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Curl — Definition, Formula & Examples

Curl is a vector calculus operation that measures how much a vector field tends to rotate around a given point. It produces a new vector whose direction is the axis of rotation and whose magnitude indicates the strength of that rotation.

For a vector field F=⟨P,Q,R⟩\mathbf{F} = \langle P, Q, R \rangle with continuously differentiable component functions on an open subset of R3\mathbb{R}^3, the curl of F\mathbf{F} is defined as ∇×F\nabla \times \mathbf{F}, yielding a vector field whose components are specific differences of partial derivatives of PP, QQ, and RR.

Key Formula

∇×F=∣ijk∂∂x∂∂y∂∂zPQR∣=⟨∂R∂y−∂Q∂z,  ∂P∂z−∂R∂x,  ∂Q∂x−∂P∂y⟩\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ P & Q & R \end{vmatrix} = \left\langle \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z},\; \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x},\; \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right\rangle
Where:
  • F=⟨P,Q,R⟩\mathbf{F} = \langle P, Q, R \rangle = A vector field with component functions P, Q, R
  • ∇×F\nabla \times \mathbf{F} = The curl of F, a new vector field
  • x,y,zx, y, z = Independent spatial variables

How It Works

To compute the curl, you set up a symbolic 3×33 \times 3 determinant with the unit vectors i,j,k\mathbf{i}, \mathbf{j}, \mathbf{k} in the first row, the partial derivative operators in the second row, and the components of F\mathbf{F} in the third row. Expand the determinant using cofactor expansion to get three components. Each component is a difference of two partial derivatives. If the curl is zero everywhere, the field is called irrotational, which often means it is a conservative (gradient) field on simply connected domains.

Worked Example

Problem: Find the curl of the vector field F=⟨2xz,  xy,  3yz⟩\mathbf{F} = \langle 2xz,\; xy,\; 3yz \rangle.
Identify components: Set P=2xzP = 2xz, Q=xyQ = xy, and R=3yzR = 3yz.
Compute the i-component: Take the partial of R with respect to y minus the partial of Q with respect to z.
∂R∂y−∂Q∂z=3z−0=3z\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = 3z - 0 = 3z
Compute the j-component: Take the partial of P with respect to z minus the partial of R with respect to x.
∂P∂z−∂R∂x=2x−0=2x\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} = 2x - 0 = 2x
Compute the k-component: Take the partial of Q with respect to x minus the partial of P with respect to y.
∂Q∂x−∂P∂y=y−0=y\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = y - 0 = y
Answer: ∇×F=⟨3z,  2x,  y⟩\nabla \times \mathbf{F} = \langle 3z,\; 2x,\; y \rangle

Why It Matters

Curl is central to electromagnetism — two of Maxwell's equations (Faraday's law and Ampère's law) are expressed using curl. It also appears in Stokes' theorem, which connects a surface integral of the curl to a line integral around the boundary, making it essential in physics and engineering fluid dynamics.

Common Mistakes

Mistake: Mixing up the sign pattern in the j-component of the determinant expansion.
Correction: The j-component carries a negative sign in standard cofactor expansion, which flips the subtraction order to ∂P∂z−∂R∂x\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} rather than the other way around. Write out the full determinant to keep signs straight.