PX275 - D3b - proof of green's theorem

this is non examinable
considering a rectangle with vertices $(x_{1}, y_{1})$ , $(x_{2}, y_{1})$ , $(x_{2}, y_{2})$ , $(x_{1,} y_{2})$
taking a term from the theorem:

\begin{aligned} - \iint \frac{\partial P}{\partial y} d y d x & = - \int_{x_{1}}^{x_{2}} \int_{y_{1}}^{y_{2}} \frac{\partial P}{\partial y} d y d x \\ = - \int_{x_{1}}^{x_{2}} [P (x, y_{2}) - P (x, y_{1})] d x \\ = \int_{x_{1}}^{x_{2}} P (x, y_{1}) d x + \int_{x_{2}}^{x_{1}} P (x, y_{2}) d x \end{aligned}

- this gives the integrals of the top and the bottom sections

similarly:

\iint \frac{\partial Q}{\partial x} = \int_{y_{1}}^{y_{2}} Q (x_{2}, y) d y + \int_{y_{2}}^{y_{1}} Q (x_{1}, y) d y

- this gives the integrals of the sides

combining the terms give the integral around the perimeter of the rectangle
for a field $\vec{F}$ in the $x - y$ plane:

\vec{\nabla} \times \vec{F} = (\frac{\partial F_{y}}{\partial x} + \frac{\partial F_{x}}{\partial y}) \hat{k}

\iint \vec{\nabla} \times \vec{F} \cdot d \vec{A} = \oint \vec{F} \cdot d \vec{r}

eg:

\vec{\nabla} \times \vec{E} = - \frac{\partial B}{\partial t}

\iint \vec{\nabla} \times \vec{E} \cdot d \vec{A} = \oint_{l o o p} \vec{E} \cdot d \vec{l} - \int \frac{\partial B}{\partial t} \cdot d \vec{A}