Dictionary Definition
curl
Noun
1 a round shape formed by a series of concentric
circles [syn: coil,
whorl, roll, curlicue, ringlet, gyre, scroll]
2 American chemist who with Richard Smalley and
Harold Kroto discovered fullerenes and opened a new branch of
chemistry (born in 1933) [syn: Robert Curl,
Robert F.
Curl, Robert
Floyd Curl Jr.]
Verb
2 shape one's body into a curl; "She curled
farther down under the covers"; "She fell and drew in" [syn:
curl up,
draw
in]
4 twist or roll into coils or ringlets; "curl my
hair, please" [syn: wave]
5 play the Scottish game of curling
User Contributed Dictionary
Pronunciation

 Rhymes: ɜː(r)l
Noun
 a piece or lock of curling hair; a ringlet
 a spin making the trajectory of an object curve
 : Any exercise performed by bending the arms or legs on the exertion, especially those that train the biceps.
 Movement of a moving rock away from a straight line
 Vector operator corresponding to the cross product of del and a given vectorial field.
Translations
piece or lock of curling hair; a ringlet
 Czech: kudrna
Verb
 to cause to curve
 to make into a curl
 A vector field denoting the rotation per unit area of a given vector field.
 To take part in curling
 I curl at my local club every weekend.
Extensive Definition
In vector
calculus, curl (or: rotor) 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.
"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.
The alternative terminology rotor and alternative
notation (used in many European countries) is
\operatorname(\mathbf) are often used for curl and
\operatorname(\mathbf).
Coordinateinvariant Definition as a Circulation Density
The component of \operatorname(\mathbf) in the direction of unit vector \mathbf is the limit of a line integral per unit area of \mathbf, namely the following integral over the closed curve \partial S^. This closed curve is in a plane normal to \mathbf:




 \mathbf_\cdot\operatorname(\mathbf) = \lim_ \frac \oint_ \mathbf \cdot d\mathbf




Now to calculate components of
\operatorname(\mathbf) for example in
Cartesian coordinates, replace \mathbf with unit vectors
i, j and k.
This defines not only the curl in a way free of
any coordinates, but makes also visible that it is a circulation
density.
Stokes's
theorem (see below) can directly be derived from it and the
representation in special coordinates can be explicitly
obtained.
Usage
In mathematics the curl is defined as:
 \operatorname(\mathbf) = \vec \times \vec
where F is the vector field to which the curl is
being applied. Although the version on the right is strictly an
abuse of
notation, it is still useful as a mnemonic if we take \nabla as a
vector differential
operator del or nabla. Such
notation involving operators
is common in physics and
algebra.
Expanded in
Cartesian coordinates, \vec \times \vec is, for F composed of
[Fx, Fy, Fz]:
 \begin
Although expressed in terms of coordinates, the
result is invariant under proper rotations of the coordinate axes
but the result inverts under reflection.
A simple representation of the expanded form of
the curl is:
 \begin
that is, del cross F, or
as the determinant
of the following matrix:
 \begin \mathbf & \mathbf & \mathbf \\ \\
where i, j, and k are the unit vectors
for the x, y, and zaxes, respectively.
In Einstein
notation, with the LeviCivita
symbol it is written as:
 (\vec \times \vec )_k = \epsilon_ \partial_\ell F_m
or as:
 (\vec \times \vec ) = \boldsymbol_k\epsilon_ \partial_\ell F_m
for unit vectors:\boldsymbol_k, k=1,2,3
corresponding to \boldsymbol, \boldsymbol, and \boldsymbol
respectively.
Using the exterior
derivative, it is written simply as:
 dF\,
Taking the exterior derivative of a vector field
does not result in another vector field, but a 2form or bivector field, properly
written as P\,(dx \wedge dy) + Q\,(dy \wedge dz) + R\,(dz \wedge
dx) .
Since bivectors are generally considered less
intuitive than ordinary vectors, the R³dual :*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 lefthanded and righthanded 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 righthand rule (four fingers of the right hand 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.
 \vec(x,y)=y\boldsymbolx\boldsymbol.
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 righthand
rule, we expect the curl to be into the page. If we are to keep
a
righthanded coordinate system, into the page will be in the
negative z direction.
If we do the math and find the curl:
 \vec \times \vec =0\boldsymbol+0\boldsymbol+ [(x)  y]\boldsymbol=2\boldsymbol
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: F(x,y)=x^2\boldsymbol.
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:
 \vec \times \vec =0\boldsymbol+0\boldsymbol+ (x^2) \boldsymbol=2x\boldsymbol
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.
Three common examples
Consider the example ∇ × [ v × F ]. Using Cartesian coordinates, it can be shown that
 \mathbf \left( \mathbf \right) = \left[ \left( \mathbf \right) + \mathbf \right] \mathbf \left[ \left( \mathbf \right) + \mathbf \right] \mathbf \ .
In the case where the vector field v and
∇ are interchanged:

 \mathbf \left( \mathbf \right) =\nabla_F \left( \mathbf \right)  \left( \mathbf \right) \mathbf \ ,
which introduces the Feynman subscript notation
∇F, which means the subscripted gradient operates on only
the factor F.
Another example is ∇ × [ ∇ ×
F ]. Using Cartesian coordinates, it can be shown that:

 \nabla \times \left( \mathbf \right) = \mathbf (\mathbf)  \nabla^2 \mathbf \ ,
which, with some headscratching, can be
construed as a special case of the first example with the
substitution v → ∇.
Descriptive examples
 In a tornado the winds are rotating about the eye, and a vector field showing wind velocities would have a nonzero 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 nonzero.
 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
References
 Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review
External links
curl in Bosnian: Rotor (matematika)
curl in Catalan: Rotacional
curl in Czech: Rotace (operátor)
curl in German: Rotation (Mathematik)
curl in Spanish: Rotacional
curl in Esperanto: Kirlo (matematiko)
curl in Persian: تاو
curl in French: Rotationnel
curl in Icelandic: Rót (virki)
curl in Italian: Rotore (matematica)
curl in Hebrew: רוטור
curl in Dutch: Rotatie (vectorveld)
curl in Japanese: 回転 (数学)
curl in Polish: Rotacja
curl in Portuguese: Rotacional
curl in Romanian: Rotor
curl in Russian: Ротор (математика)
curl in Slovak: Rotácia (operátor)
curl in Finnish: Roottori
curl in Swedish: Rotation (vektoranalys)
curl in Turkish: Rotasyonel
curl in Ukrainian: Ротор (математика)
curl in Chinese: 旋度
Synonyms, Antonyms and Related Words
arc,
arch, bend, bend back, bow, catacaustic, catenary, caustic, circle, cirrus, coil, conchoid, corkscrew, crimp, crisp, crook, curlicue, curve, decurve, deflect, diacaustic, dome, ellipse, embow, entwine, evolute, festoon, flex, frizz, frizzle, gyre, helix, hook, hump, hunch, hyperbola, incurvate, incurve, inflect, involute, kink, lituus, lock, loop, parabola, ponytail, recurve, reflect, reflex, retroflex, ringlet, roll, round, sag, screw, scroll, sinus, spiral, swag, sweep, swirl, tendril, tracery, turn, twine, twirl, twist, vault, volute, volution, vortex, whirl, whorl, wind, wreathe