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In vector calculus, 'curl' is a vector operator that shows a vector field's "rate of rotation", that is the direction of the axis of rotation and the magnitude of the rotation. It can also be described as the 'circulation density'.
In many European countries the operator is called 'rot' (short for rotor) instead of 'curl'.
"Rotation" and "circulation" are used here for properties of a vector function of position, regardless of their possible change in time.
A vector field which has zero curl everywhere is called irrotational.

Contents

Definition

Usage

Interpreting the curl

Examples

A simple vector field

A more involved example

Descriptive examples

See also

References

External links

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:
:$operatorname\{curl\}(mathbf\{F\})\; =\; lim\_\{V$
ightarrow 0} rac{1}{V} oint_{S} mathbf{n} imesmathbf{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 $mathbf\{hat\; u\}$:
:$mathbf\{hat\; u\}cdotoperatorname\{curl\}(mathbf\{F\})\; =\; lim\_\{S$
ightarrow 0} rac{1}{S} oint_{C} mathbf{F} cdot dmathbf{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
$operatorname\{curl\}(mathbf\{F\})$.

Usage

In mathematics the curl is noted by:
:
abla} imes ec{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 $$
abla as a vector differential operator del or Nabla. Such notation involving operators is common in physics and algebra.
Expanded in Cartesian coordinates,
abla} imes ec{F} is, for ''F'' composed of [''F''_''x'', ''F''_''y'', ''F''_''z'']:
:
{rac{partial F_z}{partial y}} - {rac{partial F_y}{partial z}} \ \
{rac{partial F_x}{partial z}} - {rac{partial F_z}{partial x}}\ \
{rac{partial F_y}{partial x}} - {rac{partial F_x}{partial y}}
end{bmatrix}
Although expressed in terms of coordinates, the result is invariant under proper rotations of the coordinate axes. However, the result inverses under reflection.
A simple way to remember the expanded form of the curl is to think of it as:
:
{rac{partial}{partial x}} \ \
{rac{partial}{partial y}} \ \
{rac{partial}{partial z}}
end{bmatrix} imes F
that is, del cross ''F'', or as the determinant of the following matrix:
:
{rac{partial}{partial x}} & {rac{partial}{partial y}} & {rac{partial}{partial z}} \
\ F_x & F_y & F_z end{bmatrix}
where 'i', 'j', and 'k' are the unit vectors for the ''x''-, ''y''-, and ''z''-axes, respectively.
In Einstein notation, with the Levi-Civita symbol it is written as:
:
abla} imes ec{F} )_k = epsilon_{kell m} partial_ell F_m
or as:
:
abla} imes ec{F} ) = oldsymbol{hat{e}}_kepsilon_{kell m} partial_ell F_m
for unit vectors:, k=1,2,3 corresponding to , and respectively.
Using the exterior derivative, it is written simply as:
:$dF,$
Note that taking the exterior derivative of a vector field does not result in another vector field, but a 2-form or bivector field, properly written as $P,(dx\; wedge\; dy)\; +\; Q,(dy\; wedge\; dz)\; +\; R,(dz\; wedge\; dx)$. However, since bivectors are generally considered less intuitive than ordinary vectors, the 'R'³-dual :$$
★ 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.

Interpreting the curl

The curl of vector field tells us about the rotation the field has at any point. The magnitude of the curl tells us how much rotation there is. The direction tells us, by the right-hand rule (four fingers are curled in the direction of the motion and the thumb points in the direction of the rotation) about which axis the field is rotating.
A commonly used device for thinking about curl is the paddle wheel. If we were to place a very small paddle wheel at a point in the vector field in question and treat the drawn vectors and their lengths as currents in a river with magnitude and direction, whichever way the paddle wheel would tend to turn is the direction of the curl at that point. For example, if two currents are trying to rotate the wheel in opposite directions, the stronger one (the longer vector) will win.

Examples

A simple vector field

Take the vector field
:.
Its plot looks like this:

Simply by visual inspection, we can see that the field is rotating. If we stick a paddle wheel anywhere, we see immediately its tendency to rotate clockwise. Using the right-hand rule, we expect the curl to be into the page. If we are to keep a right-handed coordinate system, into the page will be in the negative z direction.
If we do the math and find the curl:
:
abla} imes ec{F} =0oldsymbol{hat{x}}+0oldsymbol{hat{y}}+ [{rac{partial}{partial x}}(-x) -{rac{partial}{partial y}} y]oldsymbol{hat{z}}=-2oldsymbol{hat{z}}

Which is indeed in the negative z direction, as expected. In this case, the curl is actually a constant, irrespective of position. The "amount" of rotation in the above vector field is the same at any point (x,y). Plotting the curl of F isn't very interesting:

A more involved example

Suppose we now consider a slightly more complicated vector field:
:.
Its plot:

We might not see any rotation initially, but if we closely look at the right, we see a larger field at, say, x=4 than at x=3. Intuitively, if we placed a small paddle wheel there, the larger "current" on its right side would cause the paddlewheel to rotate clockwise, which corresponds to a curl in the negative z direction. By contrast, if we look at a point on the left and placed a small paddle wheel there, the larger "current" on its left side would cause the paddlewheel to rotate counterclockwise, which corresponds to a curl in the positive z direction. Let's check out our guess by doing the math:
:$$
abla imes F =0oldsymbol{hat{x}}+0oldsymbol{hat{y}}+ {rac{partial}{partial x}}(-x^2) oldsymbol{hat{z}}=-2xoldsymbol{hat{z}}

Indeed the curl is in the positive z direction for negative x and in the negative z direction for positive x, as expected. Since this curl is not the same at every point, its plot is a bit more interesting:

We note that the plot of this curl has no dependence on y or z (as it shouldn't) and is in the negative z direction for positive x and in the positive z direction for negative x.

Descriptive examples

★ 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.

See also

★ Del

★ Gradient

★ Divergence

★ Nabla in cylindrical and spherical coordinates

References

# Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, Theresa M. Korn; Korn, Granino Arthur, , , Dover Publications, ,

External links

★ The idea of divergence and curl