A function (x, y, z) is said to be harmonic in a region D in space if

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A function ƒ(x, y, z) is said to be harmonic in a region D in space if it satisfies the Laplace equationimage


throughout D.


a. Suppose that ƒ is harmonic throughout a bounded region D enclosed by a smooth surface S and that n is the chosen unit normal vector on S. Show that the integral over S of ∇ƒ · n, the derivative of ƒ in the direction of n, is zero.


b. Show that if ƒ is harmonic on D, thenimage

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Thomas Calculus Early Transcendentals

ISBN: 9780321884077

13th Edition

Authors: Joel R Hass, Christopher E Heil, Maurice D Weir

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