TEAM PROJECT. Divergence Theorem and Potential Theory. The

Chapter 10, Problem 10.8

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TEAM PROJECT. Divergence Theorem and Potential Theory. The importance of the divergence theorem in potential theory is obvious from (7)-(9) and Theorems I - 3. To emphasize it further, consider functions f and g that are harmonic in some domain D containing a region Twith boundary surface S such that T satisfies the assumptions in the divergence theorem. Prove and illustrate by examples that then: (a) II g :! dA = I IIlgrad gl2 dV. S T (b) If aglan = 0 on S, then g i8 constant in T. (c) II ( f og - g Of) dA = O. on 011 S (d) If of Ian = fJglon on S, then f = g + c in T, where c is a constant. (e) The Laplacian can be represented independently of coordinate systems in the form v2 = lim _1_ JI of dA f d(T)~O VeT) an S(T) where d(T) is the maximum distance of the points of a region T bounded by SeT) from the point at which the Laplacian is evaluated and veT) is the volume of T.