Curl
Details
- Curl is also known as rot, rotor, rotational and circulation density.
- Curl [f,x] can be input as ∇xf. The character ∇ can be typed as del or \[Del] , and the character can be typed as cross or \[Cross] . The list of variables x is entered as a subscript.
- An empty template ∇ can be entered as delx, and moves the cursor from the subscript to the main body.
- All quantities that do not explicitly depend on the variables given are taken to have zero partial derivative.
- In Curl [f,{x1,…,xn}], if f is an array with depth k<n, it must have dimensions {n,…,n}, and the resulting curl is an array with depth n-k-1 of dimensions {n,…,n}.
- If f is a scalar, Curl [f,{x1,…,xn},chart] returns an array of depth n-1 in the orthonormal basis associated with chart.
- In Curl [f,{x1,…,xn},chart], if f is an array, the components of f are interpreted as being in the orthonormal basis associated with chart.
- For coordinate charts on Euclidean space, Curl [f,{x1,…,xn},chart] can be computed by transforming f to Cartesian coordinates, computing the ordinary curl and transforming back to chart. »
- Coordinate charts in the third argument of Curl can be specified as triples {coordsys,metric,dim} in the same way as in the first argument of CoordinateChartData . The short form in which dim is omitted may be used.
- Curl works with SparseArray and structured array objects.
Examples
open all close allBasic Examples (4)
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:
Scope (6)
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:
Applications (3)
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 :
Properties & Relations (7)
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,…,xn},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:
Interactive Examples (1)
View expressions for the curl of a vector function in different coordinate systems:
See Also
Grad Div Laplacian CoordinateChartData Cross HodgeDual D DSolve NDSolve NDEigensystem NDEigenvalues
Characters: \[Del]
Tech Notes
Related Guides
Text
Wolfram Research (2012), Curl, Wolfram Language function, https://reference.wolfram.com/language/ref/Curl.html (updated 2014).
CMS
Wolfram Language. 2012. "Curl." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/Curl.html.
APA
Wolfram Language. (2012). Curl. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Curl.html
BibTeX
@misc{reference.wolfram_2025_curl, author="Wolfram Research", title="{Curl}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/Curl.html}", note=[Accessed: 16-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_curl, organization={Wolfram Research}, title={Curl}, year={2014}, url={https://reference.wolfram.com/language/ref/Curl.html}, note=[Accessed: 16-November-2025]}