Results for curl

On this page:

Dictionary:

curl

(kûrl)

v., curled, curl·ing, curls.

v.tr.

To twist (the hair, for example) into ringlets or coils.
To form into a coiled or spiral shape: curled the ends of the ribbon.
To decorate with coiled or spiral shapes.
To raise and turn under (the upper lip), as in snarling or showing scorn.
Sports. To lift (a weight) by performing a curl.

v.intr.

To form ringlets or coils.
To assume a spiral or curved shape.
To move in a curve or spiral: The wave curled over the surfer.
Sports. To engage in curling.

Something with a spiral or coiled shape.
A coil or ringlet of hair.
A treatment in which the hair is curled.
1. The act of curling: the curl of a meandering river.
2. The state of being curled.
Sports. A weightlifting exercise using one or two hands, in which a weight held at the thigh or to the side of the body is raised to the chest or shoulder and then lowered without moving the upper arms, shoulders, or back.
Any of various plant diseases in which the leaves roll up.

phrasal verb:

curl up

To assume a position with the legs drawn up: The child curled up in an armchair to read.

[Middle English crullen, curlen, from crulle, curly, perhaps of Middle Low German origin.]

#SearchBtn{margin-top:14px !important;}

Library

Computer Desktop Encyclopedia: Curl

(1) A programming environment for developing rich Internet applications (RIAs) from Curl, Inc., Cambridge, MA (www.curl.com). Conceived at MIT, Curl combines HTML markup with an object-oriented programming language. The user's machine requires the Curl runtime engine and browser plug-in for execution. The first Curl implementation was released in 2002. See RIA.

(2) (cURL) A command line utility for executing functions with URL-oriented protocols such as FTP and HTTP. Pronounced "C-URL," there are versions for Unix, Linux, Windows, Mac and other operating systems. For more information, visit http://curl.haxx.se.

Marketing Dictionary: curl

Rippling effect on paper caused inadvertently by exposure to moisture or by coating one side of the paper. See also coated paper.

Thesaurus: curl

verb

To have or cause to have a curved or sinuous form or surface: curve, undulate, wave. See straight/bent.
To move or proceed on a repeatedly curving course: coil, corkscrew, entwine, meander, snake, spiral, twine, twist, weave, wind, wreathe. See repetition, straight/bent.

Antonyms: curl

Definition: loop, ringlet, curve
Antonyms: line

v

Definition: bend, loop
Antonyms: straighten

Britannica Concise Encyclopedia: curl

In mathematics, a differential operator that can be applied to a vector-valued function (or vector field) in order to measure its degree of local spinning. It consists of a combination of the function's first partial derivatives. One of the more common forms for expressing it is:

in which v is the vector field (v₁, v₂, v₃), and v₁, v₂, v₃ are functions of the variables x, y, and z, and i, j, and k are unit vectors in the positive x, y, and z directions, respectively. In fluid mechanics, the curl of the fluid velocity field (i.e., vector velocity field of the fluid itself) is called the vorticity or the rotation because it measures the field's degree of rotation around a given point.

For more information on curl, visit Britannica.com.

Architecture: curl

A winding, swirling, or circling in the grain of wood, usually obtained from the crotch or fork of a tree; also see fiddleback.

Word Tutor: curl

IN BRIEF: To form into or grow in coils or ringlets.

After wrapping the present, I took the time to curl each ribbon.

Wikipedia: curl

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.

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 $\operatorname{curl}(\mathbf{F})$ .

Usage

In mathematics the curl is noted by:

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

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 or nabla. Such notation involving operators is common in physics and algebra.

Expanded in Cartesian coordinates, $\vec{\nabla} \times \vec{F}$ 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:

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

or as:

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

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.

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

$\vec{F}(x,y)=y\boldsymbol{\hat{x}}-x\boldsymbol{\hat{y}}$ .

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:

$\vec{\nabla} \times \vec{F} =0\boldsymbol{\hat{x}}+0\boldsymbol{\hat{y}}+ [{\frac{\partial}{\partial x}}(-x) -{\frac{\partial}{\partial y}} y]\boldsymbol{\hat{z}}=-2\boldsymbol{\hat{z}}$

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:

$F(x,y)=-x^2\boldsymbol{\hat{y}}$ .

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:

$\nabla \times F =0\boldsymbol{\hat{x}}+0\boldsymbol{\hat{y}}+ {\frac{\partial}{\partial x}}(-x^2) \boldsymbol{\hat{z}}=-2x\boldsymbol{\hat{z}}$

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:

Curl of F with the x=0 plane emphasized in dark blue

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.

References

Theresa M. Korn; Korn, Granino Arthur. Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. New York: Dover Publications, 157-160. ISBN 0-486-41147-8.

External links

The idea of divergence and curl

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

Translations: Curl

Dansk (Danish)
v. tr. - krølle, sno, fortrække i et hånligt smil
v. intr. - krølle, kruse sig, spille curling, sno sig, danne spiraler, fortrække sig i et hånligt smil
n. - krølle, snoning, krusning, krøl, krøllesyge

idioms:

curl into fortrække sig i
curl up sidde med benene trukket op under sig, krumme sig sammen

Nederlands (Dutch)
(om-/op)krullen, kringelen, curling spelen, (haar)krul, krullende beweging/handeling, krulziekte

Français (French)
v. tr. - friser, saisir (qch), se pelotonner, s'enrouler autour de qch, faire une moue dédaigneuse
v. intr. - friser, se gondoler, se racornir, monter en volutes (fumée), s'enrouler autour, faire une moue dédaigneuse
n. - boucle, anneau, copeau, volute, moue dédaigneuse

idioms:

curl into a ball s'enrouler en boule
curl up se pelotonner, se mettre en rond, se gondoler, se racornir, gondoler, racornir, se recroqueviller

Deutsch (German)
v. - sich locken, sich wellen, kräuseln, locken
n. - Locke

idioms:

curl into a ball zu Kuglen oder Ballen formen
curl up sich zusammenkugeln

Ελληνική (Greek)
v. - κατσαρώνω, σγουραίνω, συστρέφω/-ομαι, περιελίσσω/-ομαι, (καθομ.) μουτρώνω, στραβομουτσουνιάζω
n. - μπούκλα, βόστρυχος, τολύπη, τούφα, σπείρωμα, ελίκωση, (μτφ.) στραβομουτσούνιασμα

idioms:

curl into κουλουριάζομαι, συστρέφομαι σε (π.χ. μπάλα)
curl up κουλουριάζω/-ομαι, (καθομ.) χουζουρεύω (σε πολυθρόνα κ.λπ.), καταρρέω

Italiano (Italian)
arricciarsi, serpeggiare, increspare, arricciare, ricciolo

idioms:

curl into raggomitolarsi in
curl up raggomitolarsi

Português (Portuguese)
v. - enrolar
n. - cacho (m)

idioms:

curl into enrolar
curl up enrolar-se, fraquejar (gír.)

Русский (Russian)
виться, сворачиваться, завивать, локон, завиток

idioms:

curl into свернуться
curl up свернуться в клубок, уютно устроиться

Español (Spanish)
v. tr. - rizar, ensortijar
v. intr. - arrollarse, fruncirse, encresparse, ensortijarse, rizarse, hacer rollos, bucles, o rizos
n. - rizo, bucle

idioms:

curl into a ball aovillarse, hacerse un ovillo
curl up enroscarse, acurrucarse, repantigarse, ondularse

Svenska (Swedish)
v. - krulla, krusa, kröka, förvrida, sno, slå knorr på, falla ihop/till föga (vard.), spela curling
n. - hårlock, ring, krusning, lockighet, bladrullsjuka (hos potatis)

中文（简体） (Chinese (Simplified))
使卷曲, 使卷起来, 缠绕, 卷曲, 缭绕, 卷毛, 卷发, 螺旋状物

idioms:

curl into 卷起来
curl up 卷起

中文（繁體） (Chinese (Traditional))
v. tr. - 使捲曲, 使卷起來, 纏繞
v. intr. - 捲曲, 繚繞
n. - 卷毛, 捲髮, 捲曲, 螺旋狀物

idioms:

curl into 捲起來
curl up 捲起

한국어 (Korean)
v. tr. - 곱슬곱슬하게 하다, 물결이 일게 하다
v. intr. - 곱슬곱슬해지다, 비틀리다, 꽁무니를 빼다
n. - 고수머리, 소용돌이 , 비틀림

idioms:

curl into 몸을 동그랗게 말다
curl up 감아올리다, 쓰러뜨리다

日本語 (Japanese)
n. - カール, 巻毛の頭髪, らせん状のもの, 巻きひげ, カーリング病, 回転, カールすること
v. - カールさせる, ひねる, カールする, 渦巻く, しりごみする

idioms:

curl into 曲がりくねる
curl up 端から巻き上げる, 倒す, 吐き気を催させる

العربيه (Arabic)
‏(فعل) جعد, تجعد, تكور, انطوى (الاسم) تجعيدة‏

עברית (Hebrew)
v. tr. - ‮סלסל, עיוות (בבוז את השפה העליונה)‬
v. intr. - ‮התאבך, הסתלסל‬
n. - ‮תלתל, סליל, סלסול, מחלת התכדרות העלים בצמחים‬

If you are unable to view some languages clearly, click here.

To select your translation preferences click here.

Best of the Web: curl

Some good "curl" pages on the web:

Math
mathworld.wolfram.com

Shopping: curl

curl bar	Ez Curl Bar Weight
to curl hair	Bumble And Bumble Curl

Join the WikiAnswers Q&A community. Post a question or answer questions about "curl" at WikiAnswers.

Copyrights:

		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved. Read more
		Computer Desktop Encyclopedia. THIS COPYRIGHTED DEFINITION IS FOR PERSONAL USE ONLY. All other reproduction is strictly prohibited without permission from the publisher. © 1981-2008 Computer Language Company Inc. All rights reserved. Read more
		Marketing Dictionary. Dictionary of Marketing Terms. Copyright © 2000 by Barron's Educational Series, Inc. All rights reserved. Read more
		Thesaurus. Roget's II: The New Thesaurus, Third Edition by the Editors of the American Heritage® Dictionary Copyright © 1995 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved. Read more
		Antonyms. © 1999-2008 by Answers Corporation. All rights reserved. Read more
		Britannica Concise Encyclopedia. Britannica Concise Encyclopedia. © 2006 Encyclopædia Britannica, Inc. All rights reserved. Read more
		Architecture. McGraw-Hill Dictionary of Architecture and Construction. Copyright © 2003 by McGraw-Hill Companies, Inc. All rights reserved. Read more
		Word Tutor. Copyright © 2004-present by eSpindle Learning, a 501(c) nonprofit organization. All rights reserved. eSpindle provides personalized spelling and vocabulary tutoring online; free trial. Read more
		Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Curl". Read more
		Translations. Copyright © 2007, WizCom Technologies Ltd. All rights reserved. Read more

On this page: Print Link

	Where is the oil plug on 1986 Yamaha Virago?
	How do you file interstate unemployment online?
	What is the best operating system and why?
	What affect does audience fragmentation have on media costs?
	How do you set the clock on a 1986 Dodge 600?

curl

Definition

Usage

Interpreting the curl

Examples

A simple vector field

A more involved example

Descriptive examples

See also

References

External links

Keep Reading

Do you have the answers?