curl

Curl of vector field

Syntax

curl(V,X)

curl(V)

Description

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

Compute Curl of Vector Field

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

Show Curl of Gradient of Scalar Function is Zero

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

Compute Vector Laplacian of Vector Field

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

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

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

V — Input

three-dimensional symbolic vector

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

X — Variables

vector of three variables

Variables, specified as a vector of three variables

More About

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.

$$curl(V)=\nabla \times V=\left(\begin{array}{c}\frac{\partial V_3}{\partial X_2}-\frac{\partial V_2}{\partial X_3}\\ \frac{\partial V_1}{\partial X_3}-\frac{\partial V_3}{\partial X_1}\\ \frac{\partial V_2}{\partial X_1}-\frac{\partial V_1}{\partial X_2}\end{array}\right)$$