Green’s Theorem

Green’s theorem states that given a 2D dimensional vector field $\vec{F}(x,y)=[P(x,y),Q(x,y)]$ , there is a relationship between the path integral of $\vec{F}$ around a closed loop $B$ and the enclosed area $C$:

$${\oint }_B\vec{F}\cdot d\vec{l}=\int {\int }_C(\nabla \times \vec{F})\cdot \vec{\hat{z}}dA$$

Another, more popular form of Green’s Theorem is:

$${\oint }_{B}Pdx+Qdy=\int {\int }_C(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dA$$

We can derive the second form from the first by noting that $d\vec{l}=[dx,dy]$, and that the dot product inside the integral on the left hand side can be rewritten:

$$\begin{array}{rl}\vec{F}\cdot d\vec{l}& =\left[\begin{array}{c}P\\ Q\end{array}\right]\cdot \left[\begin{array}{c}dx\\ dy\end{array}\right]\\ & =Pdx+Qdy\end{array}$$

And on the right hand side, the $z$ component of $\nabla \times \vec{F}$ (which is the only non zero component when taking the dot product with $\hat{z}$) is given by the Curl:

$$(\nabla \times \vec{F})\cdot \hat{z}=\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}$$

External links

• Intuition into Green’s theorem