Divergence Theory

Gauss's Divergence Theorem

You must have heard about Quantum Theory or Matrix Theory or Probability Theory in your math class or in some sci-fi movie. But have you ever heard of Divergence Theory? Well, here we are today to learn more about this theory. Divergence Theorem is a theorem that talks about the flux of a vector field through a closed area to the volume enclosed in the divergence of the field. It is a part of vector calculus where the divergence theorem is also called  Gauss's divergence theorem or Ostrogradsky's theorem.


State and Prove the Gauss's Divergence theorem

To state the Gauss's divergence theorem in an easier way, let’s break it into parts. The surface integral of a vector field over a sealed area is known as flux through the surface. Therefore, the flux through the surface is equivalent to the volume integral of the divergence over the area inside the surface. Thus, the total sum of all sources of the field in an area gives the net flux out of the area.


This is an essential result for mathematics in engineering and physics. It is used particularly in the field of electrostatic and fluid dynamics.


Divergence Theorem is generally applied in 3 dimensions, but it can be used in any number of dimensions. When you use it in 2 dimensions, it becomes equivalent to Green’s theorem. 

When you use it in 1 dimension, it becomes equivalent to integration by parts. 


The Divergence Theorem proof 

Let’s say that “S” is a closed surface and any line drawn parallel to the coordinate axes cut S in almost 2 points. Let \[S_{1}\] and \[S_{2}\] be the surface at the top and bottom of S. These are shown by z=f(x,y)and z=ϕ(x,y) respectively.

\[\overrightarrow{F} = F_{1} \overrightarrow{i} + F_{2} \overrightarrow{j} + F_{3} \overrightarrow{k}\]

\[\iiint \frac{\partial F_{3} }{\partial z} dV = \iiint \frac{\partial F_{3} }{\partial z} dx dy dz\]

\[\iint R[ \int_{z = f(x, y)}^{z = \Phi(x, y)} \frac{\partial F_{3}}{\partial z}]dx dy\]

\[\iint R [ F_{3} (x, y, z)]z = f(x, y)z = \Phi(x, y) dx dy\]

\[\iint R [ F_{3} (x, y, f) - F_{3}(x, y, \Phi)dx dy\]         _ _ _(1)

So, for the upper surface S2,

\[\frac{dx}{dy} = cos \gamma_{2} ds = k. n_{2} dS\]

Since the normal vector \[n_{2}\] to \[S_{1}\] makes an acute angle \[\gamma_{2}\] with \[\overrightarrow{k}\] vector

\[dx dy = -cos\gamma_{2} dS_{1} = - \overrightarrow{k}.\overrightarrow{n}. dS_{1}\] 

Since the normal vector \[n_{1}\] to \[S_{1}\] makes an obtuse angle \[\gamma_{1}\] with \[\overrightarrow{k}\] vector

\[\iint_{R} F_{3} (x, y, z)dxdy = \iint _{S_{2}} F_{3} \overrightarrow{k}. n_{2} \rightarrow dS_{2}\]      - - -(2)

\[\iint _{R} F_{3}(x, y, \Phi)dxdy = \iint _{S_{1}} F_{3} \overrightarrow{k}. n_{1} \rightarrow dS_{1}\]    - - -(3)

Now, the expression (1) can be written as:

\[\iint _{r} F_{3} (x, y, z)dxdy - \iint _{R} F_{3} (x, y, \Phi) dxdy\]

\[\iint _{S_{2}} F_{3} \overrightarrow{k}. n_{2} \rightarrow dS_{2} - \iint _{S_{1}} F_{3} \overrightarrow{k}. n_{1} \rightarrow dS_{1}\]

\[\iint _{S} F_{3} \overrightarrow{k}. n \rightarrow dS\]

\[\iiint \frac{\partial F_{3} }{\partial z} dV = \iint F_{3} \overrightarrow{k}. \overrightarrow{n} dS\]

\[\iiint \frac{\partial F_{2} }{\partial y} dV = \iint F_{2} \overrightarrow{k}. \overrightarrow{n} dS\]

\[\iiint \frac{\partial F_{1} }{\partial x} dV = \iint F_{1} \overrightarrow{k}. \overrightarrow{n} dS\]

Now, add the above all three equations, we get:

\[\iint v \int [ \frac{\partial F_{1} }{\partial x} + \frac{\partial F_{2} }{\partial y} + \frac{\partial F_{3} }{\partial z}]dV = \iint s[ F_{1} \overrightarrow{i} + F_{2} \overrightarrow{j} + F_{3} \overrightarrow{k}]. \overrightarrow{n}. dS\]

Thus, the divergence theorem can be written as:

\[\iint v \int \bigtriangledown \overrightarrow{F} . dV = \iint s \overrightarrow{F} . \overrightarrow{n}. dS\]

Here we discussed the Gauss's divergence theorem proof.


Gauss's Divergence Theorem History

Lagrange was the first one to discover the Divergence Theorem in 1762. Later on in 1813, it was rediscovered independently by Gauss. He also gave the first proof of the general theorem in 1826. Many other mathematicians like Green, Simeon-Denis Poisson, and Frédéric Sarrus also discovered this theorem.


Here are some of

Gauss's divergence theorem examples

Example 1: Compute

∬SF⋅dS where

\[F = (3x + z^{77}, y^{2} - sin x^{2}z, xz + ye^{x^{5}}\] 

And the surface of the box is S.

0 ≤ x ≤ 1, 0 ≤ y ≤ 3, 0 ≤ z ≤ 2.

Answer

It would be very difficult to calculate this integral directly. However, if we use the divergence of  F it would make it very easy:

\[\bigtriangledown F = 3 + 2y + x.\]

To convert the surface integral into a triple integral, we can use the divergence theorem 

∬SF⋅dS = \[\iint B \bigtriangledown F dV\]

where B is the box

0 ≤ x ≤ 1, 0 ≤ y ≤ 3, 0 ≤ z ≤ 2.

We compute the triple integral of div⁡F=3+2y+x over the box B:

\[\iint SF . dS = \int 10 \int 30 \int 20 (3 + 2y + x)dz dy dx\]

\[\int 10 \int 30 (6 + 4y + 2x)dy dx\]

\[\int 10 (18 + 18 + 6x)dx\]

36 + 3 = 39.


Fun Facts: 

  1. This theorem was discovered by  Lagrange in 1762.

  2. It was rediscovered by Gauss in 1813

FAQ (Frequently Asked Questions)

Q) Where is the Divergence Theorem Used?

Ans) this theorem is used in fields like mathematics in engineering and physics.  It is used particularly in the field of electrostatic and fluid dynamics. Its applications are discussed below:

  1. Differential Form and an Integral Form of Physical Laws

Owing to the discovery of divergence theorem, there are many laws of physics which can be written in differential form and an integral form. Laws of physics that are an example are:

Gauss's law (in electrostatics)

Gauss's law for magnetism

Gauss's law for gravity.

  1. Inverse - Square Laws

Any inverse-square law can be written in the form of differential form and an integral form. Just like  Gauss's law. For example, Gauss's law for gravity. It follows the inverse-square Newton's law of universal gravitation.

Q) What is Gauss's Divergence Theorem? When is it Used?

Ans) Gauss's Divergence Theorem is a theorem that talks about the flux of a vector field through a closed area to the volume enclosed in the divergence of the field. It is a part of vector calculus where the divergence theorem is also called divergence theorem or Ostrogradsky's theorem. divergence theorem is used to convert the surface integral into a volume integral through the divergence of the field. When you are trying to calculate flux it is easier to bound the interior of the surface and assess a volume integral rather than assessing the surface integral directly through divergence theorem. It is mainly used for 3-dimensional space.