?For the following exercises, the heat flow vector field for conducting objects i

Chapter 6, Problem 323

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For the following exercises, the heat flow vector field for conducting objects i \(\mathbf{F}=-k \nabla T\), whereT(x, y, z) is the temperature in the object and k > 0 is a constant that depends on the material. Find the outward flux of F across the following surfaces S for the given temperature distributions and assume k = 1.

\(T(x, y, z)=-\ln \left(x^{2}+y^{2}+z^{2}\right)\) ; S is sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) .

Text Transcription:

F = -k nabla T

T(x, y, z) = -ln (x^2 + y^2 + z^2)

x^2 + y^2 + z^2 = a^2

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