Curl (mathematics)

In vector calculus, curl (also named: rotor) is a vector operator that shows a vector field's "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 $operatorname\{rot\}(mathbf\{F\})$ are often used (especially in many European countries) for curl and $operatorname\{curl\}(mathbf\{F\})$.

Definition

The curl of a vector field, denoted $operatorname\{curl\}(mathbf\{F\})$ or $vec\{nabla\}\; times\; vec\{F\}$, at a point is defined to be the limiting value of a closed line integral in a plane as the path the integral uses becomes infinitesimally close to the point, divided by the area enclosed. As such, the curl operator maps C¹ functions from R³ to R³ to C⁰ functions from R³ to R³.

Explicitly, curl is defined by:

$left(vec\{nabla\}\; times\; vec\{F\}right)\; cdot\; hat\{n\}\; overset\{underset\{mathrm\{def\}\}\{\}\}\{=\}\; lim\_\{A\; to\; 0\}\; frac\{oint\_\{C\}\; vec\{F\}\; cdot\; vec\{ds\}\}\{A\}$

where $hat\{n\}$ is a unit vector normal to the plane, $oint\_\{C\}\; vec\{F\}\; cdot\; vec\{ds\}$ is a line integral around the area in question, and A is the magnitude of the area. If $hat\{nu\}$ is an outward pointing normal to A (see caption at right) in the plane normal to $hat\{n\}$, then the orientation of C is chosen so that a vector $hat\{omega\}$ tangent to C is positively oriented if and only if $\{hat\{n\},hat\{nu\},hat\{omega\}\}$ forms a positively oriented basis for R³. This is essentially the right hand rule. If you point your right thumb in the direction of $hat\{n\}$ then C is traversed in the direction your fingers curl.

Usage

In practice, this definition is rarely used because in virtually all cases, the curl operator can be applied using some set of curvilinear coordinates, for which simpler representations have been derived.

Although the usage of $vec\{nabla\}\; times\; vec\{F\}$ is strictly an abuse of notation, it is still useful as a mnemonic in Cartesian coordinates 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 (see: Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations), $vec\{nabla\}\; times\; vec\{F\}$ is, for F composed of [F_x, F_y, F_z]:

{frac{partial}{partial x}} & {frac{partial}{partial y}} & {frac{partial}{partial z}} F_x & F_y & F_z end{vmatrix}

where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. This expands as follows:

$left(frac\{partial\; F\_z\}\{partial\; y\}\; -\; frac\{partial\; F\_y\}\{partial\; z\}right)\; mathbf\{i\}\; +\; left(frac\{partial\; F\_x\}\{partial\; z\}\; -\; frac\{partial\; F\_z\}\{partial\; x\}right)\; mathbf\{j\}\; +\; left(frac\{partial\; F\_y\}\{partial\; x\}\; -\; frac\{partial\; F\_x\}\{partial\; y\}right)\; mathbf\{k\}$

Although expressed in terms of coordinates, the result is invariant under proper rotations of the coordinate axes but the result inverts under reflection.

In Einstein notation, with the Levi-Civita symbol it is written as:

$(vec\{nabla\}\; times\; vec\{F\}\; )\_k\; =\; epsilon\_\{kell\; m\}\; partial\_ell\; F\_m$

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, the curl can be expressed as:

$vec\{nabla\}\; times\; vec\{F\}\; =\; left[star\; left(\{mathbf\; d\}\; F^flat\; right)\; right]^sharp$

Here $flat$ and $sharp$ are the musical isomorphisms, and $star$ is the Hodge dual. This formula shows how to calculate the curl of F in any coordinate system, and how to extend the curl to any oriented three dimensional Riemmannian manifold. Since this depends on a choice of orientation, curl is a chiral operation. In other words, if the orientation is reversed, then the direction of the curl is also reversed.

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.

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

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

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. The lack of x and y directions is analogous to the cross product operation.

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:

A more involved example

Suppose we now consider a slightly more complicated vector field:

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

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:

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

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:

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 identities

Consider the example ∇ × [v × F ]. 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 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).
• 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
• Vorticity
• Cross product
• Helmholtz decomposition

References

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

External links

• The idea of divergence and curl