(Average values of harmonic functions) If u.x; y/ is harmoni
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(Average values of harmonic functions) If u.x; y/ is harmonic in a domain containing a disk of radius r with boundary Cr
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(Average values of harmonic functions) If u.x; y/ is harmonic in a domain containing a disk of radius r with boundary Cr , then the average value of u around the circle is the value of u at the centre. Prove this by showing that the derivative of the average value with respect to r is zero using the Divergence Theorem and the harmonicity of u, and the fact that the limit of the average value as r ! 0 is the value of u at the centre.
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