What is the Stokes Theorem

Illustration of Gaussian and Stokes' theorem


The Gaussian and Stokes law for vector fields are indispensable in electrodynamics. They provide a tool that makes working with Maxwell's equations much easier. The Gaussian theorem represents a relation between a vector field and its divergence, the Stokesian theorem does this for the rotation.

First, let us define the divergence of a vector field as a scalar field as follows:

So it gets over the surface of a volume element in place the scalar product summed up. The divergence is a measure of the source and sink strength of a vector field (see also applet next to it).

The rotation of a vector field is again a vector field. The component of the vector field is in the direction of any unit vector defined by:

Via a surface element parallel to elsewhere with an edge becomes the scalar product here summed up.



The Gaussian theorem

The Gaussian theorem says the following:

This clearly means that the portion of a vector field that passes through the surface of the volume flows, from the sources in volume must have come or disappears in the sinks of the volume.

Now the proposition still has to be proven: Let it be the volume in small partial volumes divided up. The following applies:

If the last summation is actually carried out, the inner sides of the partial volumes are summed up in the interior of the volume, as can be seen in the applet, so that their vectors (red, green and blue in the applet) are antiparallel and therefore these inner surfaces are lifted away. All that remains is the "outer skin" of the large volume, i.e. its surface. So is
After the border crossing from the discrete, finite surface elements to infinitesimally small areas, one obtains Gauss's theorem as anticipated above.



Stokes' theorem

Stokes' theorem is similar to Gaussian:

Here the integral of a rotation of a vector field over an area is converted into an integral of the vector field itself over the edge of this area. One proceeds in a very similar way with the proof: Be the area in small partial areas divided, the following applies:
During the summation, as can be seen in the applet, the inner sides of the partial areas are summed up in the interior of the area in such a way that their inner paths (dashed in the applet with red, green, orange and blue arrows) cancel each other out. All that remains is the "outer skin" of the surface, that is, its edge. So is
After the border crossing from the discrete, finite surface and path elements or for infinitesimally small areas and paths, one gets Stokes' theorem, as it was given above.