Curl of a vector field in Cartesian coordinates:

Curl of a vector field in cylindrical coordinates:

Rotational in two dimensions:

Use del to enter ∇, for the list of subscripted variables, and cross to enter :

Use delx to enter the template ∇_, fill in the variables, press , and fill in the function:

Rotational in polar coordinates:

In a curvilinear coordinate system, even a vector with constant components may have a nonzero curl:

Curl of a rank-2 tensor:

Curl specifying metric, coordinate system, and parameters:

Curl can produce higher-rank arrays:

This is a rank-4 array:

Curl works on curved spaces:

A vector field is called irrotational or conservative if it has zero curl:

Visually, this means that the vector field's stream lines do not tend to form small closed loops:

Analytically, it means the vector field can be expressed as the gradient of a scalar function. To find this function, parameterize a curve from the origin to an arbitrary point {x,y}:

The scalar function can be found using the line integral of v along the curve:

Verify the result:

A vector field is called central if it is spherically symmetric and only has a radial component:

All central vector fields are conservative or curl free:

This means that v is a gradient field. As v only has radial dependence, the line integral for the potential u reduces to a simple one-dimensional integral:

Verify the result:

A divergence-free vector field can be expressed as the curl of a vector potential:

To find the vector potential, one must solve the underdetermined system:

The first two equations are satisfied if and are constants, and the third has the obvious solution :

Curl produces arrays that are fully antisymmetric:

The curl of a gradient is zero:

Even for non-scalar inputs, the result is zero:

This identity is respected by the Inactive form of Grad:

In dimension , Curl is only defined for tensors of rank less than :

Curl is proportional to an antisymmetrized Grad followed by a call to HodgeDual:

The proportionality constant is , where r is the rank of f:

Compute Curl in a Euclidean coordinate chart c by transforming to and then back from Cartesian coordinates:

The result is the same as directly computing Curl[f,{x₁,…,x_n},c]:

In dimension , the curl of a scalar is a tensor of rank . Thus, for the result is a rank-2 tensor:

The curl of a tensor of rank is a scalar:

The double curl of a scalar field is the Laplacian of that scalar. In two dimensions:

The same result holds in three dimensions:

View expressions for the curl of a vector function in different coordinate systems: