Curl

From Wikipedia, the free encyclopedia

For other uses, see Curl (disambiguation).

In vector calculus, curl is a vector operator that shows a vector field's rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. It can also be described as the circulation density.

"Rotation" and "circulation" are used here for properties of a vector function of position; they are not about changes with time.

A vector field which has zero curl everywhere is called irrotational.

1 Definition
2 Usage
3 Examples
4 See also
5 External links

[edit] Definition

The curl of a vector field $\mathbf{F}$ is defined as the limit of the ratio of the surface integral of the cross product of $\mathbf{F}$ with the normal $\mathbf{n}$ of closed surface $S$ , over a closed surface $S$ , to the volume $V$ enclosed by the surface $S$ , as the volume goes to zero:

$\operatorname{curl}(\mathbf{F}) = \lim_{V \rightarrow 0} \frac{1}{V} \oint_{S} \mathbf{n}\times\mathbf{F}\,dS$

More precisely, at each point $p$ in three dimensional space, $\operatorname{curl}(\mathbf{F})(p)$ is given by the above limit, where the closed surfaces $S$ all enclose $p$ and the diameter, not just the volume, of the region enclosed by $S$ tends to zero.

This definition isn't very useful, and following alternative equivalent definition gives better measures to calculate components of $\operatorname{curl}(\mathbf{F})$ .

The component of $\operatorname{curl}(\mathbf{F})$ in the direction of unit vector $\mathbf{\hat u}$ is the limit of a line integral per unit area of $\mathbf{F}$ over a closed curve C which encloses surface S, which is in a plane normal to $\mathbf{\hat u}$ :

$\mathbf{\hat u}\cdot\operatorname{curl}(\mathbf{F}) = \lim_{S \rightarrow 0} \frac{1}{S} \oint_{C} \mathbf{F} \cdot d\mathbf{l}$

Now to calculate components of $\operatorname{curl}(\mathbf{F})$ for example in Cartesian coordinates, replace $\mathbf{\hat u}$ with unit vectors i, j and k.

The alternative terminology ``rotor" and alternative notation $\operatorname{rot}(\mathbf{F})$ are often used for ``curl" and $\operatorname{curl}(\mathbf{F})$ .

[edit] Usage

In mathematics the curl is noted by:

$\operatorname{curl}(\mathbf{F}) = \nabla \times \mathbf{F}$

where F is the vector field to which the curl is being applied. Although the version on the right is simply an abuse of notation, it is still useful as a mnemonic if we take $\nabla$ as a vector differential operator del. Such notation involving operators is common in physics and algebra.

Expanded in Cartesian coordinates, $\nabla \times F$ is, for F composed of [F_x, F_y, F_z]:

$\begin{bmatrix} {\frac{\partial F_z}{\partial y}} - {\frac{\partial F_y}{\partial z}} \\ \\ {\frac{\partial F_x}{\partial z}} - {\frac{\partial F_z}{\partial x}}\\ \\ {\frac{\partial F_y}{\partial x}} - {\frac{\partial F_x}{\partial y}} \end{bmatrix}$

Although expressed in terms of coordinates, the result is invariant under proper rotations of the coordinate axes. However, the result inverses under reflection.

A simple way to remember the expanded form of the curl is to think of it as:

$\begin{bmatrix} {\frac{\partial}{\partial x}} \\ \\ {\frac{\partial}{\partial y}} \\ \\ {\frac{\partial}{\partial z}} \end{bmatrix} \times F$

that is, del cross F, or as the determinant of the following matrix:

$\begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \\ {\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}} \\ \\ F_x & F_y & F_z \end{bmatrix}$

where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively.

In Einstein notation, with the Levi-Civita symbol it is written as:

$(\nabla \times F)_k = \epsilon_{k\ell m} \partial_\ell F_m$

or as:

$(\nabla \times F) = \boldsymbol{\hat{e}}_k\epsilon_{k\ell m} \partial_\ell F_m$

for unit vectors: $\boldsymbol{\hat{e}}_k$ , k=1,2,3 corresponding to $\boldsymbol{\hat{x}}, \boldsymbol{\hat{y}}$ , and $\boldsymbol{\hat{z}}$ respectively.

Using the exterior derivative, it is written simply as:

$dF\,$

Note that taking the exterior derivative of a vector field does not result in another vector field, but a 2-form or bivector field, properly written as $P\,(dx \wedge dy) + Q\,(dy \wedge dz) + R\,(dz \wedge dx)$ . However, since bivectors are generally considered less intuitive than ordinary vectors, the R³-dual : $*dF\,$ is commonly used instead (where $*\,$ denotes the Hodge star operator). This is a chiral operation, producing a pseudovector that takes on opposite values in left-handed and right-handed coordinate systems.

[edit] Examples

If we were to place a very small paddle wheel or impeller into a turbulent liquid (described by a vector field), the curl of the field will tell us, for each point in the liquid, which way to point the impeller so as to get the fastest right-hand rotation.
In a tornado the winds are rotating about the eye, and a vector field showing wind velocities would have a non-zero curl at the eye, and possibly elsewhere (see vorticity).
In a vector field that describes the linear velocities of each individual part of a rotating disk, the curl will have a constant value on all parts of the disk.
If velocities of cars on a freeway were described with a vector field, and the lanes had different speed limits, the curl on the borders between lanes would be non-zero.
Faraday's law of induction, one of Maxwell's equations, can be expressed very simply using curl. It states that the curl of an electric field is equal to the opposite of the time rate of change of the magnetic field.