The Divergence Theorem broadly connects the surface integration and the volume integration in case of the closed surface. It is one of the important mathematical tools that are required in the Electromagnetics.

What is the Divergence?

Technically the Divergence of a vector field at a given point is defined as the net outward flux per unit volume as the volume shrinks (tends to) zero at that point. Go through the following article to get an in-depth view.

What is the Divergence of vector field?

In simple words it the measure of the ‘outgoingness’ of the field at that point within the field.

The Divergence Theorem

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

The Divergence Theorem in detail

Consider the vector field A is present and within the field, say, a closed surface preferably a cube is present as shown below at point P.

Now, what is the closed surface? It is the surface of any 3-d body. In other words, the close surface encloses the volume inside it. Hence along with the surface area, it also possesses the volume. In our illustration, we have considered the cube which has six surfaces enclosing the volume. You can read more about the closed surface in the following article.

What is close surface in Electromagnetics?

So we need to find the flux coming out of each of these six surface in order to calculate the net outward flux coming out of the closed surface i.e. cube. Isn’t it?

Now you also know that we calculate the flux of the vector field for the given surface using the surface integration. Correct? So the net flux through this cube or closed surface would be the addition of surface integrations of all the six surfaces. This can be represented as shown below. You can see a small circle at the centre of the integration sign. This indicates that this term is the addition of all the individual integrals of each surface or the net flux that is coming out of this closed surface (cube).

Now, according to the Divergence Theorem, the net flux of the field that is coming out of the closed surface is equal to volume integration of the divergence of that vector field. The volume for the integration must be obviously the volume that is enclosed by the given closed surface. In our case, the volume enclosed by the cube.

So we can easily equate the above equations and lead to as shown below.

In simple words, we can say that the theorem relates the surface integration with the volume integration. But most important! The surface on which we are applying the theorem must be closed surface. Otherwise, it wouldn’t be possible to apply it. It is very simple. If it is not the closed surface then we won’t get the volume inside. Hence no volume integration.



So the Divergence Theorem can be used only for the closed surfaces.

Thus the flux of the field can be calculated using surface integration in any case (open or closed surface). In addition, it can also be found out by volume integration in case of the closed surfaces.

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