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{{math-h}}

The '''curl''' is a type of [[derivative]] of a [[vector]] that corresponds to its rate of rotation in three-dimensional space.  It is particularly important in electromagnetism and fluid mechanics.  The curl of a vector field is another vector field.

The '''curl''' is a type of [[derivative]] of a [[vector field]] that corresponds to its rate of rotation in three-dimensional space.  It is particularly important in electromagnetism and fluid mechanics.  The curl of a vector field is another vector field.

More precisely, it is defined<ref>[http://mathworld.wolfram.com/Curl.html Curl] at Wolfram Mathworld</ref> as the limiting value of rotation per unit area.  Written explicitly,

More precisely, it is defined<ref>[http://mathworld.wolfram.com/Curl.html Curl] at Wolfram Mathworld</ref> as the limiting value of rotation per unit area.  Written explicitly,

==Cartesian coordinates==

==Cartesian coordinates==

In [[Cartesian coordinates]], the curl is written as though it were the [[cross product]] of the special symbol "<math>\nabla</math>" (which is commonly called "del" or "nabla"), with the given vector field, like this: <math>\nabla \times \vec V</math>.  This is usually pronounced "curl V" or "del cross V".

In [[Cartesian coordinates]], the curl is written as though it were the [[cross product]] of the special symbol "<math>\nabla</math>" (which is commonly called "del" or "nabla"), with the given vector field, like this: <math>\nabla \times \vec V</math>.  This is usually pronounced "curl V" or "del cross V".

In 3-D [[Cartesian coordinates]], the curl is calculated as:

In 3-D [[Cartesian coordinates]], the curl is calculated as:

If one thinks of <math>\nabla</math> as being a fictional vector field with components <math>(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})</math>, one can sort of see that the cross product notation makes sense.  This is also useful for remembering how to calculate a curl.

If one thinks of <math>\nabla</math> as being a fictional vector field with components <math>(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})</math>, one can sort of see that the cross product notation makes sense.  This is also useful for remembering how to calculate a curl.

The curl is a true vector field operation&mdash;the result is independent of the coordinate system that is used.  The proof of that, and its ramifications, are beyond the scope of this page.

The curl is a true vector field operation&mdash;the result is independent of the coordinate system that is used.

The curl operation has an intrinsic "handedness" to it.  Any physical phenomenon described by the curl operation (for example, magnetic fields), involves some kind of "right-hand rule".

The curl operation has an intrinsic "handedness" to it.  Any physical phenomenon described by the curl operation (for example, magnetic fields), involves some kind of "right-hand rule."

The curl is an extremely important operation in physics, mathematics, and engineering.  It is perhaps most famous for its appearance in [[Maxwell's Equations]].

The curl is an extremely important operation in physics, mathematics, and engineering.  It is perhaps most famous for its appearance in [[Maxwell's Equations]].