pi z / 3 )$
C) $u = J _ { 0 } \left( n ^ { 2 } \pi ^ { 2 } r / 9 \right) \sin ( n \pi z / 3 )$
D) $u = J _ { 0 } \left( z _ { n } r \right) \cosh \left( z _ { n } ( z - 3 ) \right)$
E) $u = J _ { 0 } \left( z _ { n } r / 2 \right) \sinh \left( z _ { n } ( z - 3 ) / 2 \right)$

Question 27

Consider the steady-state temperature distribution in a circular cylinder of radius 2 and height 3, with zero temperature at $r = 2$ and at $z = 3$ and a temperature of 10 at $z = 0$ . The mathematical model for this problem is

A) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { \partial ^ { 2 } u } { \partial z ^ { 2 } } = 0 , u ( 2 , z ) = 0 , u ( r , 3 ) = 0 , u ( r , 0 ) = 10$
B) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { \partial ^ { 2 } u } { \partial z ^ { 2 } } = 0 , u ( 2 , z ) = 0 , u ( r , 3 ) = 0 , u ( r , 0 ) = 10$
C) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { \partial ^ { 2 } u } { \partial z ^ { 2 } } = 0 , u ( 2 , z ) = 0 , u ( r , 3 ) = 0 , u ( r , 0 ) = 10$
D) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { \partial ^ { 2 } u } { \partial z ^ { 2 } } = 0 , u ( 2 , z ) = 0 , u ( r , 3 ) = 0 , u ( r , 0 ) = 10$
E) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { \partial ^ { 2 } u } { \partial z ^ { 2 } } = 0 , u ( 2 , z ) = 0 , u ( r , 3 ) = 0 , u ( r , 0 ) = 10$

Question 28

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

A) $r ^ { 2 } R ^ { \prime \prime } + 2 r R ^ { \prime } + \lambda R = 0 , R ( 0 )$ is bounded; $\sin ( \theta ) \Theta ^ { \prime \prime } + \cos ( \theta ) \Theta ^ { \prime } + \lambda \sin ( \theta ) \Theta = 0,\Theta$ is bounded on $[ 0 , \pi ]$ .
B) $r ^ { 2 } R ^ { \prime \prime } + 2 r R ^ { \prime } - \lambda R = 0 , R ( 0 )$ is bounded; $\sin ( \theta ) \Theta ^ { \prime \prime } + \cos ( \theta ) \Theta ^ { \prime } + \lambda \sin ( \theta ) \Theta = 0,\Theta$ is bounded on $[ 0 , \pi ]$ .
C) $r ^ { 2 } R ^ { \prime \prime } + 2 r R ^ { \prime } - \lambda R = 0 , R ( 0 )$ is bounded; $\sin ( \theta ) \Theta ^ { \prime \prime } + \cos ( \theta ) \Theta ^ { \prime } - \lambda \sin ( \theta ) \Theta = 0,\Theta$ is bounded on $[ 0 , \pi ]$ .
D) $r ^ { 2 } R ^ { \prime \prime } - 2 r R ^ { \prime } - \lambda R = 0 , R ( 0 )$ is bounded; $\sin ( \theta ) \Theta ^ { \prime \prime } + \cos ( \theta ) \Theta ^ { \prime } + \lambda \sin ( \theta ) \Theta = 0,\Theta$ is bounded on $[ 0 , \pi ]$ .
E) $r ^ { 2 } R ^ { \prime \prime } + 2 r R ^ { \prime } - \lambda R = 0 , R ( 0 )$ is bounded; $\sin ( \theta ) \Theta ^ { \prime \prime } - \cos ( \theta ) \Theta ^ { \prime } + \lambda \sin ( \theta ) \Theta = 0,\Theta$ is bounded on $[ 0 , \pi ]$ .

Question 29

The Laplacian in polar coordinates is

A) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { 1 } { r } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } }$
B) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } }$
C) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } }$
D) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } }$
E) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } }$

Question 30

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. The mathematical model of this situation is

A) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$
B) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$
C) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } - \frac { 1 } { r } \frac { \partial u } { \partial r } - \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$
D) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r ^ { 2 } } \frac { \partial u } { \partial r } + \frac { 1 } { r } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$
E) $\frac { \partial ^ { 2 } u } { \partial r ^ { 2 } } + \frac { 1 } { r } \frac { \partial u } { \partial r } + \frac { 1 } { r ^ { 2 } } \frac { \partial ^ { 2 } u } { \partial \theta ^ { 2 } } = 0 , u ( c , \theta ) = f ( \theta )$

Question 31

In the previous four problems, the infinite series solution is (for certain values of the constants)

A) $u = \sum _ { n - 1 } ^ { \infty } c _ { n } J _ { 0 } ( n \pi r / 3 ) \cos ( n \pi z / 3 )$
B) $u = \sum _ { n - 1 } ^ { \infty } c _ { n } J _ { 0 } \left( n ^ { 2 } \pi ^ { 2 } r / 9 \right) \cos ( n \pi z / 3 )$
C) $u = \sum _ { n - 1 } ^ { \infty } c _ { n } J _ { 0 } \left( n ^ { 2 } \pi ^ { 2 } r / 9 \right) \sin ( n \pi z / 3 )$
D) $u = \sum _ { n - 1 } ^ { \infty } c _ { n } J _ { 0 } \left( z _ { n } r / 2 \right) \sinh \left( z _ { n } ( z - 3 ) / 2 \right)$
E) $u = \sum _ { n = 1 } ^ { \infty } c _ { n } J _ { 0 } \left( z _ { n } r \right) \cosh \left( z _ { n } ( z - 3 ) \right)$

Question 32

In the three previous problems, the product solutions are

A) $u _ { n } = \left( a _ { n } r ^ { n } + b _ { n } r ^ { - n } \right) \left( c _ { n } \cos ( n \theta ) + d _ { n } \sin ( n \theta ) \right) , n = 0,1,2 , \ldots$
B) $u _ { n } = r ^ { n } \left( c _ { n } \cos ( n \theta ) + d _ { n } \sin ( n \theta ) \right) , n = 1,2,3 , \ldots$
C) $u _ { n } = r ^ { n } \left( c _ { n } \cos ( n \theta ) + d _ { n } \sin ( n \theta ) \right) , n = 0,1,2 , \ldots$
D) $u _ { n } = r ^ { n } \left( c _ { n } e ^ { n \theta } + d _ { n } e ^ { - n \theta } \right) , n = 0,1,2 , \ldots$
E) $u _ { n } = r ^ { n } \left( c _ { n } e ^ { n \theta } + d _ { n } e ^ { - n \theta } \right) , n = 1,2,3 , \ldots$

Question 33

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

A) $\lambda = n \pi , R = \sin ( n \pi \ln r )$
B) $\lambda = ( n \pi ) ^ { 2 } , R = \sin ( n \pi \ln r )$
C) $\lambda = ( n \pi ) ^ { 2 } , R = \sin ( n \pi \ln r ) + \cos ( n \pi \ln r )$
D) $\lambda = n \pi , R = \sin ( n \pi \ln r ) - \cos ( n \pi \ln r )$
E) none of the above

Question 34

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

A) $\lambda = n ( n + 1 ) , R = r ^ { n } , n = 1,2,3 , \ldots$
B) $\lambda = n ( n - 1 ) , R = r ^ { n } , n = 1,2,3 , \ldots$
C) $\lambda = n ( n - 1 ) , \Theta = P _ { n } ( \cos \theta ) , n = 1,2,3 , \ldots$
D) $\lambda = n ( n + 1 ) , \Theta = P _ { n } ( \cos \theta ) , n = 1,2,3 , \ldots$
E) $\lambda = n ( n + 1 ) , \Theta = P _ { n } ( \sin \theta ) , n = 1,2,3 , \ldots$

Question 35

In the previous two problems, the product solutions are

A) $u _ { n } = r ^ { n } P _ { n } ( \cos \theta )$
B) $u _ { n } = r ^ { - n } P _ { n } ( \cos \theta )$
C) $u _ { n } = r ^ { n } P _ { n } ( \sin \theta )$
D) $u _ { n } = r ^ { - n } P _ { n } ( \sin \theta )$
E) none of the above

Question 36

In the previous problem, after separating variables, the resulting problems are

A) $r R ^ { \prime \prime } - R ^ { \prime } + \lambda r R = 0 , R ( 0 )$ is bounded, $R ( 2 ) = 0 , Z ^ { \prime \prime } + \lambda Z = 0 , Z ( 3 ) = 0$
B) $r R ^ { \prime \prime } + R ^ { \prime } - \lambda r R = 0 , R ( 0 )$ is bounded, $R ( 2 ) = 0 , Z ^ { \prime \prime } - \lambda Z = 0 , Z ( 3 ) = 0$
C) $r R ^ { \prime \prime } - R ^ { \prime } - \lambda r R = 0 , R ( 0 )$ is bounded, $R ( 2 ) = 0 , Z ^ { \prime \prime } - \lambda Z = 0 , Z ( 3 ) = 0$
D) $r R ^ { \prime \prime } + R ^ { \prime } + \lambda r R = 0 , R ( 0 )$ is bounded, $R ( 2 ) = 0 , Z ^ { \prime \prime } - \lambda Z = 0 , Z ( 3 ) = 0$
E) $r R ^ { \prime \prime } + R ^ { \prime } + \lambda r R = 0 , R ( 0 )$ is bounded, $R ( 2 ) = 0 , Z ^ { \prime \prime } + \lambda Z = 0 , Z ( 3 ) = 0$

Question 37

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

A) $- \sin \phi / ( r \sin \theta )$
B) $\sin \phi / ( r \sin \theta )$
C) $- \cos \phi / ( r \sin \theta )$
D) $\cos \phi / ( r \cos \theta )$
E) $\sin \phi / ( r \cos \theta )$

Question 38

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.

A) $A _ { 0 } = \int _ { 0 } ^ { 2 \pi } f ( \theta ) d \theta l ( 2 \pi )$
B) $A _ { n } = \int _ { 0 } ^ { 2 \pi } f ( \theta ) \sin ( n \theta ) d \theta / \left( c ^ { n } \pi \right)$
C) $A _ { n } = \int _ { 0 } ^ { 2 \pi } f ( \theta ) \cos ( n \theta ) d \theta / \left( c ^ { n } \pi \right)$
D) $B _ { n } = \int _ { 0 } ^ { 2 \pi } f ( \theta ) \sin ( n \theta ) d \theta / \left( c ^ { n } \pi \right)$
E) $B _ { n } = \int _ { 0 } ^ { 2 \pi } f ( \theta ) \cos ( n \theta ) d \theta / \left( c ^ { n } \pi \right)$

Question 39

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

A) $\lambda = z _ { n } ^ { 2 } / 4$ , where $J _ { 0 } \left( z _ { n } \right) = 0 ; R = J _ { 0 } \left( z _ { n } r / 2 \right)$
B) $\lambda = z _ { n } ^ { 2 }$ , where $J _ { 0 } \left( z _ { n } \right) = 0 ; R = J _ { 0 } \left( z _ { n } r \right)$
C) $\lambda = n \pi / 9 , Z = \cos ( n \pi z / 3 )$
D) $\lambda = n ^ { 2 } \pi ^ { 2 } / 9 , Z = \cos ( n \pi z / 3 )$
E) $\lambda = n ^ { 2 } \pi ^ { 2 } / 9 , Z = \sin ( n \pi z / 3 )$

Question 40

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

A) $\cos \phi$
B) $\cos \theta$
C) $\tan \theta$
D) $\sin \phi$
E) $\sin \theta$