Multivariable calculus and vector analysis

More details about the components of the curl

Once you’ve learned about line integrals, you may be able make sense of the following description about the origin of the formula for the curl.

In the previous reading, we denoted the components of the curl by

$$\begin{array}{lll}\hfill \mathrm{curl}\mathbf{F}=\mathbf{v}=v_1\mathbf{i}+v_2\mathbf{j}+v_3\mathbf{k}.& & \hfill \end{array}$$

We visualized the component of the curl in the $x$ direction as the rotation of a ball on a rod parallel to the $x$-axis.

The component of the curl in the $x$ direction is $v_1=\mathbf{v}\cdot \mathbf{i}=\mathrm{curl}\mathbf{F}\cdot \mathbf{i}$. We could derive an expression for this component of the curl just like we derived an expression for the “microscopic circulation” used in Green’s theorem. To see this, rotate the above animation so that the $x$-axis is coming straight out of the screen and the $yz$-plane is parallel to the screen. You can see that the rotation of the sphere is affected only by the components of $\mathbf{F}$ that are parallel to the $yz$-plane (and perpendicular to the $x$-axis), i.e., $F_2$ and $F_3$. We have reduced the situation to a two-dimensional case of rotation parallel to the $yz$-plane. We simply need to find the “microscopic circulation” of $\left(F_2,F_3\right)$.

To estimate this “microscopic circulation,” we can construct a curve $C$ (shown in red below) centered at the sphere’s location, and parallel to the $yz$-plane. The circulation of $\mathbf{F}$ around $C$ is just the line integral ${\int \; <!--nolimits\; -->}_C\mathbf{F}\cdot d\mathbf{s}$.

The “microscopic circulation” or “circulation per unit area,” is just the circulation around $C$, divided by the the area of the region inside $C$, in the limit where $C$ shrinks down to a point (drag the red point on the slider to the left). If we repeat the calculation used for Green’s theorem, we could conclude that this microscopic circulation is

$$\begin{array}{lll}\hfill v_1=\mathrm{curl}\mathbf{F}\cdot \mathbf{i}=\frac{\partial F_3}{\partial y}-\frac{\partial F_2}{\partial z}.& & \hfill \end{array}$$

One can perform similar calculations to determine the formulas for the other components of the curl, as given in the previous reading.

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