   Chapter 14.6, Problem 31E

Chapter
Section
Textbook Problem

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°.(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 toward the origin.

(a)

To determine

To find: The rate of change of T at (1,2,2) in the direction towards the point (2,1,3) .

Explanation

Given:

The temperature at the point (1,2,2) is 120° .

Theorem used:

“Suppose f is a differentiable function of two variables. The rate of change is defined as Duf(x,y)=f(x,y)u where u is unit vector.”

Calculation:

Let the temperature be T=kx2+y2+z2 (1)

To obtain the T value at (1,2,2) is 120° .

T(1,2,2)=k(1)2+(2)2+(2)2120=k1+4+4120=k3k=360

The unit vector from the point (1,2,2) towards (2,1,3) is, 21,12,32=1,1,1 .

Then the unit vector u is computed as follows,

u=112+(1)2+121,1,1=11+1+11,1,1=131,1,1

Thus, the unit vector u is 131,1,1 .

The gradient of T(x,y,z) is as follows.

T(x,y,z)=Tx,Ty,Tz=x(360x2+y2+z2),y(kx2+y2+z2),z(kx2+y2+z2)=[(3602)(x2+y2+z2)32(2x)],[(3602)(x2+y2+z2)32(2y)],[(3602)(x2+y2+z2)32(2z)]=[360x(x2+y2+z2)32],[360y(x2+y2+z2)32],[360z(x2+y2+

(b)

To determine

To show: The vector from (x,y,z) to the origin (0,0,0) passes through which point.

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