Editing Curl (mathematics) (section)

==Examples==

=== Example 1 ===
Suppose the vector field describes the [[velocity field]] of a [[fluid flow]] (such as a large tank of [[liquid]] or [[gas]]) and a small ball is located within the fluid or gas (the center of the ball being fixed at a certain point). If the ball has a rough surface, the fluid flowing past it will make it rotate. The rotation axis (oriented according to the right hand rule) points in the direction of the curl of the field at the center of the ball, and the angular speed of the rotation is half the magnitude of the curl at this point.<ref>{{citation|first1=Josiah Willard | last1=Gibbs|author-link1=Josiah Willard Gibbs| first2=Edwin Bidwell|last2=Wilson| author-link2=Edwin Bidwell Wilson | title=Vector analysis |series=Yale bicentennial publications| url = http://hdl.handle.net/2027/mdp.39015000962285?urlappend=%3Bseq=179 | year=1901 | publisher=C. Scribner's Sons | hdl=2027/mdp.39015000962285?urlappend=%3Bseq=179}}</ref>
The curl of the vector field at any point is given by the rotation of an infinitesimal area in the ''xy''-plane (for ''z''-axis component of the curl), ''zx''-plane (for ''y''-axis component of the curl) and ''yz''-plane (for ''x''-axis component of the curl vector). This can be seen in the examples below.

=== Example 2 ===
{{Multiple image
| align             = 
| direction         = 
| total_width       = 400
| image1            = Uniform curl.svg
| image2            = Curl of uniform curl.png
| alt1              = 
| caption1          = 
| caption2          = 
| footer            = Vector field {{math|1='''F'''(''x'',''y'')=[''y'',−''x'']}} (left) and its curl (right).
}}

The [[vector field]]
<math display="block">\mathbf{F}(x,y,z)=y\boldsymbol{\hat{\imath}}-x\boldsymbol{\hat{\jmath}}</math>
can be decomposed as
<math display="block">F_x =y, F_y = -x, F_z =0.</math>

Upon visual inspection, the field can be described as "rotating". If the vectors of the field were to represent a linear [[force]] acting on objects present at that point, and an object were to be placed inside the field, the object would start to rotate clockwise around itself. This is true regardless of where the object is placed.

Calculating the curl:
<math display="block">\nabla \times \mathbf{F} =0\boldsymbol{\hat{\imath}}+0\boldsymbol{\hat{\jmath}}+ \left({\frac{\partial}{\partial x}}(-x) -{\frac{\partial}{\partial y}} y\right)\boldsymbol{\hat{k}}=-2\boldsymbol{\hat{k}}
</math>

The resulting vector field describing the curl would at all points be pointing in the negative {{Math|''z''}} direction. The results of this equation align with what could have been predicted using the [[Right-hand rule#A rotating body|right-hand rule]] using a [[Cartesian coordinate system#In three dimensions|right-handed coordinate system]]. Being a uniform vector field, the object described before would have the same rotational intensity regardless of where it was placed.

{{clear}}
=== Example 3 ===
{{Multiple image
| align             = 
| direction         = 
| total_width       = 400
| image1            = Nonuniform curl.svg
| image2            = Curl of nonuniform curl.png
| alt1              = 
| caption1          = 
| caption2          = 
| footer            = Vector field {{math|1='''F'''(''x'', ''y'') = [0, −''x''<sup>2</sup>]}} (left) and its curl (right).
}}

For the vector field
<math display="block">\mathbf{F}(x,y,z) = -x^2\boldsymbol{\hat{\jmath}}</math>

the curl is not as obvious from the graph. However, taking the object in the previous example, and placing it anywhere on the line {{math|1=''x'' = 3}}, the force exerted on the right side would be slightly greater than the force exerted on the left, causing it to rotate clockwise. Using the right-hand rule, it can be predicted that the resulting curl would be straight in the negative {{math|''z''}} direction. Inversely, if placed on {{math|1=''x'' = −3}}, the object would rotate counterclockwise and the right-hand rule would result in a positive {{math|''z''}} direction.

Calculating the curl:
<math display="block">{\nabla} \times \mathbf{F} = 0 \boldsymbol{\hat{\imath}} + 0\boldsymbol{\hat{\jmath}} + {\frac{\partial}{\partial x}}\left(-x^2\right) \boldsymbol{\hat{k}} = -2x\boldsymbol{\hat{k}}.</math>

The curl points in the negative {{math|''z''}} direction when {{math|''x''}} is positive and vice versa. In this field, the intensity of rotation would be greater as the object moves away from the plane {{math|1=''x'' = 0}}.

{{clear}}

===Further examples===
* In a vector field describing the linear velocities of each part of a rotating disk in [[uniform circular motion]], the curl has the same value at all points, and this value turns out to be exactly two times the vectorial [[angular velocity]] of the disk (oriented as usual by the [[right-hand rule]]). More generally, for any flowing mass, the linear velocity vector field at each point of the mass flow has a curl (the [[vorticity]] of the flow at that point) equal to exactly two times the ''local'' vectorial angular velocity of the mass about the point.
* For any solid object subject to an external physical force (such as gravity or the electromagnetic force), one may consider the vector field representing the infinitesimal force-per-unit-volume contributions acting at each of the points of the object. This force field may create a net ''[[torque]]'' on the object about its center of mass, and this torque turns out to be directly proportional and vectorially parallel to the (vector-valued) integral of the ''curl'' of the force field over the whole volume.
* Of the four [[Maxwell's equations]], two—[[Faraday's law of induction|Faraday's law]] and [[Ampère's circuital law|Ampère's law]]—can be compactly expressed using curl. Faraday's law states that the curl of an [[electric field]] is equal to the opposite of the time rate of change of the magnetic field, while Ampère's law relates the curl of the magnetic field to the current and the time rate of change of the electric field.