I'm having a hard time understanding where exactly the formula for computing area using Green's theorem comes from. It is typically:
\begin{equation} \int_C x\,dy = \int_C -y\,dx = Area \end{equation}
Or the equation with:
\begin{equation} \frac{1}{2}\int_C x\,dy -ydx= Area \end{equation}
But I don't understand and can't seem to find anywhere where this is proven or from where it is derived. I do know that when you do a little algebraic manipulation on the first equation, you can get:
\begin{equation} \int_C x\,dy + y\,dx = Area \end{equation}
And then using Green's theorem, I seem to get the partial derivative of x with respect to x and the partial derivative of y with respect to y to subtract each other, which gives me Area = 0. This is where I'm stuck. Any and all help is appreciated.
Thanks.