Curl Calculator

MiniWebtool’s Curl Calculator differentiates a 2D or 3D vector field to find ∇×F. Enter the components and an optional evaluation point to see symbolic steps, with a rotational-flow visualization for 2D fields.

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Help me understand MiniWebtool's Curl Calculator: https://miniwebtool.com/curl-calculator/. What can I use it for, what inputs or settings do I need, and how do I use it and understand the output? Walk me through one simple example. Use this page as your source and answer in English.

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Curl is the spin of a field: for F = ⟨−y, x⟩, ∂Q/∂x − ∂P/∂y = 2

Enter the components of a 2D or 3D vector field and an optional point to get the curl with the steps, and to see if the field is irrotational.

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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.

At a Glance

Curl Calculator computes the curl (∇×F) of a 2D or 3D vector field. You enter the component functions, and it returns the symbolic curl with steps and, optionally, the value at a chosen point.

Curl Calculator: features at a glance
Feature	What you get	Good for
2D and 3D fields	Choose 2D for F = <P, Q>, which gives a scalar curl, or 3D for F = <P, Q, R>, which gives a curl vector.	Calculus problems in the plane or in space
Function entry	Type components using ^ for powers and functions such as sin(x), cos(y), exp(x), ln(x) and sqrt(x). Implicit multiplication like 2x is supported.	Entering fields the way they are written in class
Cross product steps	Shows the determinant expansion step by step before giving the symbolic curl.	Checking each partial derivative
Evaluation point	Enter comma-separated coordinates, such as 1, 2, 3, to get a numeric value and a rotation classification.	Finding whether a field turns clockwise or counterclockwise at a point
Example fields	Buttons load six samples, including counterclockwise, clockwise and irrotational 2D fields and two 3D fields.	Trying the tool before typing your own field
2D visualization	For 2D fields, a vorticity heat map and an optional animated particle flow show rotation: orange for counterclockwise, purple for clockwise.	Seeing the rotational flow of a 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

Curl Calculator: 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

Curl Calculator: 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

Curl Calculator: 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

Curl Calculator: 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

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

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Gemini (copy & open)

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Help me understand MiniWebtool's Curl Calculator: https://miniwebtool.com/curl-calculator/. What can I use it for, what inputs or settings do I need, and how do I use it and understand the output? Walk me through one simple example. Use this page as your source and answer in English.

Curl Calculator

About Curl Calculator

At a Glance

What Is Curl?

Physical Meaning of Curl

Curl Formulas in Different Coordinate Systems

Important Identities Involving Curl

Applications of Curl

How to Use the Curl Calculator

Worked Example

Curl vs. Divergence

FAQ

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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 \(h_r=1, h_\theta=r, h_\phi=r\sin\theta\)

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	\(\displaystyle\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}\)

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