# Mathematics

## Divergence Theorem or Gauss Theorem or Greens Theorem (Gausscher Integralsatz)

This theorem states that the "volume integral over the divergence of a vector function" is equal to the "closed surface integral over the vector function".

## Stokes Theorem (Stokescher Integralsatz)

This theorem states that "a surface integral over the rotation of a vector function" is equal to the "closed path integral along the surface thru the vector function".

# Electrostatics

The typical problem is to find the electric field for a given charge or given charge distribution.

## Couloms Law

## The electrical Field

## Electric Flux

The flux thru a closed surface that contains the enclosed charge $Q_e$ is defined as $1/\epsilon_0*Q_e$.

## Gauss Law

There are two forms of gauss law - the integral form ans the differential form.

The differential form states that the "divergence of the electric field" equals the "charge distribution".

The integral form states that the "volume integral over the divergence of the electric field" equals the "closed surface integral over an electric field" which is equal to "the electric flux thru the closed surface"

The advantage of gauss law applies only to problems with the given symmetry:

- spherical symmetry
- cylindric symmetry
- plane symmetry

With the above symmetries, the surface normal vector points in the same direction as the electrical field vector. Their vector product is zero.

This simplifies the calculation of the surface integral significantly, because the integral reduces to a one dimensional integral.

As the direct application of gauss law is limited to the three symmetries, the more complex objects can be separated into one or more of the three symmetries.

The resulting electrical field is the superposition of fields of the individual symmetries.

## The rotation of the *static electric field*

The rotation of any ** static electric field** is zero.

This is not true for changing electrical fields !