# 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) 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.

Code to add this calci to your website  Hence Gauss - Ostrogradsky Divergence Theorem is proved with this simple example.