\[C\]
with interior \[B\]
defined on \[{\mathbb{R}}^2\]
has a smooth parametrization \[\mathbf{p}(s) = (p_1 (s) , p_2 (s)) \]
where \[s\]
is the distance along the curve.The unit tangent to
\[C\]
is \[\mathbf{T} = ( \frac{p'_1(s)}{\sqrt{p'^2_1(s)+p'^2_2(s)}} , \frac{p'_2(s)}{\sqrt{p'^2_1(s)+p'^2_2(s)}} )\]
The unit normal to
\[C\]
is given by \[\mathbf{n(s)} =\mathbf{T'} = ( \frac{p''_1(s)}{\sqrt{p''^2_1(s)+p''^2_2(s)}} , \frac{p''_2(s)}{\sqrt{p''^2_1(s)+p''^2_2(s)}} )\]
The circulation of a force
\[\mathbf{F}\]
around \[C\]
is defined as \[\oint_C F_1 \: dx + F_2 \: dy = \oint_C \mathbf{F} \cdot d \mathbf{r} ds \]
\[curl \mathbf{F} = (\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}) \mathbf{k} \]
Hence Green's Theorem
\[\oint_C F_1 \: dx + F_2 \: dy = \int_B (\frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y})dx \: dy\]
can be written
\[\oint_C \mathbf{F} \cdot \mathbf{T} ds = \int \int_B curl \mathbf{F} \; dx \: dy\]
This is called Stokes Theorem.