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Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter2: Working With Real Numbers

Section2.3: Rules For Addition

Problem 41WE

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Heat flux Suppose a solid object in ℝ³ has a temperature distribution given by T(x, y, z). The heat flow vector field in the object is F = -k∇T, where the conductivity k > 0 is a property of the material. Note that the heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is
∇ ⋅ F = -k∇⋅ ∇T = -k∇²T (the Laplacian of T). Compute the heat flow vector field and its divergence for the following temperature distributions.

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