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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
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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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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