## Gauss's Theorem From Green's Theorem

Let\[B\]

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.