Deck 13: Boundary-Value Problems in Other Coordinate Systems

Question 1

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

A)
B)
C) , where
D) , where
E) , where

Answer 1

C

Question 2

Using the separation of the previous problem, the equation for becomes

A)
B)
C)
D)
E)

Answer 2

C

Question 3

The two dimensional wave equation in polar coordinates is

A)
B)
C)
D)
E)

Answer 3

C

Question 4

Consider the vibrations of a circular membrane of radius 2 clamped along the circumference, with an initial displacement of and an initial velocity of zero. The mathematical model for this situation is

A)
B)
C)
D)
E)

Question 5

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

A)
B)
C)
D)
E)

Question 6

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

A)
B)
C)
D)
E)

Question 7

In the previous problem, separate variables using . The resulting problems are

A)
B)
C)
D)
E)

Question 8

The solution of is

A)
B)
C)
D)
E) none of the above

Question 9

In the previous four problems, the infinite series solution of the original problem is where Select all that apply.

A)
B)
C)
D)
E)

Question 10

In changing from Cartesian to spherical coordinates, becomes

A)
B)
C)
D)
E)

Question 11

In the three previous problems, the product solutions are

A)
B)
C)
D)
E)

Question 12

In the previous three problems, the solution for is

A)
B)
C)
D)
E)

Question 13

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

A)
B)
C)
D)
E)

Question 14

In separating variables, using, , in the equation , the resulting equation for and is

A)
B)
C)
D)
E)

Question 15

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

A)
B)
C)
D)
E)

Question 16

In changing from Cartesian to polar coordinates, is

A)
B)
C)
D)
E)

Question 17

In changing from Cartesian to polar coordinates, is

A)
B)
C)
D)
E)

Question 18

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

A) is bounded
B) is bounded
C) is bounded
D) is bounded
E) is bounded

Question 19

Consider the steady-state temperature distribution in a circular disc of radius centere at the origin, with temperature given as a function, on the boundary and zero on the boundaries and The mathematical model of this situation is

A)
B)
C)
D)
E)

Question 20

In the previous three problems, the product solutions are

A)
B)
C)
D)
E)

Question 21

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

A)
B)
C)
D)
E)

Question 22

In changing from Cartesian to polar coordinates, is

A)
B)
C)
D)
E)

Question 23

In the previous problem, separate variables using . The resulting problems are

A)
B)
C)
D)
E)

Question 24

In changing from Cartesian to polar coordinates, is

A)
B)
C)
D)
E)

Question 25

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

A)
B)
C)
D)
E)

Question 26

In the previous three problems, the product solutions are

A)
B)
C)
D)
E)

Question 27

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

A)
B)
C)
D)
E)

Question 28

In the problem , separate variables, using . The resulting problems for and are

A) is bounded; is bounded on .
B) is bounded; is bounded on .
C) is bounded; is bounded on .
D) is bounded; is bounded on .
E) is bounded; is bounded on .

Question 29

The Laplacian in polar coordinates is

A)
B)
C)
D)
E)

Question 30

Consider the steady-state temperature distribution in a circular disc of radius centere at the origin, with temperature given as a function, on the boundary. The mathematical model of this situation is

A)
B)
C)
D)
E)

Question 31

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

A)
B)
C)
D)
E)

Question 32

In the three previous problems, the product solutions are

A)
B)
C)
D)
E)

Question 33

The solution of is

A)
B)
C)
D)
E) none of the above

Question 34

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

A)
B)
C)
D)
E)

Question 35

In the previous two problems, the product solutions are

A)
B)
C)
D)
E) none of the above

Question 36

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

A) is bounded,
B) is bounded,
C) is bounded,
D) is bounded,
E) is bounded,

Question 37

When changing from Cartesian to spherical coordinates,

A)
B)
C)
D)
E)

Question 38

In the previous four problems, the infinite series solution of the original problem is where Select all that apply.

A)
B)
C)
D)
E)

Question 39

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

A) , where
B) , where
C)
D)
E)

Question 40

When changing from Cartesian to spherical coordinates,

A)
B)
C)
D)
E)