# The simple proof for the Divergence Theorem.

Divergence theorem, in simple language, connects the surface and volume integrations for the

## What is Divergence Theorem?

It states that the total outward flux of vector field **A**, through the closed surface, say *S,* is the same as the volume integration of the divergence of **A**. Mathematically it can be written as,

Suggested Read: – *What is the Divergence Theorem?*

## The Proof in simple words

Consider any vector field A is present in the region. Assume any **S** is present within the field. We know that the closed surface encloses the volume within it. Let **V** be the volume enclosed.

Now the Divergence theorem needs following two to be equal: –

1) The net flux of the **A **through this **S**

2) Volume integration of the **divergence of A** over volume **V**.

As I have explained in the Surface Integration, the flux of the field through the given surface can be calculated by taking the surface integration over that surface. I have considered the cube as a closed surface for our illustration. So the total flux would be the addition of all the six terms each for one surface. And the net flux coming out of the cube can be represented with a little circle present in the integration sign as follows: –

Note that, for calculating the surface integration of each surface, the **ds** vector is taken as coming out of the volume. The reason for this is very simple. Our Divergence Theorem needs the outward flux i.e. flux coming out of the volume (cube).

Now let us assume that we have subdivided our cube i.e. volume into a **dv**. Let us say there are total **k** cells forming the whole cube with **k**, a very large number.

Now, the total flux coming out of the total surface S can be written as follows: –

It is simple. We are saying that the total flux coming out the cube is the addition individual fluxes of each tiny cell inside. This is obvious since the outward flux to one cell is inwards to some other neighbouring cells resulting in the cancellation on every interior surface. So adding all the individual fluxes would eventually lead to the total flux coming out of the final surface **S** i.e. out of the cube. Let us rearrange the above equation as shown below: –

I have reframed the equation purposefully. Just check out the term from the right-hand side of the equation. You will come to know this is exactly the same as the definition of the Divergence of vector field **A**. Right?

Suggested Read: – *What is the Divergence of vector field?*

Now for the infinitesimal case, as we have assumed initially, the summation can be replaced by the integration. Hence the above equation becomes as follows: –

This is nothing but the Divergence Theorem, isn’t it? Note that, at the starting of the Divergence Theorem Proof, we have assumed the surface to be a closed surface which encloses the volume so that we could find the volume integration. In other words, it is applicable only for the

Can you comment few applications of the Divergence Theorem in Electromagnetics?

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Tag:Electromagnetism