\[\mathbf{F}=\mathbf{F}(x,y,z)\]
with continuous partial derivatives defined on a surface \[S\]
with a boundary curve \[C\]
, \[\oint_C \mathbf{F} \cdot d \mathbf{r} = \int \int_S (\mathbf{\nabla} \times \mathbf{F}) \cdot \mathbf{n} dS\]
.Suppose we can write
\[x=x(u,v), \; y=y(u,v), \; z=z(u,v)\]
where \[u,v\]
representing a change of coordinates. We can write \[\mathbf{F}=\mathbf{F}(u,v)\]
.A unit normal to
\[S\]
is
\[\mathbf{n}= \frac { \frac{\partial \mathbf{F}}{\partial u} \times \frac{\partial \mathbf{F}}{\partial v} }{ \left| \frac{\partial \mathbf{F}}{\partial u} \times \frac{\partial \mathbf{F}}{\partial v} \right| } ,\; \left| \frac{\partial \mathbf{F}}{\partial u} \times \frac{\partial \mathbf{F}}{\partial v} \right| \neq 0. \]
and an element of surface area is
\[\mathbf{n} dS =( \frac{\partial \mathbf{F}}{\partial u} \times \frac{\partial \mathbf{F}}{\partial v}) du dv \]
The right hand side of Stoke's equation becomes
\[ \int \int_S (\mathbf{\nabla} \times \mathbf{F}) \cdot \mathbf{n} dS = \int \int_{S'} (\mathbf{\nabla} \times \mathbf{F}) \cdot ( \frac{\partial \mathbf{F}}{\partial u} \times \frac{\partial \mathbf{F}}{\partial v}) du dv \]
.where
\[S'\]
is the image of \[S\]
.
ON the left hand side \[d \mathbf{r} = \frac{\partial \mathbf{F}}{\partial u}du + \frac{\partial \mathbf{F}}{\partial v}dv \]
The left hand side becomes
\[\oint_{C'} \mathbf{F} \cdot (\frac{\partial \mathbf{F}}{\partial u}du + \frac{\partial \mathbf{F}}{\partial v}dv)\]
Stoke's Theorem is now
\[\oint_{C'} \mathbf{F} \cdot (\frac{\partial \mathbf{F}}{\partial u}du + \frac{\partial \mathbf{F}}{\partial v}dv=\int \int_{S'} (\mathbf{\nabla} \times \mathbf{F}) \cdot ( \frac{\partial \mathbf{F}}{\partial u} \times \frac{\partial \mathbf{F}}{\partial v}) du dv \]