Gauss Divergence Theorem

What is Gauss Divergence theorem? State and Prove Gauss Divergence Theorem.

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Gauss Divergence Theorem

According to the Gauss Divergence Theorem, the surface integral of a vector field A over a closed surface is equal to the volume integral of the divergence of a vector field A over the volume (V) enclosed by the closed surface.

Gauss Divergence Theorem

Proof of Gauss Divergence Theorem

Consider a surface S which encloses a volume V. Let vector A be the vector field in the given region. Let this volume is made up of a large number of elementary volumes in the form of parallelopipeds.

Gauss Divergence Theorem

Consider jth parallelopiped of volume Δ Vj and bounded by a surface Sj of area d vector Sj. The surface integral of vector A over the surface Sj is given by

Gauss Divergence Theorem

For simplicity, consider the whole volume is divided into elementary volumes I, II, and III as shown in figure 1. The outward of elementary volume I is inward of elementary volume II and the outward of elementary volume II is inward of elementary volume III and so on.

Therefore, the sum of integrals of elementary volumes will cancel each other and we are left with the surface integral which arises from the surface of S.

Gauss Divergence Theorem

Multiply and divide left hand side of eqn. (1) by Δ Vi , we get

Gauss Divergence Theorem

Now, let us suppose the volume of surface S is divided into infinite elementary volumes so that ΔVi – 0

Gauss Divergence Theorem

Now,

Gauss Divergence Theorem

Hence eqn. (2) becomes

Gauss Divergence Theorem

Since Δ Vi – 0, therefore Σ Δ Vi becomes integral over volume V

Gauss Divergence Theorem

Which is the Gauss divergence theorem.