Solved: TEAM PROJECT. Divergence Theorem and Potential

Chapter 10, Problem 10.1.169

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Divergence Theorem and Potential Theory. The importance of the divergence theorem in potential theory is obvious from (7)-(9) and Theorems 1 - 3. To emphasize it further, consider functions f and g that are harmonic in some domain D containing a region T with boundary surface S such that T satisfies the assumptions in the divergence theorem. Prove and illustrate by examples that then:

(a) \(\iint_{S} g \frac{\partial g}{\partial n} d A=\iiint_{T}|g \operatorname{rad} g|^{2} d V\)

(b) If \(\partial g / \partial n=0\) on S, then g is constant in T.

(c) \(\iint_{S}\left(f \frac{\partial g}{\partial n}-g \frac{\partial f}{\partial n}\right) d A=0\)

(d) If \(\partial f / \partial n=\partial g / \partial n\) 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

\(\nabla^{2} f=\lim _{d(T) \rightarrow 0} \frac{1}{V(T)} \iint_{S(T)} \frac{\partial f}{\partial n} d A\)

where d(T) is the maximum distance of the points of a region T bounded by S(T) from the point at which the Laplacian is evaluated and V(T) is the volume of T.

Text Transcription:

iint_S g partial g / partial n dA = iiint_T |g rad g|^2 dV

partial g/partial n = 0

iint_S (f partial g / partial n - g partial f / partial n) dA = 0

partial f/partial n = partial g/partial n

nabla^2 f = lim _d(T) rightarrow 0} 1 / V(T) iint_S(T) partial f / partial n dA