Divergence theorem

The “Divergence Theorem” is the most important theorem in Calculus. This theorem is used to solve a variety of difficult integral problems. The surface integral is compared to the volume integral. It denotes the relationship between the two. This article will go over the divergence theorem statement, proof, Gauss divergence theorem, and instances in depth. The divergence theorem, commonly known as Gauss’ theorem or Ostrogradsky’s theorem in vector calculus, is a theorem that connects the flux of a vector field through a closed surface to the field’s divergence in the volume enclosed.

• Definition

The surface integral of a vector field across a closed surface, known as the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface, according to the divergence theorem.. Intuitively, it states that the sum of all field sources in a region (with sinks viewed as negative sources) yields the net flux out of the region. 

The surface integral of the normal component of a vector point function “F” over a closed surface “S” equals the volume integral of the function’s divergence, according to the divergence theorem.

  F→   

taken over the surface S-enclosed volume “V”.As a result, the divergence theorem is symbolically denoted as follows: 

∬V. F .dV = ∬S. F. n.ds

• Divergence Theorem Proof

Consider a surface denoted by S that covers a volume denoted by V.

Assume vector A represents the vector field in the specified region. Assume that this volume is composed of a large number of parallelepipeds (1- 6 parallelepipeds) that represent elementary volumes. 

Consider the volume of the jth parallelepiped is is ΔVj which is bounded by a surface  Sj of area  dSj.

The surface integral of A over the surface Sj will be-

∯sA. dSj 

Consider dividing the entire volume into elementary volumes I, II, and III.

The elementary volume I outward corresponds to the elementary volume II within, and the elementary volume II outward corresponds to the elementary III inward, and so on.

As a result, the total of the constituent volume integrals cancels out, leaving only the surface integral coming from the surface S. 

 ∑ ∮∮Sj  A.dSj =∮∮S A.dS ….  (1) 

We get by multiplying and dividing the left-hand side of the equation(1) by ΔVi 

         sA.dS = ∑ 1Vi (∮∮Si   A .dSi )ΔVi

Assume that the volume of surface S is partitioned into infinite constituent volumes in such a way that   ΔVi →0.

Now,

Therefore, eq (2) becomes

We know that ΔVi→0 thus ∑ΔVi will become the integral over volume V.

Conclusion

The divergence theorem is the one that relates the surface integral to the volume integral. The Divergence theorem, in further detail, connects the flux through the closed surface of a vector field to the divergence in the field’s enclosed volume.It states that the outward flux via a closed surface is equal to the integral volume of the divergence over the area within the surface. 

The net flow of a region is obtained by subtracting the sum of all sources from the sum of all sinks.The outcome defines the flow of a vector by a surface and the behaviour of the vector field within it. 

Divergence Theorem is commonly employed in three dimensions, but it can be applied in any number of dimensions. In two dimensions, it is equivalent to Green’s theorem, which states that the line integral around any simple closed curve is equal to the double integral over the plane region. When used in a single dimension, it is equivalent to integration by parts.