Multivariable calculus and vector analysis

Line integrals as circulation

Recall that a line integral ${\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}$ of a vector field $\mathbf{F}$ over an oriented curve $C$ “adds up” the component of the vector field that is tangent to the curve. Hence, in some sense, the line integral measures how much the vector field is aligned with the curve. If the curve $C$ is a closed curve, then the line integral indicates how much the vector field tends to circulate around the curve $C$. In fact, for oriented close curve $C$, we call the line integral the “circulation” of $\mathbf{F}$ around $C$:

$$\begin{array}{lll}\hfill {\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}=\text{ circulation of }\mathbf{F}\text{ around }C.& & \hfill \end{array}$$

Sometimes one might write the integral as

$$\begin{array}{lll}\hfill {\oint }_C\mathbf{F}\cdot d\mathbf{s}& & \hfill \end{array}$$

to emphasize that the integral is around a closed curve, but we tend to omit the circle decoration on the integral sign since it is redundant.

Note that the circulation can be positive or negative, depending on the orientation of $C$ compared to the flow of $\mathbf{F}$. For example, if $\mathbf{F}\left(x,y\right)=\left(y,-x\right)$ and $C$ is the ellipse

$$\begin{array}{lll}\hfill \frac{x^2}{4}+\frac{y^2}{9}=1& & \hfill \end{array}$$

oriented counterclockwise, then, as shown in the graph, we expect the circulation ${\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}$ to be negative.

The vector field appears to circulate in the clockwise direction so tends to point in the opposite direction of the orientation of the curve.

We can compute the circulation by parametrizing $C$ by

$$\begin{array}{lll}\hfill \mathbf{c}\left(t\right)=\left(2cost,3sint\right)& & \hfill \end{array}$$

for $0\le t\le 2\pi$. Then, since ${\mathbf{c}}^{\text{'}}\left(t\right)=\left(-2sint,3cost\right)$, the line integral is

$$\begin{array}{llll}\hfill {\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}& ={\int \; <!--nolimits\; -->}_0^{2\pi }\mathbf{F}\left(\mathbf{c}\left(t\right)\right)\cdot {\mathbf{c}}^{\text{'}}\left(t\right)dt& \hfill & \\ \hfill & ={\int \; <!--nolimits\; -->}_0^{2\pi }\mathbf{F}\left(2cost,3sint\right)\cdot \left(-2sint,3cost\right)dt& \hfill & \\ \hfill & ={\int \; <!--nolimits\; -->}_0^{2\pi }\left(3sint,-2cost\right)\cdot \left(-2sint,3cost\right)dt& \hfill & \\ \hfill & ={\int \; <!--nolimits\; -->}_0^{2\pi }\left(-6{sin}^{2}t-6{cos}^{2}t\right)dt={\int \; <!--nolimits\; -->}_0^{2\pi }-6dt=-12\pi .& \hfill & \end{array}$$

The circulation may not be so obvious in a picture. For example, let $\mathbf{F}\left(x,y\right)=\left(-y,0\right)$ and let $C$ be the counterclockwise oriented unit circle, as pictured below.

In this case, there is no obvious circulation of $\mathbf{F}$. However, if you look closely at the alignment of the vector field, you will see that it tends to align with the orientation of the curve. The circulation of $\mathbf{F}$ around $C$ is positive.

We verify this by calculating directly the circulation. Parametrizing the unit circle by $\mathbf{c}\left(t\right)=\left(cost,sint\right)$ for $0\le t\le 2\pi$, the circulation is

$$\begin{array}{llll}\hfill {\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}& ={\int \; <!--nolimits\; -->}_0^{2\pi }\mathbf{F}\left(\mathbf{c}\left(t\right)\right)\cdot {\mathbf{c}}^{\text{'}}\left(t\right)dt& \hfill & \\ \hfill & ={\int \; <!--nolimits\; -->}_0^{2\pi }\mathbf{F}\left(cost,sint\right)\cdot \left(-sint,cost\right)dt& \hfill & \\ \hfill & ={\int \; <!--nolimits\; -->}_0^{2\pi }\left(-sint,0\right)\cdot \left(-sint,cost\right)dt& \hfill & \\ \hfill & ={\int \; <!--nolimits\; -->}_0^{2\pi }{sin}^{2}tdt={\int \; <!--nolimits\; -->}_0^{2\pi }\frac{1-cos2t}{2}dt=\pi .& \hfill & \end{array}$$

Circulation plays a big role in vector calculus. It is circulation defined by line integrals that is the “macroscopic circulation” in our discussion of the curl and its subtleties. Three of the four fundamental theorems of vector calculus involve circulation. The link between the “microscopic ciculation” of the curl and this circulation form the basis of Green’s theorem and Stokes’ theorem. Lack of circulation can be thought of as the defining proporty of conservative vector fields.

Contents by topic

1. Geometry in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

4a. Vector field basics

4b. Parametrized curves

4c. Parametrized surfaces

4d. Path integrals and line integrals

• Path integral of a scalar-valued function
• Path integral examples
• Line integral of a vector field
• Alternate notation for line integrals
• Line integrals as circulation
• Line integral examples

4e. Surface integrals

5. The fundamental theorems of vector calculus

6. Review material