NationMaster - Encyclopedia: 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. Look up Curl in Wiktionary, the free dictionary. ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... A vector operator is a type of differential operator used in vector calculus. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... A sphere rotating around its axis. ... The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs. ... In fluid dynamics, circulation is the path integral around a closed curve of the fluid velocity. ...

"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. In vector calculus, an irrotational or conservative vector field is a vector field whose curl is zero. ...

1 Definition
2 Usage
3 Interpreting the curl
4 Examples
5 See also
6 References
7 External links

Definition

Parts of this article may be confusing or unclear.
Please help clarify the article. Suggestions may be on the talk page. (September 2007)

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: Image File history File links Broom_icon. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... In mathematics, a surface integral is a definite integral taken over some surface that may be a curved set in space; it can be thought of as the double integral analog of the path integral. ... A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ...

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

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 $mathbf{hat u}$ : This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ...

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

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. Fig. ... In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...

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. In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition (while being unlikely to introduce errors or cause confusion). ... For other uses, see Mnemonic (disambiguation). ... In mathematics, a differential operator is an operator defined as a function of the differentiation operator. ... In vector calculus, del is a vector differential operator represented by the nabla symbol: âˆ‡. Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember. ... Nabla is a symbol, shown as . ... In mathematical formulations of quantum mechanics, an operator is a linear transformation from a Hilbert space to itself. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... This article is about the branch of mathematics. ...

Expanded in Cartesian coordinates, $vec{nabla} times vec{F}$ is, for F composed of [F_x, F_y, F_z]: Fig. ...

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

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: For the cross product in algebraic topology, see KÃ¼nneth theorem. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

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

where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...

In Einstein notation, with the Levi-Civita symbol it is written as: This article or section does not adequately cite its references or sources. ... The Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in tensor calculus. ...

$(vec{nabla} times vec{F} )_k = epsilon_{kell m} partial_ell F_m$

or as:

$(vec{nabla} times vec{F} ) = boldsymbol{hat{e}}_kepsilon_{kell m} partial_ell F_m$

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: In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...

$dF,$

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. A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... A bivector is an element of the antisymmetric tensor product of a tangent space with itself. ... Look up Dual in Wiktionary, the free dictionary A dual is a pair or a grouping of two. ... In mathematics, the Hodge star operator on an oriented inner product space V is a linear operator on the exterior algebra of V, interchanging the subspaces of k-vectors and n−k-vectors where n = dim V, for 0 ≤ k ≤ n. ... In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion followed by a proper rotation). ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ...

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. The left-handed orientation is shown on the left, and the right-handed on the right. ...

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 Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

$vec{F}(x,y)=yboldsymbol{hat{x}}-xboldsymbol{hat{y}}$ .

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. Image File history File links Uniform_curl. ... The left-handed orientation is shown on the left, and the right-handed on the right. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

If we do the math and find the curl:

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

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:

Image File history File links No higher resolution available. ...

A more involved example

Suppose we now consider a slightly more complicated vector field:

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

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: Image File history File links No higher resolution available. ...

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

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. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

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.

This article is about the weather phenomenon. ... Vorticity is a mathematical concept used in fluid dynamics. ... A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ... Faradays law of induction (more generally, the law of electromagnetic induction) states that the induced emf (electromotive force) in a closed loop equals the negative of the time rate of change of magnetic flux through the loop. ... For thermodynamic relations, see Maxwell relations. ...

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

Categories: Cleanup from September 2007 | Wikipedia articles needing clarification | Vector calculus

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