   Chapter 16.7, Problem 47E

Chapter
Section
Textbook Problem

The temperature at the point (x, y, z) in a substance with conductivity K = 6.5 is u(x, y, z) = 2y2 + 2z2. Find the rate of heat flow inward across the cylindrical surface y2 + z2 - 6, 0 ≤ x ≤ 4.

To determine

To find: The rate of heat flow inward across the cylindrical surface y2+z2=6 , 0x4 .

Explanation

Given:

K=6.5 , u(x,y,z)=2y2+2z2 and cylindrical surface y2+z2=6 , 0x4 .

Formula used:

SFdS=SKudS (1)

rr=xri+yrj+zrk (2)

=ix+jy+kz

Let x=x and y=6cosθ and z=6sinθ with limits 0x4 and 0θ2π .

r(x,θ)=xi+6cosθj+6sinθk

Find u .

u=(ix+jy+kz)(2y2+2z2)=ix(2y2+2z2)+jy(2y2+2z2)+kz(2y2+2z2)=i(0)+j(4y)+k(4z)=4yj+4zk

Substitute 6cosθ for y and 6sinθ for z ,

u=4(6cosθ)j+4(6sinθ)k=46cosθj+46sinθk

Find rx .

Modify equation (2).

rx=xxi+yxj+zxk

Substitute x for x , 6cosθ for y and 6sinθ for z ,

rx=(x)xi+(6cosθ)xj+(6sinθ)xk=(1)i+(0)j+(0)k=i

Find rθ .

Modify equation (2).

rθ=xθi+yθj+zθk

Substitute x for x , 6cosθ for y and 6sinθ for z ,

rθ=(x)θi+(6cosθ)θj+(6sinθ)θk=(0)i+6(sinθ)j+6(cosθ)k=6sinθj+6cosθk

Find rx×rθ

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