Explanation Given information: Let D be the region bounded by the simple closed curve C .A function φ ( x , y ) on D (whose second-order partial derivative exists and are continuous) is called harmonic if Δ φ = 0 where Δ φ is the Laplace operator. Let P = ( a , b ) and and let C r be the circle of radius r centered at P . Formula Used: The average value of a continuous function φ on C r is defined as the integral I φ ( r ) = 1 2 π ∫ 0 2 π φ ( a + r cos θ , b + r sin θ ) d θ The continuity of the function φ lim I φ ( r ) = φ ( P ) r → 0 Proof: By using the integral of the average value of a continuous function φ ; I φ ( r ) = 1 2 π ∫ 0 2 π φ ( a + r cos θ , b + r sin θ ) d θ Now consider the limits r → 0 on both sides of the equation since

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 18.1, Problem 46E

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To prove:

The mean value property of harmonic functions: If φ is harmonic then, Iφ(r)=φ(P) for all r.

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Chapter 18.1, Problem 45E

Chapter 18.1, Problem 47E

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Chapter 18 Solutions

Calculus - Standalone book

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To prove: The mean value property of harmonic functions: If φ is harmonic then, I φ ( r ) = φ ( P ) for all r .

Solution Summary: The author demonstrates the mean value property of harmonic functions by using the integral of the average value of a continuous function phi .

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To prove:

The mean value property of harmonic functions: If φ is harmonic then, Iφ(r)=φ(P) for all r.

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Chapter 18 Solutions

To prove: The mean value property of harmonic functions: If φ is harmonic then, I φ ( r ) = φ ( P ) for all r .

Solution Summary: The author demonstrates the mean value property of harmonic functions by using the integral of the average value of a continuous function phi .

Videos

To prove: The mean value property of harmonic functions: If φ is harmonic then, Iφ(r)=φ(P) for all r.

Want to see the full answer?

Chapter 18 Solutions

To prove:

The mean value property of harmonic functions: If φ is harmonic then, Iφ(r)=φ(P) for all r.