pg. 811, #25

The temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (1, 2, 2) is 120 degrees.

a. Find the rate of change of T at (1, 2, 2) in the direction toward the point (2, 1, 3).
b. Show that at any point in the ball the direction of greatest increase in temperature is given by a vector that points at the origin

ans:

Solution:

a. The first sentence of the problem states that if (x,y,z) is a point within the sphere then the temperature at this point is given by where k is some constant.
We are also given that , so k = 360. In sum, .
Computing the gradient,


Next, the direction from (1, 2, 2) to (2, 1, 3) is < 1, -1, 1> so
Consequently, .

b. The direction of the greatest increase in temperature is given by any vector parallel to and having the same direction as . By our above calculation of , <-x, -y, -z>, the vector from (x,y,z) to the origin, is the direction of greatest increase.