Exercises for Section 11.5.  Steady State Temperatures

Exercise 1 (a).  Show that    satisfies Laplace's equation    in three-dimensional Cartesian space,

and that      does not satisfy Laplace's equation      in two-dimensional Cartesian space.

Solution 1 (a).

Exercise 1 (b).  Show that    does not satisfy Laplace's equation    in three-dimensional Cartesian space,

and that      satisfies Laplace's equation      in two-dimensional Cartesian space.

Solution 1 (b).

Exercise 2.  Find the temperature function in the infinite strip bounded by the lines   ,

that satisfies the following boundary values (shown in Figure 11.22).

Solution 2.

Exercise 3.  Find the temperature function in the first quadrant  ,

that satisfies the following boundary values (shown in Figure 11.23).

Solution 3.

Exercise 4.  Find the temperature function inside the unit disk  ,

that satisfies the following boundary values (shown in Figure 11.24).

Hint.  Use  .

Solution 4.

Exercise 5.  Find the temperature function in the semi-infinite strip  ,

that satisfies the following boundary values (shown in Figure 11.25).

Solution 5.

Exercise 6.  Find the temperature function in the domain  ,

that satisfies the following boundary values (shown in Figure 11.26).

Hint.  Use   .

Solution 6.

Exercise 7.  Find the temperature function in the domain  ,

that satisfies the following boundary conditions (shown in Figure 11.27).

Solution 7.

Exercise 8.  Find the temperature function in the domain  ,

that satisfies the following boundary conditions (shown in Figure 11.28).

Hint.  Use  .

Solution 8.

Exercise 9.  Find the temperature function in the first quadrant  ,

that satisfies the following boundary conditions (shown in Figure 11.29).

Solution 9.

Exercise 10.  Find the temperature function in the infinite strip  ,

that satisfies the following boundary conditions (shown in Figure 11.30).

Hint.  Use  .

Solution 10.

Exercise 11.  Find the temperature function in the upper half-plane  ,

that satisfies the following boundary conditions (shown in Figure 11.31).

Solution 11.

Exercise 12.  Find the temperature function in the first quadrant  ,

that satisfies the following boundary conditions (shown in Figure 11.32).

Solution 12.

Exercise 13.  For the temperature    in the upper half-disk  ,

show that the isothermals   ,   for  ,   are portions of circles that pass through the points    and  ,  as illustrated in Figure 11.33.

Solution 13.

Exercise 14.  For the temperature function    in the upper half-plane  ,

show that the isothermals    are portions of hyperbolas that have foci at the points    and  ,  as illustrated in Figure 11.34.

Solution 14.

Exercise 15.  Find the temperature function in the portion of the upper half-plane

that lies inside the ellipse    and satisfies the following boundary conditions (shown in Figure 11.35).

Hint.  Use   .

Solution 15.

(c) 2008 John H. Mathews, Russell W. Howell