Curl Calculator

Calculate the curl of any 2D or 3D vector field with a step-by-step determinant expansion. Enter the component functions, get the symbolic curl, evaluate it at a point, spot irrotational fields and view a vorticity overlay.

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About Curl Calculator

The Curl Calculator computes the curl ∇×F of any 2D or 3D vector field with full step-by-step cross product determinant expansion. Enter your vector field components P, Q (and R for 3D), optionally evaluate at a specific point, and get the symbolic curl, rotation classification, and for 2D fields, an interactive visualization with a vorticity heat map and animated particle flow showing the rotational behavior of the field.

What Is Curl?

The curl of a vector field $\mathbf{F}$ measures the infinitesimal rotation of the field at each point. For a 3D field $\mathbf{F} = \langle P, Q, R \rangle$, the curl is computed as a cross product:

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

Expanding the determinant gives the curl vector:

$$\nabla \times \mathbf{F} = \left\langle \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z},\; \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x},\; \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right\rangle$$

For a 2D field $\mathbf{F} = \langle P, Q \rangle$, the curl reduces to the scalar $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$, which represents rotation in the xy-plane.

Physical Meaning of Curl

↺

CCW Rotation (curl > 0)

A tiny paddle wheel placed here would spin counterclockwise. The field circulates in the positive direction.

↻

CW Rotation (curl < 0)

A tiny paddle wheel placed here would spin clockwise. The field circulates in the negative direction.

⊘

Irrotational (curl = 0)

No net rotation — the field is conservative. A potential function φ exists with F = ∇φ.

♻

Stokes' Theorem

The surface integral of curl equals the line integral around the boundary: circulation = total rotation.

Curl Formulas in Different Coordinate Systems

Coordinate System	Curl Formula
Cartesian 2D	$\text{curl}\,\mathbf{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$ (scalar)
Cartesian 3D	$\nabla \times \mathbf{F} = \left\langle \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z},\; \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x},\; \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right\rangle$
Cylindrical	$\nabla \times \mathbf{F} = \left\langle \frac{1}{r}\frac{\partial F_z}{\partial \theta} - \frac{\partial F_\theta}{\partial z},\; \frac{\partial F_r}{\partial z} - \frac{\partial F_z}{\partial r},\; \frac{1}{r}\frac{\partial(rF_\theta)}{\partial r} - \frac{1}{r}\frac{\partial F_r}{\partial \theta} \right\rangle$
Spherical	See full expansion using scale factors $h_r=1, h_\theta=r, h_\phi=r\sin\theta$

Important Identities Involving Curl

Identity	Formula
Curl of gradient	$\nabla \times (\nabla f) = \mathbf{0}$ (always zero — gradients are irrotational)
Divergence of curl	$\nabla \cdot (\nabla \times \mathbf{F}) = 0$ (always zero — curls are solenoidal)
Linearity	$\nabla \times (a\mathbf{F} + b\mathbf{G}) = a(\nabla \times \mathbf{F}) + b(\nabla \times \mathbf{G})$
Product rule	$\nabla \times (f\mathbf{F}) = f(\nabla \times \mathbf{F}) + (\nabla f) \times \mathbf{F}$
Stokes' theorem	$\displaystyle\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}$

Applications of Curl

Field	Application	What Curl Represents
Electromagnetism	Faraday's Law	$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$ — changing magnetic fields create circulating electric fields
Electromagnetism	Ampere's Law	$\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$ — electric currents create circulating magnetic fields
Fluid Dynamics	Vorticity	$\boldsymbol{\omega} = \nabla \times \mathbf{v}$ — measures how the fluid spins locally
Mechanics	Angular velocity	For rigid body rotation $\mathbf{v} = \boldsymbol{\omega} \times \mathbf{r}$, the curl gives $2\boldsymbol{\omega}$
Conservative fields	Path independence	If $\nabla \times \mathbf{F} = 0$, line integrals are path-independent and a potential exists

How to Use the Curl Calculator

Choose dimension: Select 2D for fields F = ⟨P, Q⟩ (scalar curl) or 3D for F = ⟨P, Q, R⟩ (vector curl) using the toggle buttons.
Enter component functions: Type each component function (P, Q, and optionally R) using standard notation. Use ^ for exponents, * for multiplication, and functions like sin(x), cos(y), exp(x), ln(x), sqrt(x). Implicit multiplication is supported (e.g., 2x = 2*x).
Enter an evaluation point (optional): Provide comma-separated coordinates to evaluate the curl numerically and classify the rotation direction.
Click Compute Curl: View the symbolic curl, step-by-step cross product determinant expansion, numerical evaluation, and rotation classification.
Explore the visualization: For 2D fields, view the vector field arrows with a vorticity heat map (orange = counterclockwise, purple = clockwise) and animated particle flow.

Worked Example

Find the curl of $\mathbf{F}(x, y, z) = \langle y z,\; x z,\; x y \rangle$ at the point $(1, 2, 3)$:

Step 1: Write the determinant: $\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \partial_x & \partial_y & \partial_z \\ yz & xz & xy \end{vmatrix}$

Step 2: Expand: $\mathbf{i}(x - x) - \mathbf{j}(y - y) + \mathbf{k}(z - z) = \langle 0, 0, 0 \rangle$

Step 3: The curl is identically zero — this field is irrotational (conservative). In fact, $\mathbf{F} = \nabla(xyz)$, confirming a potential function exists.

Curl vs. Divergence

Property	Curl (∇×F)	Divergence (∇·F)
Operator type	Cross product with ∇	Dot product with ∇
Output	Vector (3D) / Scalar (2D)	Scalar
Measures	Rotation / circulation	Expansion / contraction
Zero means	Irrotational / conservative	Solenoidal / incompressible
Theorem	Stokes' theorem	Divergence (Gauss) theorem

FAQ

What is the curl of a vector field?

The curl of a vector field measures the rotation or circulation tendency at each point. For a 3D field F = (P, Q, R), the curl is a vector computed via the cross product of the nabla operator with F: ∇×F = (∂R/∂y − ∂Q/∂z, ∂P/∂z − ∂R/∂x, ∂Q/∂x − ∂P/∂y). Positive curl indicates counterclockwise rotation, negative indicates clockwise rotation.

How do you calculate curl?

To calculate the curl of a 3D vector field, set up a 3×3 determinant with unit vectors i, j, k in the first row, partial derivative operators ∂/∂x, ∂/∂y, ∂/∂z in the second row, and the field components P, Q, R in the third row. Expand the determinant using cofactor expansion along the first row to get each component of the curl vector.

What does it mean when curl is zero?

When the curl is zero everywhere, the vector field is called irrotational or conservative. This means a scalar potential function φ exists such that F = ∇φ, and the line integral of F around any closed loop is zero. Gravitational fields and electrostatic fields are examples of irrotational fields.

What is the difference between curl and divergence?

Curl measures rotation and produces a vector (in 3D), while divergence measures expansion or contraction and produces a scalar. Curl uses the cross product of nabla with F (∇×F), while divergence uses the dot product (∇·F). A field can have nonzero curl but zero divergence (like F = (−y, x, 0)), or zero curl but nonzero divergence (like F = (x, y, z)).

What is the physical meaning of curl?

Physically, curl measures the tendency of a vector field to circulate or rotate around a point. Imagine placing a tiny paddle wheel in a fluid flow — the curl tells you how fast and in which direction the paddle wheel would spin. In electromagnetism, curl appears in Faraday's law (changing magnetic fields create circulating electric fields) and Ampere's law (currents create circulating magnetic fields). In fluid dynamics, the curl of velocity is called vorticity.

Reference this content, page, or tool as:

"Curl Calculator" at https://MiniWebtool.com/curl-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-08

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