Stokes' Integral Theorem

In Sec. 2.4,

curlwas identified as that vector function which had an integral over a surfaceASthat could be reduced to an integral onover the enclosing contourAC. This was done by applying (2.4.1) to an incremental surface. But does this relation hold forSandCof finite size and arbitrary shape?The generalization to an arbitrary surface begins by subdividing

Sinto differential area elements, each enclosed by a contourC. As shown in Fig. 2.5.1, each differential contour coincides in direction with the positive sense of the original contour. We shall now prove that

where the sum is over all contours bounding the surface elements into which the surface

Shas been subdivided.

Figure 2.5.1.Arbitrary surface enclosed by contourCis subdivided into incremental elements, each enclosed by a contour having the same sense asC.Because the segments are followed in opposite senses when evaluated for the adjacent area elements, line integrals along those segments of the contours which separate two adjacent surface elements add to zero in the sum of (1). Only those line integrals remain which pertain to the segments coinciding with the original contour. Hence, (1) is demonstrated.

Next, (1) is written in the slightly different form.

We can now appeal to the definition of the component of the curl in the direction of the normal to the surface element, (2.4.2), and replace the summation by an integration.

Another way of writing this expression is to take advantage of the vector character of the

curland the definition of a vector area element,d:a=nda

This is

Stokes' integral theorem. If a vector function can be written as thecurlof a vector, then the integral of that function over a surfaceAScan be reduced to an integral ofon the enclosing contourAC.