21.1 Definition and properties of curl

Curl measures how much a vector field rotates at each point in space. It's central to vector calculus and shows up throughout fluid dynamics and electromagnetism. This section covers how to compute curl using the nabla operator and cross product, its key algebraic properties, and its connections to conservative fields and Stokes' theorem.

Definition and Properties of Curl

Curl as a Vector Operator

Curl is a vector operator that measures the infinitesimal rotation of a vector field. Given a 3D vector field $\mathbf{F}$, the curl produces a new vector field describing the axis and magnitude of rotation at each point.

The notation is $\nabla \times \mathbf{F}$, where $\nabla$ is the nabla (del) operator:

$\nabla = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right)$

You can think of $\nabla$ as a "vector" whose components are partial derivative operators. The curl is then the formal cross product of this operator with $\mathbf{F}$.

Calculating Curl Using the Cross Product

To compute $\nabla \times \mathbf{F}$ for $\mathbf{F} = P\hat{i} + Q\hat{j} + R\hat{k}$, set up the determinant just like you would for any cross product:

$\nabla \times \mathbf{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix}$

Expanding this determinant gives the component formula:

$\nabla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right)\hat{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right)\hat{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\hat{k}$

Step-by-step process:

1. Write out the components $P$, $Q$, $R$ of your vector field.
2. Compute the six partial derivatives that appear in the formula.
3. For each component, subtract the appropriate pair (notice the cyclic pattern: each component involves the two other variables).
4. Assemble the result as a vector.

A common mistake is getting the signs wrong on the $\hat{j}$ component. Remember that the cofactor expansion of a $3 \times 3$ determinant gives a minus sign on the middle term, which is already accounted for in the formula above ($\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}$, not the other way around).

Properties of Curl

• Locality: Curl depends only on the behavior of the vector field in a neighborhood of the point. Values of $\mathbf{F}$ far away don't matter.
• Irrotational fields: A vector field with $\nabla \times \mathbf{F} = \mathbf{0}$ everywhere is called irrotational (or curl-free).
• Pseudovector behavior: Curl is a pseudovector (axial vector), meaning it changes sign under a reflection (parity transformation). This is the same behavior as angular velocity or torque.
• Linearity: Curl is linear: $\nabla \times (a\mathbf{F} + b\mathbf{G}) = a(\nabla \times \mathbf{F}) + b(\nabla \times \mathbf{G})$ for constants $a, b$.
• Scalar-field product rule: When a scalar function $f$ multiplies a vector field $\mathbf{F}$:

$\nabla \times (f\mathbf{F}) = (\nabla f) \times \mathbf{F} + f(\nabla \times \mathbf{F})$

This is analogous to the ordinary product rule but uses the cross product for the gradient term.

• Curl of a gradient vanishes: For any sufficiently smooth scalar function $\phi$:

$\nabla \times (\nabla \phi) = \mathbf{0}$

This identity is worth memorizing. It's the reason conservative fields are irrotational.

Curl in Different Dimensions

Curl in 2D and 3D

In 2D, rotation can only happen about the axis perpendicular to the plane, so the curl reduces to a single scalar. For $\mathbf{F}(x, y) = P(x, y)\hat{i} + Q(x, y)\hat{j}$:

$\nabla \times \mathbf{F} = \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\hat{k}$

You'll recognize the expression $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$ from Green's theorem. That's not a coincidence: Green's theorem is the 2D special case of Stokes' theorem.

In 3D, the curl is a full vector with three components, each describing the tendency to rotate about the corresponding axis. The magnitude $|\nabla \times \mathbf{F}|$ gives the rate of rotation, and the direction points along the axis of rotation (following the right-hand rule).

Irrotational Fields

An irrotational field satisfies $\nabla \times \mathbf{F} = \mathbf{0}$ everywhere in its domain. The key facts:

• Every irrotational field on a simply connected domain is conservative, meaning $\mathbf{F} = \nabla \phi$ for some scalar potential $\phi$.
• The converse always holds: if $\mathbf{F} = \nabla \phi$, then $\nabla \times \mathbf{F} = \mathbf{0}$ (by the curl-of-gradient identity).
• On domains that are not simply connected (e.g., a region with a hole), a field can be irrotational without being conservative. The classic example is $\mathbf{F} = \frac{-y}{x^2+y^2}\hat{i} + \frac{x}{x^2+y^2}\hat{j}$, which has zero curl away from the origin but a nonzero circulation around any loop enclosing the origin.

Physical examples of irrotational fields include gravitational fields and electrostatic fields.

Curl and Vector Calculus

Conservative Fields and Curl

A conservative field is one that can be written as $\mathbf{F} = \nabla \phi$ for some scalar potential $\phi$. The defining properties are:

• $\nabla \times \mathbf{F} = \mathbf{0}$ everywhere
• Line integrals depend only on endpoints, not on the path
• The integral around any closed loop is zero: $\oint_C \mathbf{F} \cdot d\mathbf{r} = 0$

These three conditions are equivalent on simply connected domains. The simply connected requirement matters: it guarantees there are no "hidden" topological obstructions.

For example, the gravitational field $\mathbf{g} = -\nabla \phi$ (where $\phi$ is the gravitational potential) is conservative. The work done by gravity depends only on the starting and ending heights, not on the path taken.

Stokes' Theorem

Stokes' theorem connects the curl of a field over a surface to the circulation of the field around the surface's boundary:

$\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r}$

Here $S$ is an oriented surface, $\partial S$ is its boundary curve (oriented consistently with $S$ via the right-hand rule), and $\mathbf{F}$ is a vector field with continuous partial derivatives on $S$.

The theorem generalizes in both directions:

• Green's theorem is the special case where $S$ is a flat region in the $xy$-plane.
• The fundamental theorem of line integrals is the 1D analogue.

Stokes' theorem shows up directly in physics. Ampère's law in electromagnetism, for instance, relates the curl of the magnetic field to the current density, and integrating both sides over a surface gives the familiar integral form.