Maths - Curl

Curl takes a vector field as an input and gives another vector field as an output. Curl is a measure of rotation where the which points away from the vortex according to the right-hand rule. So divergence tells us how much is going in or out of our infinitesimally small volume, curl tells how much things are going around the volume.

If the vector field represents velocity, then curl of this will be 2 times the angular velocity. This page shows this for a rotating solid object.

We have to be a bit careful about what we mean by rotation here, it is not just circular movement that will produce a non-zero curl value, imagine we have a fluid flowing through a pipe in such a way that it is flowing faster in the centre of the pipe than outside, this will also produce a non-zero curl value. To visualise this imagine an infinitesimally small paddle wheel in the fluid, if this would be rotated then curl is non-zero.

Another way to look at it is, curl tells you how much the rate of change of a function in one direction changes as you move in a different direction.

Curl is defined, in the 3D case (there are relatives to curl for more or less dimensions), as follows:

curl()=∇× =
vz - vy
∂y ∂z
vx - vz
∂z ∂x
vy - vx
∂x ∂y


∇× is an alternative way to denote curl(v), where:

Or in alternative notation:

curl(v) = i( vz - vy )+ k( vx - vz )+ j ( vy - vx )
∂y ∂z ∂z ∂x ∂x ∂y

Another way to write this is as a determinant as follows:

curl(v)=∇×v =
i j k
∂x vy vz


curl(s v) = s curl(v) + (curl(s)) × v

curl(v1× v2) = (v2∇) v1 - (v1∇) v2 + v1 div v2 + v2 div v1

curl(grad s) = 0

stokes's theorem


metadata block
see also:
Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.


cover Div, Grad, Curl and All That: An Informal Text on Vector Calculus

Other Math Books

Commercial Software Shop

Where I can, I have put links to Amazon for commercial software, not directly related to the software project, but related to the subject being discussed, click on the appropriate country flag to get more details of the software or to buy it from them.


This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2014 Martin John Baker - All rights reserved - privacy policy.