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Divergence Theorem

Divergence Theorem

The line integral with a unit normal n of the closed curve is . Hereby the path  is given by

.

The tangent vector is the vector composed of the derivatives.

The unit tangent vector is the tangent vector divided by the length of the vector.

The unit normal vector n is perpendicular to this unit tangent vector.

The line integral is

Put in the vectors for F and n.

Multiply the compenents and shorten the unit vector r’.

Switch the terms and substitute

Remember the Green’s Theorem:

Divergence Theorem or Gauss Theorem