Theory

Introduction

In 2D, it turns out that the divergence theorem and Green’s Theorem are actually restatements of the same fact about pairs of scalar functions.

Suppose $\mathbf{F}=(F_1,F_2)$. Then Green’s Theorem says that:

$${\iint }_{\mathcal{D}}\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}dA={\oint }_{\partial \mathcal{D}}{F}_{1}dx+F_{2}dy.$$

Now rotate the vector field clockwise by $90^{\circ }$ around each individual point $(x,y)$, and call the new field $\mathbf{G}$. As we have seen with the components of $\mathbf{n}$, the components of $\mathbf{G}$ are given by $\mathbf{G}=(F_2,-F_1)$. Now the divergence of $\mathbf{G}$ is the same as the curl of $\mathbf{F}$:

$$\nabla \cdot \mathbf{G}=\frac{\partial F_2}{\partial x}+\frac{\partial (-F_1)}{\partial y}=\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}=\nabla {\times }_z\mathbf{F}.$$

Furthermore, ${\mathbf{e}}_{\mathbf{n}}dt=(dy,-dx)$, so:

$$\mathbf{G}\cdot {\mathbf{e}}_{\mathbf{n}}ds=(F_2,-F_1)\cdot (dy,-dx)=F_{2}dy+F_{1}dx=\mathbf{F}\cdot d\mathbf{s}.$$

Putting this together, the two sides of the divergence theorem become:

$$\begin{array}{rl}{\iint }_{\mathcal{D}}\nabla \cdot \mathbf{G}dA& ={\iint }_{\mathcal{D}}\nabla {\times }_z\mathbf{F}\\ {\oint }_{\partial \mathcal{D}}\mathbf{G}\cdot {\mathbf{e}}_{\mathbf{n}}ds& ={\oint }_{\partial \mathcal{D}}\mathbf{F}\cdot d\mathbf{s}.\end{array}$$

So, in 2D, Green’s Theorem and the divergence theorem express the same relationship about the functions $F_1$ and $F_2$ on the region $\mathcal{D}$ and its boundary $\partial \mathcal{D}$. Green’s Theorem packages this relationship in terms of a vector field $(F_1,F_2)$, and the divergence theorem packages this relationship in terms of the rotated vector field $(F_2,-F_1)$.