12. TEAM PROJECT. Divergence Theorem and Poten-tial Theory. The importance of the divergence...

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12. TEAM PROJECT. Divergence Theorem and Poten-tial Theory. The importance of the divergence theo-rem 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) ifg an —andA = Igrad gl2 dV. (b) If ag/an = 0 on S. then g is constant in T. (c) an ag – g 2. an )dA – O.
(d) If 4/an = agian on S. then/ = g + c in T. where c is a constant. (e) The Laplacian can be represented independently of coordinate systems in the form
V2f = a(14).°10 V(IT) aafn
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 7'.

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David · Accepted Answer

a. Substitute f=g in green’s first formula, i.e. in the formula, 
∭(                )
 ∬ 
  
  

This yields us, 
∬ 
  
  

 ∭(                )
 ∭(    )
 ∭
 ∭|     |
                      (                             ) 
b. If 
  
  
   on S, the LHS of the above integral becomes 0.

12. TEAM PROJECT. Divergence Theorem and Poten-tial Theory. The importance of the divergence theo-rem in potential theory is obvious from (7)–(9) and Theorems 1-3. To emphasize it further, consider...

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