Gauss's Theorem From Green's Theorem
\[B\]
be a region of the plane with smooth boundary \[C\]
. Since \[C\]
is smooth it has a smooth parametrization \[p(t)=(p_1 (t), p_2 (t)), \: a \leq t \leq b\]
The unit normal vector to
\[C\]
is \[p'(t)=( \frac{p'_2 (t)}{p'(t)} , - \frac{p'_1 (t)}{p'(t)} ) \]
Apply Green's Theorem to
\[\mathbf{F} = ( -F_2 (t), F_1 (t))\]
to obtain \[\oint_C -F_2 \: dx_1 + F_1 \: dx_2 = \int \int_B \frac{\partial F_1}{\partial x_1} + \frac{\partial F_2}{\partial x_2} \: dx_1 \: dx_2 = \int \int_B div \mathbf{F} \: dx_1 \: dx_2\]
Hence
\[\oint_C \mathbf{F} \cdot d \mathbf{n} = \int \int_B div \mathbf{F} \: dx_1 \: dx_2\]
Green's Theorem in this form is called Gauss's Theorem.
