Curl

For other uses, see Curl (disambiguation).

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. Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner A vector operator is a type of Differential operator used in Vector calculus. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation 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 It can also be described as the circulation density. In Fluid dynamics, circulation is the Line 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 a conservative vector field is a Vector field which is the Gradient of a Scalar potential.

The alternative terminology rotor and alternative notation (used in many European countries) is $\operatorname{rot}(\mathbf{F})$ are often used for curl and $\operatorname{curl}(\mathbf{F})$ .

1 Coordinate-invariant Definition as a Circulation Density
2 Usage
3 Interpreting the curl
4 Examples
5 See also
6 References
7 External links

Coordinate-invariant Definition as a Circulation Density

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 $\mathbf{F}$ , namely the following integral over the closed curve $\partial S^{(2)}$ . In Mathematics, a line integral (sometimes called a path integral or curve integral) is an Integral where the function to be integrated This closed curve is in a plane normal to $\mathbf{\hat u}$ :

$\mathbf{\hat u}_{|\,(\,S^{(2)}\perp\mathbf \hat u\,)}\cdot\operatorname{curl}(\mathbf{F}) = \lim_{S^{(2)} \rightarrow 0} \frac{1}{|S^{(2)}|} \oint_{\partial S^{(2)}} \mathbf{F} \cdot d\mathbf{l}$

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. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length

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. In Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from

Usage

In mathematics the curl is defined as:

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

where F is the vector field to which the curl is being applied. Although the version on the right is strictly an abuse of notation, it is still useful as a mnemonic if we take $\nabla$ as a vector differential operator del or nabla. 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 A mnemonic device (nəˈmɒnɪk is a Memory aid Commonly met mnemonics are often verbal something such as a very short poem or a special word used to help a person remember In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator &nablaDel Nabla is the Symbol \nabla The name comes from the Greek word for a Hebrew Harp, which had a similar shape Such notation involving operators is common in physics and algebra. In physics an operator is a function acting on the space of Physical states As a resultof its application on a physical state another physical state is obtained Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity.

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 but the result inverts under reflection. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

A simple representation of the expanded form of the curl is:

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

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 Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length

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. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational The Levi-Civita symbol, also called the Permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in Tensor

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 2-form or bivector field, properly written as $P\,(dx \wedge dy) + Q\,(dy \wedge dz) + R\,(dz \wedge dx)$ . In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is In Differential geometry, a p -vector is the Tensor obtained by taking Linear combinations of the Wedge product of p

Since bivectors are generally considered less intuitive than ordinary vectors, the R³-dual : $*dF\,$ is commonly used instead (where $*\,$ denotes the Hodge star operator). In Mathematics, the Hodge star operator or Hodge dual is a significant Linear map introduced in general by W This is a chiral operation, producing a pseudovector that takes on opposite values in left-handed and right-handed coordinate systems. In Physics and Mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation but gains an In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point

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 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. For the related yet different principle relating to electromagnetic coils see Right hand grip rule.

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 constructed using unit vectors

$\vec{F}(x,y)=y\boldsymbol{\hat{x}}-x\boldsymbol{\hat{y}}$ . In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length

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. For the related yet different principle relating to electromagnetic coils see Right hand grip rule. If we are to keep a right-handed coordinate system, into the page will be in the negative z direction. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

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:

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

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.

Three common examples

Main article: Vector calculus identities

Consider the example ∇ × [ v × F ]. The following identities are important in Vector calculus: Single operators (summary This section explicitly lists what some symbols mean for clarity Using Cartesian coordinates, it can be shown that

$\mathbf{ \nabla \times} \left( \mathbf{v \times F} \right) = \left[ \left( \mathbf{ \nabla \cdot F } \right) + \mathbf{F \cdot \nabla} \right] \mathbf{v}- \left[ \left( \mathbf{ \nabla \cdot v } \right) + \mathbf{v \cdot \nabla} \right] \mathbf{F} \ .$

In the case where the vector field v and ∇ are interchanged:

$\mathbf{v \ \times } \left( \mathbf{ \nabla \times F} \right) =\nabla_F \left( \mathbf{v \cdot F } \right) - \left( \mathbf{v \cdot \nabla } \right) \mathbf{ F} \ ,$

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:

$\nabla \times \left( \mathbf{\nabla \times F} \right) = \mathbf{\nabla} (\mathbf{\nabla \cdot F}) - \nabla^2 \mathbf{F} \ ,$

which, with some head-scratching, 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 non-zero curl at the eye, and possibly elsewhere (see vorticity). A tornado is a violent rotating column of air which is in contact with both the surface of the earth and a Cumulonimbus cloud or in rare cases the base of a Cumulus Vorticity is a mathematical concept used in Fluid dynamics. It can be related to the amount of " circulation " or "rotation" (or more strictly the
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. A mathematical constant is a number usually a Real number, that arises naturally in Mathematics.
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. Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric 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. &nablaDel In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the This is a list of some Vector calculus formulae of general use in working with various Coordinate systems Table Vorticity is a mathematical concept used in Fluid dynamics. It can be related to the amount of " circulation " or "rotation" (or more strictly the In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which In Mathematics, in the area of Vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently 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

Dictionary

-noun

a piece or lock of curling hair; a ringlet
a spin making the trajectory of an object curve
(weightlifting): Any exercise performed by bending the arms or legs on the exertion, especially those that train the biceps.
(curling) Movement of a moving rock away from a straight line
(mathematics) Vector operator corresponding to the cross product of del and a given vectorial field.

-verb

to cause to curve
to make into a curl
(calculus) A vector field denoting the rotation per unit area of a given vector field.
To take part in curling

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