Stokes' Theorem explained in simple words with an intuitive proof.

What is Stokes’ Theorem?

What is Stokes' Theorem and its Proof?

Stokes’ Theorem broadly connects the line integration and surface integration in case of the closed line. It is one of the important terms for deriving Maxwell’s equations in Electromagnetics.

What is the Curl?

Before starting the Stokes’ Theorem, one must know about the Curl of a vector field. Technically, it is a vector whose magnitude is the maximum circulation of the given field per unit area (tending to zero) and whose direction is normal to the area when it is oriented for maximum circulation.

In simple words, curl of a vector field at a given point depicts the whirling or rotating nature of that field. Go through the following article to get an in-depth discussion for the curl.

What is the Curl of a vector field?

Stokes’ Theorem

It states that the circulation of a vector field, say A, around a closed path, say L, is equal to the surface integration of the Curl of A over the surface bounded by L.

Stokes’ Theorem in detail

Consider a vector field A and within that field, a closed loop is present as shown in the following figure.

Explanation of Stokes' Theorem

Now, what is the closed loop or close line? Normal line or open line posses only a length. But the closed line doesn’t have any end and posses both the length and the enclosed area.

What is the Close Line in Electromagnetics?

Now, again go through the statement of the theorem. It requires the circulation of the field around this closed path. The circulation of the vector field along any path is nothing but the line integration of the given field along that path. [Ref: What is the line integration?]

So circulation of the vector field A around the closed path is given as \oint\limits_L\;\overrightarrow A\cdot d\overrightarrow l

Now according to the theorem, this circulation of the field around closed path is equal to the surface integration of the curl of that field over the enclosed surface.

Now as we can see from our figure above, the surface enclosed by the closed line L is S. This will be normal surface and not a closed surface and the surface integration is performed on this enclosed surface. So the theorem is mathematically written as follows –

\oint\limits_L\;\overrightarrow A\cdot d\overrightarrow l=\int\limits_S\left(\nabla\times\overrightarrow A\right)\cdot d\overrightarrow s

Proof of Stokes’ Theorem

Consider the same vector field A and a closed loop L, from the above figure. Let me present the similar figure again. Note from the figure that, I have taken a certain direction for the closed loop. Because for finding the circulation of the field around the loop the nature of circulation is necessary.

Proof for Stokes's Theorem

According to Stokes’s Theorem, we need to prove the two things equal: –
1) The circulation of the field A around L i.e. \oint\limits_L\;\overrightarrow A\cdot d\overrightarrow l and
2) The surface integration of the curl of A over the closed surface S i.e. \int\limits_S\left(\nabla\times\overrightarrow A\right)\cdot d\overrightarrow s.

Now, let us subdivide the surface S into very small subdivisions as shown in the following figure. Let each small portion of the surface is Sk.

Relation between closed line and surface

Now we want the circulation of the A around the loop L. But we have divided that major loop into many tiny sub-loops each for tiny subdivision surfaces Sk. And the total circulation i.e. the line integration along this closed loop L can be considered as the sum of the circulations of the field for each tiny loop. Can you guess why?

\oint\limits_L\;\overrightarrow A\cdot d\overrightarrow l=\sum_k\oint\limits_{L_k}\;\overrightarrow A\cdot d\overrightarrow l

The reason is very simple. There is a cancellation of the line integration effect on each interior path as shown in the above figure. As for any interior side that you consider, the direction of integration is in opposite direction for the adjacent loops forming that side. So, the sum of all these line integrations around all tiny Lks is the same as the line integration of the field around L. Now, let us adjust the above step little bit.

\oint\limits_L\;\overrightarrow A\cdot d\overrightarrow l=\sum_k\oint\limits_{L_k}\;\overrightarrow A\cdot d\overrightarrow l=\sum_k\frac{\oint\limits_{L_k}\;\overrightarrow A\cdot d\overrightarrow l}{S_k}S_k

Now just recall the definition of Curl of a vector field. It is the maximum circulation of the vector field per unit infinitesimal area. And at the beginning of the proof, we have already considered Sk to be a very small or an infinitesimal element, so we can say,

\frac{\oint\limits_{L_k}\;\overrightarrow A\cdot d\overrightarrow l}{S_k}=\nabla\times\overrightarrow A

Also for the infinitesimal case, the summation can be replaced by the integration. So that term is written as shown below. The integration must be surface integration. Why? Because we have divided the original enclosed surface S into a number of small elements i.e. infinitesimal surfaces and now we are trying summing them up. Isn’t it?

\sum_k\frac{\oint\limits_{L_k}\;\overrightarrow A\cdot d\overrightarrow l}{S_k}S_k=\int\limits_s\left(\nabla\times\overrightarrow A\right)\cdot d\overrightarrow s

Finally connecting all the links from the initial step, we can say that:

\oint\limits_L\;\overrightarrow A\cdot d\overrightarrow l=\int\limits_s\left(\nabla\times\overrightarrow A\right)\cdot d\overrightarrow s

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