Stokes Theorem

Statement. The line integral of a vector A taken around a closed curve C, which bounds a surface S, is equal to the surface integral of the curl of A taken over S, i.e.,
            ∫c A.dr = ∫∫s curl A.dS
Explanation. Stoke’s theorem enables us to transform the surface integral of the curl of the vector field A into the line integral of that field A over the boundary C of that surface and vice-versa.

Proof. Consider an open surface S in a vector field A. The bounding edge of the surface is a closed curve. The line integral of A round the closed curve when it is traced in the anticlock wise sense.
                Φc A.dr.                                … (1)

Divide the surface S into a large number of small areas. Consider one such element of area dS. Let n be a unit positive (outward) normal upon dS. The vector area of the element is
                n dS = dS.


By definition, the curl of a (non-lamellar) vector field at any point is defined as the maximum line integral of the vector computed per unit area along the boundary of an infinitesimal area at the point.
Therefore, the line integral of A around the boundary of the area dS is
                Curl A. dS.
This applies to all surface elements. Hence the sum of the line integrals of A around the boundaries of all the elements is
                ∫s∫curl A.dS,                            … (2)
which is the surface integral of curl A.
    Now, it is clear from the figure that the line integrals along the common sides of the adjacent elements mutually cancel because they are traversed in opposite direction. Thus only those sides of the element which lie in the periphery of the surface contribute to the line integral. Therefore, the surface integral (2) gives the line integral of A around the periphery of the surface, which is also given by (1).
                Φc A. dr = ∫∫s curl A. dS
This proves Stokes’s theorem.


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