Curl
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 6.5: “Divergence and Curl” |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/6-5-divergence-and-curl |
| Textbook used in class | Stewart, Calculus, Section 16.5: “Curl and Divergence” (Examples 1, 2) |
Quick Reference
For $\mathbf{F} = \langle P, Q, R\rangle$: $$\text{curl}\,\mathbf{F} = \nabla\times\mathbf{F} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ \partial/\partial x & \partial/\partial y & \partial/\partial z \\ P & Q & R\end{vmatrix} = \langle R_y - Q_z,\; P_z - R_x,\; Q_x - P_y\rangle.$$
Key identity: $\text{curl}(\nabla f) = \mathbf{0}$ (gradient fields have zero curl).
Conservative test: $\mathbf{F}$ conservative (on simply-connected domain) $\Leftrightarrow$ $\text{curl}\,\mathbf{F} = \mathbf{0}$.
Motivation
Curl measures the tendency of a vector field to rotate: if you placed a tiny paddle wheel in a fluid flow $\mathbf{F}$, the curl at that point tells you how fast the wheel would spin and around which axis. A field with nonzero curl has rotational flow; a field with zero curl (irrotational) has no local rotation -- which is exactly what conservative fields are.
In 2D, $\text{curl}\,\mathbf{F}$ reduces to the scalar $Q_x - P_y$, which is the integrand in Green’s theorem.
Key Concepts
1. Computing Curl
The $3\times 3$ determinant is a mnemonic, not a literal determinant (the entries are operators, not numbers). Expand along the first row:
$$\text{curl}\,\mathbf{F} = \left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)\mathbf{i} - \left(\frac{\partial R}{\partial x}-\frac{\partial P}{\partial z}\right)\mathbf{j} + \left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\mathbf{k}.$$
2. Geometric Interpretation
$\text{curl}\,\mathbf{F}(x_0,y_0,z_0)$ is a vector. Its direction is the axis around which a fluid element at $(x_0,y_0,z_0)$ tends to rotate; its magnitude is the rate of rotation (angular speed times 2).
Worked Example
Find $\text{curl}\,\mathbf{F}$ for $\mathbf{F}(x,y,z) = \langle xz, xyz, -y^2\rangle$. (Stewart 16.5, Example 1.)
$P = xz$, $Q = xyz$, $R = -y^2$.
$R_y = -2y$, $Q_z = 0$. First component: $-2y - 0 = -2y$.
$P_z = x$, $R_x = 0$. Second component (with minus sign): $-(x - 0) = -x$.
$Q_x = yz$, $P_y = 0$. Third component: $yz - 0 = yz$.
$$\text{curl}\,\mathbf{F} = \langle -2y, -x, yz\rangle.$$
curl of a 2D field $\mathbf{F} = \langle P, Q\rangle$ is $Q_x - P_y$. The curl of a 2D field (treated as $\mathbf{F} = \langle P, Q, 0\rangle$ in 3D) is the vector $\langle 0, 0, Q_x - P_y\rangle$ -- only the $z$-component is nonzero. The quantity $Q_x - P_y$ is the $z$-component of $\text{curl}\,\mathbf{F}$, which is why it appears in Green’s theorem as a scalar. Curl is a vector quantity in 3D, not a scalar.
Common Misconceptions
the curl of a vector field is a scalar quantity.
This is the multiplicative-not-additive error. Curl is defined as the cross product $\nabla \times \mathbf{F}$, which is a vector. In three dimensions, $\operatorname{curl}\mathbf{F} = \langle R_y - Q_z, P_z - R_x, Q_x - P_y\rangle$ has three components. Only in the special case where $\mathbf{F}$ is a 2D field embedded in 3D does the curl reduce to having only a nonzero $z$-component, and the scalar $Q_x - P_y$ is that single component -- not the whole curl. Confusing the $z$-component with the curl itself leads to errors when applying Stokes’ theorem in 3D.
Leveled Practice
Problem 1. Compute $\text{curl}\,\mathbf{F}$ for $\mathbf{F} = \langle e^x\sin y, e^x\cos y, 0\rangle$ and determine if $\mathbf{F}$ is conservative.
Show answer
$P = e^x\sin y$, $Q = e^x\cos y$, $R = 0$.
$R_y = 0$, $Q_z = 0$. First: $0$.
$P_z = 0$, $R_x = 0$. Second: $0$.
$Q_x = e^x\cos y$, $P_y = e^x\cos y$. Third: $0$.
$\text{curl}\,\mathbf{F} = \mathbf{0}$. So $\mathbf{F}$ is conservative (on a simply-connected domain).