# curl

Curl of vector field

## Syntax

``curl(V,X)``
``curl(V)``

## Description

example

````curl(V,X)` returns the curl of the vector field `V` with respect to the vector `X`. The vector field `V` and the vector `X` are both three-dimensional.```
````curl(V)` returns the curl of the vector field `V` with respect to the vector of variables returned by `symvar(V,3)`.```

## Examples

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Compute the curl of this vector field with respect to vector X = (x, y, z) in Cartesian coordinates.

```syms x y z V = [x^3*y^2*z, y^3*z^2*x, z^3*x^2*y]; X = [x y z]; curl(V,X)```
```ans = x^2*z^3 - 2*x*y^3*z x^3*y^2 - 2*x*y*z^3 - 2*x^3*y*z + y^3*z^2```

Compute the curl of the gradient of this scalar function. The curl of the gradient of any scalar function is the vector of 0s.

```syms x y z f = x^2 + y^2 + z^2; vars = [x y z]; curl(gradient(f,vars),vars)```
```ans = 0 0 0```

The vector Laplacian of a vector field V is defined as follows.

`${\nabla }^{2}V=\nabla \left(\nabla \cdot V\right)-\nabla ×\left(\nabla ×V\right)$`

Compute the vector Laplacian of this vector field using the `curl`, `divergence`, and `gradient` functions.

```syms x y z V = [x^2*y, y^2*z, z^2*x]; vars = [x y z]; gradient(divergence(V,vars)) - curl(curl(V,vars),vars)```
```ans = 2*y 2*z 2*x```

## Input Arguments

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Input, specified as a three-dimensional vector of symbolic expressions or symbolic functions.

Variables, specified as a vector of three variables

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### Curl of a Vector Field

The curl of the vector field V = (V1, V2, V3) with respect to the vector X = (X1, X2, X3) in Cartesian coordinates is this vector.

`$curl\left(V\right)=\nabla ×V=\left(\begin{array}{c}\frac{\partial {V}_{3}}{\partial {X}_{2}}-\frac{\partial {V}_{2}}{\partial {X}_{3}}\\ \frac{\partial {V}_{1}}{\partial {X}_{3}}-\frac{\partial {V}_{3}}{\partial {X}_{1}}\\ \frac{\partial {V}_{2}}{\partial {X}_{1}}-\frac{\partial {V}_{1}}{\partial {X}_{2}}\end{array}\right)$`