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Line 1: + − The curl is written as though it were the [[cross product]] of the special symbol "<math>\nabla</math>" (which is commonly called "del" or "nabla"), with the given vector field, like this: <math>\nabla \times \vec V</math>. This is usually pronounced "curl V" or "del cross V".+ − + + + + + + + + + + Line 26:
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{{math-h}}
The '''curl''' is a way of expressing a certain type of [[derivative]] of a [[vector field]]. It is defined for fields of 3-dimensional vectors on 3-dimensional space. The curl of a vector field is another vector field.
The '''curl''' is a way of expressing a certain type of [[derivative]] of a [[vector field]]. It is defined for fields of 3-dimensional vectors on 3-dimensional space. The curl of a vector field is another vector field.
More precisely, it is defined<ref>[http://mathworld.wolfram.com/Curl.html Curl] at Wolfram Mathworld</ref> as the limiting value of rotation per unit area. Written explicitly,
In ordinary [[Cartesian coordinates]], the curl is calculated as:
:<math>(\nabla \times \mathbf{F}) \cdot \mathbf{\hat{n}} \equiv \lim_{A \to 0} \frac{\oint_{C} \mathbf{F} \cdot d\mathbf{s}}{A}</math>
where <math>\mathbf{\hat{n}}</math> is the unit [[normal vector]] of the area element A and C is the boundary of the area element (<math>C =\partial A</math>). The right side of the definition is a path integral along the boundary of the area element A, which is allow to shrink in the limiting process.
==Cartesian coordinates==
In [[Cartesian coordinates]], the curl is written as though it were the [[cross product]] of the special symbol "<math>\nabla</math>" (which is commonly called "del" or "nabla"), with the given vector field, like this: <math>\nabla \times \vec V</math>. This is usually pronounced "curl V" or "del cross V".
In 3-D [[Cartesian coordinates]], the curl is calculated as:
:<math>\nabla \times \vec V = (\ \ \frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z},\ \ \ \ \frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x},\ \ \ \ \frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y}\ \ )</math>
:<math>\nabla \times \vec V = (\ \ \frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z},\ \ \ \ \frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x},\ \ \ \ \frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y}\ \ )</math>
Vector fields with a curl of zero are called ''irrotational''.
Vector fields with a curl of zero are called ''irrotational''.
==References==
{{reflist}}
[[Category:Calculus]]
[[Category:Calculus]]
[[Category:Physics]]
[[Category:Physics]]