Multivariable calculus and vector analysis

Subtleties about curl

Macroscopic versus microscopic circulation

We introduced the curl of a vector field as the microscopic circulation of the vector field. In that introductory reading we attempted to keep things as simple as possible, so we didn’t make a big fuss over the difference between macroscopic circulation of the vector around in circles and the microscopic circulation that curl measures. For the example vector field we showed, the microscopic and macroscopic circulations were essentially the same, so we didn’t have to worry about the difference between them.

The vector field $\mathbf{F}\left(x,y,z\right)=\left(-y,x,0\right)$ is similar to the vector field of the introductory reading except that the rotation is around the $z$-axis. In this case, the macroscopic circulation is counterclockwise (when viewed from the positive $z$-axis) around the $z$-axis. Although this method isn’t a surefire way to detect circulation, in this case, we can see circulation by placing a sphere in the fluid and allowing it to move freely with the fluid. The macroscopic circulation is evident by the sphere’s circling around the $z$-axis. Since this vector field has no $z$-component and is independent of $z$, we may be able to get a better picture of the action by looking at the projection in the $xy$-plane (viewed from the positive $z$-axis), which is shown to the right. In all the figures on this page, we show the three-dimensional graph at the left along with its two-dimensional projection at the right.

Of course, this macroscopic circulation isn’t the curl of the vector field. Recall that in order to measure the microscopic circulation of the curl, one needs to fix the center of the sphere and allow it to rotate freely around the center. In this example, there is nothing tricky, and the rotation is again counterclockwise when viewed from the positive $z$-axis. Using the right hand rule, the curl points in the positive $z$ direction, as shown by the green arrow. Indeed, $\mathrm{curl}\mathbf{F}=\left(0,0,2\right)$.

However, you may or may not be surprised to learn that for some vector fields, the microscopic and macroscopic circulations could be quite different. The definition of curl as microscopic circulation is a little more subtle than it just being a measure of the rotation of the vector field.

Curl-free macroscopic circulation

In the vector field pictured below, there is clear macroscopic circulation of the vector field around the $z$-axis. If we placed a sphere in a fluid with velocity given by the vector field, and allowed the sphere to move freely with the fluid, it would circulate around the $z$-axis as shown.

Does this circulation correspond to the microscopic circulation of curl? Again, one needs to fix the center of the sphere to measure curl, as the motion of a free-floating sphere does not correspond to curl. If sphere still spins when its center is fixed, such spinning around its center would measure the microscopic circulation of curl. As shown below, the sphere with its center fixed does not rotate, indicating the absence of curl.

This vector field is

$$\begin{array}{lll}\hfill \mathbf{F}\left(x,y,z\right)=\frac{\left(-y,x,0\right)}{x^2+y^2}& & \hfill \end{array}$$

for $\left(x,y\right)\ne \left(0,0\right)$. Along circles $x^2+y^2=$ constant, the vector field is similar to the first vector field, above, giving the vector field the macroscopic circulation around the $z$-axis. However, due to the denominator, the vector field shrinks rapidly as one moves away from the $z$-axis. Since for $\left(x,y\right)\ne \left(0,0\right)$,

$$\begin{array}{lll}\hfill \frac{\partial F_2}{\partial x}=\frac{\partial F_1}{\partial y}=\frac{y^2-x^2}{{\left(x^2+y^2\right)}^2}& & \hfill \end{array}$$

one can compute that away from the $z$-axis, $\mathrm{curl}\mathbf{F}\left(x,y,z\right)=\left(0,0,0\right)$.

How can one see from the picture that the microscopic circulation is zero? The general rotation of the flow around the $z$-axis in the counterclockwise direction (as viewed from the positive $z$-axis) would indeed tend to make the sphere spin in that direction. But, since the arrows are longer closer to the $z$-axis, the vector field tends to push the sphere more strongly on the side closest to the $z$-axis. This second effect would tend to make the sphere spin in the clockwise direction (viewed from the positive $z$-axis). In this example, I have chosen the vector field so that the two effects precisely cancel each other out so that the curl is zero.

Of course, the effects need not balance. For the vector field

$$\begin{array}{lll}\hfill \mathbf{F}\left(x,y,z\right)=\frac{\left(-y,x,0\right)}{{\left(x^2+y^2\right)}^{3∕2}},& & \hfill \end{array}$$

for $\left(x,y\right)\ne \left(0,0\right)$, the length of the arrows diminishes even faster as one moves away from the $z$-axis. In this case, the microscopic circulation is opposite of the macroscopic circulation, and the sphere with a fixed center rotates clockwise (when viewed from the positive $z$-axis). Using the right hand rule, $\mathrm{curl}\mathbf{F}$ must point downward, as shown by the green arrow. We can calculate that for $\left(x,y\right)\ne \left(0,0\right)$,

$$\mathrm{curl}\mathbf{F}\left(x,y,z\right)=\frac{\left(0,0,-1\right)}{{\left(x^2+y^2\right)}^{3∕2}}.$$

Circulation may not be obvious from picture

For the vector field $\mathbf{F}\left(x,y,z\right)=\left(y,0,0\right)$, there is no rotation of the vector field, as the vector field is always parallel to the $x$-axis. However, if we place a sphere in the flow and fix its center, it does indeed spin. The spin is clockwise when viewed from the positive $z$-axis, so the curl points in the negative $z$ direction as shown by the green arrow. Indeed, one can easily calculate that $\mathrm{curl}\mathbf{F}\left(x,y,z\right)=\left(0,0,-1\right)$.

The sphere spins because the flow on the positive $y$ side the sphere pushes the sphere more strongly in the positive $x$ direction, compared with the flow on the negative $y$ side of the sphere.

In this case, the subtle issue isn’t between microscopic and macroscopic circulation. The macroscopic circulation, as one might define it via line integrals, would also be clockwise when viewed from the positive $z$ axis. (As such macroscopic circulation cannot be measured by letting a sphere float freely in the flow, this example shows how measuring macroscopic circulation with spheres doesn’t always work.)

Contents by topic

1. Geometry in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

4a. Vector field basics

• Vector fields
• The idea of divergence and curl
• The components of the curl
• More details on the components of the curl
• Divergence and curl notation
• Divergence and curl example
• Subtleties about divergence
• Subtleties about curl

4b. Parametrized curves

4c. Parametrized surfaces

4d. Path integrals and line integrals

4e. Surface integrals

5. The fundamental theorems of vector calculus

6. Review material