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What is the Stokes theorem?

According to the Stokes theorem, “The surface integral of the curl of a vector field  over the surface S is equal to the line integral of that field along the boundary C of the surface S. i.e.

Thus, the Stokes theorem equates a surface integral with the line integral along the boundary of the surface. We may find the direction of surface element vector da by using right hand rule, which states that if the figures of right hand are curl along the direction of line integral, then the thumb will give the direction of vector da.

Proof:

If we divide the area enclosed by the curve C into two parts by a line , we get two closed curve C1 and C2 and the line integral of vector along the boundary of curve C will be equal to the sum of line integral of along C1 closed curve and C2 closed curve. The line integral along the line for the curve C1 is cancelled by the line integral along opposite direction of line of curve C2.

This is Stokes theorem.

Physical interpretation of Stokes Theorem:

Let us consider that a vector field F that represents the velocity field of a fluid flow. Then the curl of vector field measures circulation or rotation. Thus, the surface integral of the curl over some surface represents the total amount of whirl. On the other side, the line integral of that field along the boundary of the surface represents the net flow of fluid along the boundary.

Now there may be infinite number of surfaces of different shapes with the same boundary line. In this case since the value of line integral will be same for each surface, therefore according to stokes` law the value of surface integral of the curl over the surface will also be same for each surface. That is total amount of whirl will be same for each surface. Similarly, since the boundary line of a closed surface shrinks down to a point (i.e. the value of line integral is zero for a closed surface) therefore the value of surface integral will also be equal to zero (i.e. the total circulation is zero).

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