Theory

Introduction

In what follows we study a 2D version of the Fundamental Theorem of Calculus called Green’s Theorem. The 1D integral ${\int }_a^b{F}^{\prime }(x)dx$ over the $x$-axis is upgraded to a double integral ${\iint }_{\mathcal{D}}\nabla {\times }_z\mathbf{F}\cdot d\mathbf{s}$ over a 2D region $\mathcal{D}$. The derivative $F^{\prime }$ is replaced by a new kind of derivative $\nabla {\times }_z\mathbf{F}$ called ‘curl’. The difference $F(b)-F(a)$ taken on the boundary of the 1D interval $[a,b]$ is upgraded to the line integral ${\oint }_{\mathcal{C}}\mathbf{F}\cdot d\mathbf{s}$ taken around the boundary $\mathcal{C}$ of the 2D region $\mathcal{D}$.

Curl as derivative operator

Start with a vector field $\mathbf{F}$ with vector components $\mathbf{F}=(F_1,F_2)$, where $F_1(x,y)$ and $F_2(x,y)$ are functions of points in space.

From this $\mathbf{F}$ we create a scalar field called the scalar curl of $\mathbf{F}$, written $\nabla {\times }_z\mathbf{F}$. This field is calculated with the formula:

$$\nabla {\times }_z\mathbf{F}=\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}.$$

One way to remember this formula is to create a vector-derivative-operator with components $\nabla =(\frac{\partial }{\partial x},\frac{\partial }{\partial y})$, and then take the cross product “$\nabla \times \mathbf{F}$”, applying $\frac{\partial }{\partial x}$ and $\frac{\partial }{\partial y}$ to the function components of $\mathbf{F}$. (Interpret $\nabla$ and $\mathbf{F}$ as 3D vectors with zero $z$-component.) The result is a vector with only a $z$-component. The $z$ subscript indicates that we are just taking the scalar value of that $z$-component.

Curl as measure of circulation

The scalar curl field can be explained as a measure of circulation of the vector field around each point in space.

It is good to understand how this works. Below we calculate the circulation of $\mathbf{F}$ around a small box. In a Problem, you calculate the circulation of $\mathbf{F}$ around a small circular loop.

Consider a small box $\mathcal{C}$ with bottom left corner at a point $\mathbf{p}=(a,b)$ in the plane, and sides of length $\Delta x=h$ and $\Delta y=k$. Label the sides with subscripts for left, right, top, bottom as in the picture:

The ‘circulation’ we wish to calculate is the aligned vector line integral around this box:

$${\oint }_{\mathcal{C}}\mathbf{F}\cdot d\mathbf{s}.$$

Break this into four integrals, one over each side of the box. Define parametric curves traversing each side, with $t\in [0,1]$ for each case:

• For side $C_B$, take ${\mathbf{r}}_B(t)=\mathbf{p}+th{\mathbf{e}}_{\mathbf{i}}$.
  • Observe that ${\mathbf{r}}_B^{\prime }(t)=h{\mathbf{e}}_{\mathbf{i}}$ and $d\mathbf{s}=(h,0)dt$.
• For side $C_R$, take ${\mathbf{r}}_R(t)=\mathbf{p}+h{\mathbf{e}}_{\mathbf{i}}+tk{\mathbf{e}}_{\mathbf{j}}$.
  • Observe that ${\mathbf{r}}_R^{\prime }(t)=k{\mathbf{e}}_{\mathbf{j}}$ and $d\mathbf{s}=(0,k)dt$.
• For side $C_T$, take ${\mathbf{r}}_T(t)=\mathbf{p}+(1-t)h{\mathbf{e}}_{\mathbf{i}}+k{\mathbf{e}}_{\mathbf{j}}$.
  • Observe that ${\mathbf{r}}_T^{\prime }(t)=-h{\mathbf{e}}_{\mathbf{i}}$ and $d\mathbf{s}=(-h,0)dt$.
• For side $C_L$, take ${\mathbf{r}}_L(t)=\mathbf{p}+k(1-t){\mathbf{e}}_{\mathbf{j}}$.
  • Observe that ${\mathbf{r}}_L^{\prime }(t)=-k{\mathbf{e}}_{\mathbf{j}}$ and $d\mathbf{s}=(0,-k)dt$.

Notice that $\mathbf{F}\cdot d\mathbf{s}=F_1({\mathbf{r}}_B(t))hdt$ on $C_B$, and the others simplify in a similar way because of the zeros in the $d\mathbf{s}$ vectors.

We are considering small boxes $\mathcal{C}$, and we will in fact take a limit as the sides $h,k$ go to zero. It is appropriate to use the linear approximation for each component $F_1$ and $F_2$ on the box, with differentials from the point $\mathbf{p}$. This lets us write the circulation integral in terms of $\frac{\partial F_i}{\partial x}$ and $\frac{\partial F_i}{\partial y}$, evaluated at the point $\mathbf{p}$; these appear in the curl formula we are trying to explain. Write $(q_1,q_2)$ for the components of $\mathbf{F}(\mathbf{p})$.

• For side $C_B$, we have $F_1({\mathbf{r}}_B(t))\approx q_1+\frac{\partial F_1}{\partial x}th$.
• For side $C_R$, we have $F_2({\mathbf{r}}_R(t))\approx q_2+\frac{\partial F_2}{\partial x}h+\frac{\partial F_2}{\partial y}tk$.
• For side $C_T$, we have $F_1({\mathbf{r}}_T(t))\approx q_1+\frac{\partial F_1}{\partial x}(1-t)h+\frac{\partial F_1}{\partial y}k$.
• For side $C_L$, we have $F_2({\mathbf{r}}_L(t))\approx q_2+\frac{\partial F_2}{\partial y}(1-t)k$.

Now plug everything in and compute four integrals in $t$:

$$\begin{array}{rl}{\oint }_{\mathcal{C}}\mathbf{F}\cdot d\mathbf{s}& \approx {\int }_0^1\left(q_1+\frac{\partial F_1}{\partial x}th\right)hdt\\ & +{\int }_0^1\left(q_2+\frac{\partial F_2}{\partial x}h+\frac{\partial F_2}{\partial y}tk\right)kdt\\ & +{\int }_0^1\left(q_1+\frac{\partial F_1}{\partial x}(1-t)h+\frac{\partial F_1}{\partial y}k\right)(-h)dt\\ & +{\int }_0^1\left(q_2+\frac{\partial F_2}{\partial y}(1-t)k\right)(-k)dt.\end{array}$$

Simplify this by cancelling some terms and combining some others:

$$\begin{array}{rl}{\oint }_{\mathcal{C}}\mathbf{F}\cdot d\mathbf{s}& \approx {\int }_0^1{h}^2\frac{\partial F_1}{\partial x}(2t-1)+k^2\frac{\partial F_2}{\partial y}(2t-1)+hk\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dt\\ & =hk\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)=hk(\nabla {\times }_z\mathbf{F}).\end{array}$$

Notice that $hk=\Delta x\Delta y$ is the area of the box. The bigger the box, the greater the effect of circulation.

The scalar curl is the limit of the ratio of circulation to the enclosed area, for boxes shrinking to zero:

$$\nabla {\times }_z\mathbf{F}=\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}=\underset{h,k\to 0}{\lim }\frac{1}{\text{Area}(\mathcal{C})}{\oint }_{\mathcal{C}}\mathbf{F}\cdot d\mathbf{s}.$$