Explanation 1) Concept: i) If u x , y , z is a scalar field and ∂ u ∂ x , ∂ u ∂ y , ∂ u ∂ z exists, then the divergence of u is ∇ u = ∂ u ∂ x i + ∂ u ∂ y j + ∂ u ∂ z k ii) The rate of heat flow across the surface S in the body is given by the surface integral ∬ S □ F · d S = F · n ∬ S □ d S 2) Given: The temperature at a point in the ball is inversely proportional to the distance from the center of the ball 3) Calculations: The equation of sphere with radius a is, x 2 + y 2 + z 2 = a 2 And let the temperature at a point in a ball be u ( x , y , z ) Therefore, u x , y , z ∝ 1 x 2 + y 2 + z 2 Let c be the proportionality constant. So, u x , y , z = c x 2 + y 2 + z 2 By using the concept i), ∇ u = ∂ ∂ x c x 2 + y 2 + z 2 i + ∂ ∂ y c x 2 + y 2 + z 2 j + ∂ ∂ z c x 2 + y 2 + z 2 k ∇ u = - c x x 2 + y 2 + z 2 3 2 i + - c y x 2 + y 2 + z 2 3 2 j + - c z x 2 + y 2 + z 2 3 2 k Therefore, F = - K · ∇ u = - K - c x

8th Edition

ISBN: 9781305770430

Author: Stewart

Publisher: CENGAGE C

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Chapter 16.7, Problem 48E

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The rate of the heat flow across a sphere S of radius a with center at the center of ball

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Chapter 16.7, Problem 47E

Chapter 16.7, Problem 49E

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Chapter 16 Solutions

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Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - 110 Sketch the vector field F by drawing a diagram...Ch. 16.1 - Prob. 7E Ch. 16.1 - Prob. 8E Ch. 16.1 - Prob. 9E Ch. 16.1 - Prob. 10E

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The rate of the heat flow across a sphere S of radius a with center at the center of ball

Solution Summary: The author explains how the rate of heat flow across a sphere S of radius A is inversely proportional to the distance from the center of the ball.

To find: The rate of the heat flow across a sphere S of radius a with center at the center of ball

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Chapter 16 Solutions

To find:

The rate of the heat flow across a sphere S of radius a with center at the center of ball