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18. A function f(x, y, z) is called harmonic in a region D in space if it satisfies the Laplace equation: Vf = v. Vf = as + 32s əx2 əy2 = 0 throughout D. a) Suppose that f is a harmonic function throughout a bounded region enclosed by a smooth surface S and that n is the chosen unit normal vector on S. Show that the integral over S of Vf n, the derivative of f in the direction of n is zero. b) Show that if f is harmonic on D, then ff, fVf n do = SSS, IVfl? dV.

18. A function f(x, y, z) is called harmonic in a region D in space if it satisfies the Laplace equation: Vf = v. Vf = as + 32s əx2 əy2 = 0 throughout D. a) Suppose that f is a harmonic function throughout a bounded region enclosed by a smooth surface S and that n is the chosen unit normal vector on S. Show that the integral over S of Vf n, the derivative of f in the direction of n is zero. b) Show that if f is harmonic on D, then ff, fVf n do = SSS, IVfl? dV.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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18. A function f(x, y, z) is called harmonic in a region D in space if it satisfies the Laplace equation: a² f a² f v²f = V.Vf: = a²f + + = 0 throughout D. əx² дуг ?z2 a) Suppose that f is a harmonic function throughout a bounded region enclosed by a smooth surface S and that n is the chosen unit normal vector on S. Show that the integral over S of Vf. n, the derivative of f in the direction of n is zero. b) Show that if f is harmonic on D, then ff ƒVƒ ·ndo = SSS, Vƒ|² dv.
Transcribed Image Text:18. A function f(x, y, z) is called harmonic in a region D in space if it satisfies the Laplace equation: a² f a² f v²f = V.Vf: = a²f + + = 0 throughout D. əx² дуг ?z2 a) Suppose that f is a harmonic function throughout a bounded region enclosed by a smooth surface S and that n is the chosen unit normal vector on S. Show that the integral over S of Vf. n, the derivative of f in the direction of n is zero. b) Show that if f is harmonic on D, then ff ƒVƒ ·ndo = SSS, Vƒ|² dv.
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