# Curl

 $\frac{d}{dx} \sin x=?\,$ This article/section deals with mathematical concepts appropriate for a student in late high school or early university.

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[1] as the limiting value of rotation per unit area. Written explicitly,

$(\nabla \times \mathbf{F}) \cdot \mathbf{\hat{n}} \equiv \lim_{A \to 0} \frac{\oint_{C} \mathbf{F} \cdot d\mathbf{s}}{A}$

where $\mathbf{\hat{n}}$ is the unit normal vector of the area element A and C is the boundary of the area element ($C =\partial A$). The right side of the definition is a path integral along the boundary of the area element A, which is allow to shrink in the limiting process.

## Cartesian coordinates

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

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

$\nabla \times \vec V = (\ \ \frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z},\ \ \ \ \frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x},\ \ \ \ \frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y}\ \ )$

or, using a notation like the determinant notation for cross product,

$\nabla \times \vec V = \begin{vmatrix} \hat x & \hat y & \hat z \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ V_x & V_y & V_z \end{vmatrix}$

where $\hat x$, $\hat y$, and $\hat z$ are the unit basis vectors.

If one thinks of $\nabla$ as being a fictional vector field with components $(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})$, 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—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 is an extremely important operation in physics, mathematics, and engineering. It is perhaps most famous for its appearance in Maxwell's Equations.

Intuitively, the curl measures the degree to which the vector field rotates around a given point. If you were to measure the curl of the vector field of wind speed in the vicinity of a meteorological low pressure area, then, keeping in mind that the Coriolis force makes the wind move in a counterclockwise vortex in the Northern hemisphere, it would be a vector pointing upward. (The reason for the Coriolis force is not important here, we're just talking about the observation that the air moves in a counterclockwise vortex.) The way to see that the curl points upward is to visualize a giant right hand with the fingers curled counterclockwise—the thumb would point upward.

Vector fields with a curl of zero are called irrotational.

## References

1. Curl at Wolfram Mathworld