Gauss-Ostrogradsky Divergence Theorem Proof, Example

The Divergence theorem in vector calculus is more commonly known as Gauss theorem. It is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field.

Gauss-Ostrogradsky Divergence Theorem Proof, Example

Statement:

If V is the volume bounded by a closed surface S and M is a vector function of position with continuous derivatives, then


∫∫∫ V . M dV = ∫∫ M.n.dS
v s


Proof:

Let M = Pi + Qj + Rk.



Then,
 ∂P +  ∂Q +  ∂R
div F =


 ∂x  ∂y  ∂z
Hence,

If n is the unit outward normal of S, then the surface integral on the left side.


∫∫ M . dS = ∫∫ M.n.dS
S S

So, to prove the theorem,


Hence we prove that,

(or)


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Hence Gauss - Ostrogradsky Divergence Theorem is proved with this simple example.

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