Module 17 : Surfaces, Surface Area, Surface integrals, Divergence Theorem and applications
Lecture 51 :  Applications of Divergence theorem [Section 51.2]
 

and we have

   
51.2.3 Green's Identity and properties of Harmonic functions:
 

Let be two scalar-fields which are twice continuously differentiable in a region which includes a solid and its boundary surface . Let . Then,


where


called the Laplacian operator . Thus, by the divergence theorem applied to over we get

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