Multivariable calculus and vector analysis

Other ways of writing Green’s theorem

In the introductory reading on Green’s theorem, we wrote Green’s theorem as

$$\begin{array}{lll}\hfill {\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}={\iint }_D\left(\mathrm{curl}\mathbf{F}\right)\cdot \mathbf{k}dA& & \hfill \end{array}$$

or

$$\begin{array}{lll}\hfill {\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}={\iint }_D\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dA,& & \hfill \end{array}$$

where $\mathbf{F}\left(x,y\right)=\left(F_1\left(x,y\right),F_2\left(x,y\right)\right)$ is a two-dimensional vector field, $D$ is a region in plane, and $C$ is its positively oriented boundary. I used this notation to tie Green’s theorem in with the concept of the curl of a vector field.

We often denote $C$ by $\partial D$ to make it explicit that the curve $C$ is the (positively oriented) boundary of $D$. This notation is also more natural when the region $D$ has more than one boundary component. Then, Green’s theorem can look like, for example,

$$\begin{array}{lll}\hfill {\int \; <!--nolimits\; -->}_{\partial D}\mathbf{F}\cdot d\mathbf{s}={\iint }_D\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dA,& & \hfill \end{array}$$

Note that the notation $\partial D$ simply means the boundary of $D$. It has nothing to do with a partial derivative.

However, people frequently write Green’s theorem differently. First, they like to change the formula by writing the line integral at the left in terms of components:

$$\begin{array}{lll}\hfill {\int \; <!--nolimits\; -->}_{\partial D}{F}_{1}dx+F_{2}dy={\iint }_D\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dA.& & \hfill \end{array}$$

Then, they like to let the vector field be $\mathbf{F}\left(x,y\right)=\left(P\left(x,y\right),Q\left(x,y\right)\right)$, so that Green’s theorem becomes

$$\begin{array}{lll}\hfill {\int \; <!--nolimits\; -->}_{\partial D}Pdx+Qdy={\iint }_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA.& & \hfill \end{array}$$

Sometimes, $\mathbf{F}=\left(P,Q\right)$ won’t be referred to as a vector field. Instead, one can discuss the above version of Green’s theorem as applied to the two scalar valued functions $Q:D\to \mathbf{R}$ and $P:D\to \mathbf{R}$.

Contents by topic

1. Geometry in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

5. The fundamental theorems of vector calculus

5a. Green's theorem

• The idea behind Green's theorem
• Other ways of writing Green's theorem
• Green's theorem with multiple boundary components
• Using Green's theorem to find area
• Sketch of proof for circulation per unit area
• Green's theorem examples

5b. The theorem for conservative vector fields

5c. Stokes' theorem

5d. The divergence theorem

6. Review material