Stokes Theorem Statement
According to this theorem, the line integral of a vector field A vector around any closed curve is equal to the surface integral of the curl of A vector taken over any surface S of which the curve is a bounding edge.
Stokes Theorem Proof
Let A vector be the vector field acting on the surface enclosed by closed curve C. Then the line integral of vector A vector along a closed curve is given by
where dl vector is the length of a small element of the path as shown in fig.
Now let us divide the area enclosed by the closed curve C into two equal parts by drawing a line ab as shown in fig. We have now two closed curves C1 and C2.
Therefore, the line integral of vector A vector along a closed curve C can be written as
If the area enclosed by the curve C is divided into a large number of small areas such as dS1, dS2, dS3……………… dSn bounded by the curves C1, C2……………Cn as shown in fig.
According to the definition of curl
Put this value in (2), we get
where, dSn is the surface area in the case under consideration.
Hence eqn. (4) can be written as
which is the Stokes Theorem.