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)

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

(c)

(d) If ˆ‚f/ˆ‚n = ˆ‚g/ˆ‚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

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.

Transcribed Image Text:

ag Igrad g|2 dV. - dA Әn af дg dA = 0. дn) дn