Curl: Difference between revisions

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The '''curl''' is a differential operator defined in [[vector analysis]]. Two important applications of the curl are (i) in [[Maxwell equations]] for electromagnetic fields and (ii) in the [[Helmholtz decomposition]] of arbitary vector fields.
==Definition==
Given a 3-dimensional [[vector field]] '''F'''('''r'''), the '''curl''' (also known as '''rotation''')  of '''F'''('''r''') is the differential [[vector operator]] [[nabla]] (symbol '''∇''')  applied to '''F'''. The application of '''∇'''  is in the form of a [[cross product]]:
Given a 3-dimensional [[vector field]] '''F'''('''r'''), the '''curl''' (also known as '''rotation''')  of '''F'''('''r''') is the differential [[vector operator]] [[nabla]] (symbol '''∇''')  applied to '''F'''. The application of '''∇'''  is in the form of a [[cross product]]:
:<math>
:<math>
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\end{vmatrix}
\end{vmatrix}
</math>
</math>
As a vector-matrix-vector product  
As a vector-matrix-vector product:
:<math>
:<math>
\boldsymbol{\nabla}\times \mathbf{F}(\mathbf{r}) = \left(\mathbf{e}_x, \; \mathbf{e}_y,\; \mathbf{e}_z\right)\;  
\boldsymbol{\nabla}\times \mathbf{F}(\mathbf{r}) = \left(\mathbf{e}_x, \; \mathbf{e}_y,\; \mathbf{e}_z\right)\;  
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\end{pmatrix}
\end{pmatrix}
</math>
</math>
In terms of the antisymmetric [[Levi-Civita symbol]] &epsilon;<sub>&alpha;&beta;&gamma;</sub>
In terms of the antisymmetric [[Levi-Civita symbol]] &epsilon;<sub>&alpha;&beta;&gamma;</sub>:
:<math>
:<math>
\Big(\boldsymbol{\nabla}\times \mathbf{F}(\mathbf{r}) \Big)_\alpha
\Big(\boldsymbol{\nabla}\times \mathbf{F}(\mathbf{r}) \Big)_\alpha
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</math>
</math>
(the component of the curl along the Cartesian &alpha;-axis).
(the component of the curl along the Cartesian &alpha;-axis).
 
==Irrotational vector field==
Two important applications of the curl are (i) in [[Maxwell equations]] for electromagnetic fields and (ii) in the [[Helmholtz decomposition]] of arbitary vector fields.
From the [[Helmholtz decomposition]] follows that any ''curl-free vector field'' (also known as ''irrotational field'') '''F'''('''r'''), i.e., a vector field for which
 
From the Helmholtz decomposition follows that any ''curl-free vector field'' (also known as ''irrotational field'') '''F'''('''r'''), i.e., a vector field for which
:<math>
:<math>
\boldsymbol{\nabla}\times \mathbf{F}(\mathbf{r}) = \mathbf{0}
\boldsymbol{\nabla}\times \mathbf{F}(\mathbf{r}) = \mathbf{0}
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</math>
</math>


==Orthogonal curvilinear coordinate systems==
==Curl in orthogonal curvilinear coordinates==
In a general 3-dimensional orthogonal [[curvilinear coordinate system]] ''u''<sub>1</sub>,
In a general 3-dimensional orthogonal [[curvilinear coordinate system]] ''u''<sub>1</sub>,
''u''<sub>2</sub>, and ''u''<sub>3</sub>, characterized by the [[scale factors]] ''h''<sub>1</sub>,
''u''<sub>2</sub>, and ''u''<sub>3</sub>, characterized by the [[scale factors]] ''h''<sub>1</sub>,

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The curl is a differential operator defined in vector analysis. Two important applications of the curl are (i) in Maxwell equations for electromagnetic fields and (ii) in the Helmholtz decomposition of arbitary vector fields.

Definition

Given a 3-dimensional vector field F(r), the curl (also known as rotation) of F(r) is the differential vector operator nabla (symbol ) applied to F. The application of is in the form of a cross product:

where ex, ey, and ez are unit vectors along the axes of a Cartesian coordinate system of axes.

As any cross product the curl may be written in a few alternative ways.

As a determinant (evaluate along the first row):

As a vector-matrix-vector product:

In terms of the antisymmetric Levi-Civita symbol εαβγ:

(the component of the curl along the Cartesian α-axis).

Irrotational vector field

From the Helmholtz decomposition follows that any curl-free vector field (also known as irrotational field) F(r), i.e., a vector field for which

can be written as minus the gradient of a scalar potential Φ

Curl in orthogonal curvilinear coordinates

In a general 3-dimensional orthogonal curvilinear coordinate system u1, u2, and u3, characterized by the scale factors h1, h2, and h3, (also known as Lamé factors, the square roots of the elements of the diagonal g-tensor) the curl takes the form of the following determinant (evaluate along the first row):

For instance, in the case of spherical polar coordinates r, θ, and φ

the curl is

Definition through Stokes' theorem

Stokes' theorem is

where dS is a vector of length the infinitesimal surface dS and direction perpendicular to this surface. The integral is over a surface S encircled by a contour (closed non-intersecting path) C. The right-hand side is an integral along C. If we take S so small that the integrand of the integral on the left-hand side may be taken constant, the integral becomes

where is a unit vector perpendicular to ΔS. The right-hand side is an integral over a small contour, say a small circle, and in total the curl may be written as

The line integral along the infinitesimally small circle C is the total "circulation" of F at the center of the circle. This leads to the following interpretation of the curl: It is a vector with a component oriented perpendicular to the plane of circulation. The perpendicular component has length equal to the circulation per unit surface.

External link

MathWorld curl