Curl

The curl (also known as rotation) is a differential operator acting on a vector field. It is defined in the branch of mathematics known as vector analysis. Important applications of the curl are in the Maxwell equations for electromagnetic fields, in the Helmholtz decomposition of arbitrary vector fields, and in the equation of motion of fluids.

Three notations are in common use:

$\mathrm {curl} \;\mathbf {F} \equiv \mathrm {rot} \;\mathbf {F} \equiv {\boldsymbol {\nabla }}\times \mathbf {F} ,$

where F is a vector field.

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:

${\boldsymbol {\nabla }}\times \mathbf {F} (\mathbf {r} )\;{\stackrel {\mathrm {def} }{=}}\;\mathbf {e} _{x}\left({\frac {\partial F_{y}}{\partial z}}-{\frac {\partial F_{z}}{\partial y}}\right)+\mathbf {e} _{y}\left({\frac {\partial F_{z}}{\partial x}}-{\frac {\partial F_{x}}{\partial z}}\right)+\mathbf {e} _{z}\left({\frac {\partial F_{x}}{\partial y}}-{\frac {\partial F_{y}}{\partial x}}\right),$

where e_x, e_y, and e_z are unit vectors along the axes of a Cartesian coordinate system.

As with any cross product, the curl may be written in several alternative ways as follows.

As a determinant (evaluate along the first row):

${\boldsymbol {\nabla }}\times \mathbf {F} (\mathbf {r} )={\begin{vmatrix}\mathbf {e} _{x}&\mathbf {e} _{y}&\mathbf {e} _{z}\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\F_{x}&F_{y}&F_{z}\end{vmatrix}}$

As a vector-matrix-vector product:

${\boldsymbol {\nabla }}\times \mathbf {F} (\mathbf {r} )=\left(\mathbf {e} _{x},\;\mathbf {e} _{y},\;\mathbf {e} _{z}\right)\;{\begin{pmatrix}0&{\frac {\partial }{\partial z}}&-{\frac {\partial }{\partial y}}\\-{\frac {\partial }{\partial z}}&0&{\frac {\partial }{\partial x}}\\{\frac {\partial }{\partial y}}&-{\frac {\partial }{\partial x}}&0\\\end{pmatrix}}{\begin{pmatrix}F_{x}\\F_{y}\\F_{z}\end{pmatrix}}$

In terms of the antisymmetric Levi-Civita symbol ε_αβγ:

${\Big (}{\boldsymbol {\nabla }}\times \mathbf {F} (\mathbf {r} ){\Big )}_{\alpha }=\sum _{\beta ,\gamma =x,y,z}\epsilon _{\alpha \beta \gamma }{\frac {\partial F_{\beta }}{\partial \gamma }},\qquad \alpha =x,y,z$

(This gives 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

${\boldsymbol {\nabla }}\times \mathbf {F} (\mathbf {r} )=\mathbf {0}$

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

$\mathbf {F} (\mathbf {r} )=-{\boldsymbol {\nabla }}\Phi (\mathbf {r} ).$

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):

${\boldsymbol {\nabla }}\times \mathbf {F} (\mathbf {r} )={\frac {1}{h_{1}h_{2}h_{3}}}{\begin{vmatrix}h_{1}\mathbf {e} _{1}&h_{2}\mathbf {e} _{2}&h_{3}\mathbf {e} _{3}\\{\frac {\partial }{\partial u_{1}}}&{\frac {\partial }{\partial u_{2}}}&{\frac {\partial }{\partial u_{3}}}\\h_{1}F_{1}&h_{2}F_{2}&h_{3}F_{3}\end{vmatrix}}$

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

$h_{r}=1,\qquad h_{\theta }=r,\qquad h_{\phi }=r\sin \theta$

the curl is

$\nabla \times \mathbf {F} ={\frac {1}{r^{2}\sin \theta }}{\begin{vmatrix}\mathbf {e} _{r}&r\mathbf {e} _{\theta }&r\sin \theta \mathbf {e} _{\phi }\\{\frac {\partial }{\partial r}}&{\frac {\partial }{\partial \theta }}&{\frac {\partial }{\partial \phi }}\\F_{r}&rF_{\theta }&r\sin \theta F_{\phi }\\\end{vmatrix}},$

Definition through Stokes' theorem

Stokes' theorem is

$\iint _{S}\,({\boldsymbol {\nabla }}\times \mathbf {F} )\cdot d\mathbf {S} =\oint _{C}\mathbf {F} \cdot d\mathbf {s} ,$

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

$({\boldsymbol {\nabla }}\times \mathbf {F} )\cdot {\hat {\mathbf {n} }}\;\Delta S$

where ${\hat {\mathbf {n} }}$ 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

$({\boldsymbol {\nabla }}\times \mathbf {F} )\cdot {\hat {\mathbf {n} }}=\lim _{\Delta S\rightarrow 0}{\frac {1}{\Delta S}}\;\oint _{C}\mathbf {F} \cdot d\mathbf {s} ,$

The line integral is the circulation of F with respect to C. The expression 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.

Properties

${\begin{aligned}{\boldsymbol {\nabla }}\times {\boldsymbol {\nabla }}\Phi &=0\\{\boldsymbol {\nabla }}\cdot ({\boldsymbol {\nabla }}\times \mathbf {F} )&=0\\{\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}\times \mathbf {F} )&={\boldsymbol {\nabla }}({\boldsymbol {\nabla }}\cdot \mathbf {F} )-({\boldsymbol {\nabla }}\cdot {\boldsymbol {\nabla }})\mathbf {F} \\\end{aligned}}$

The operator

${\boldsymbol {\nabla }}\cdot {\boldsymbol {\nabla }}\equiv \nabla ^{2}$

is the Laplace operator. Further properties:

${\begin{aligned}{\boldsymbol {\nabla }}\times (\mathbf {A} \times \mathbf {B} )&=(\mathbf {B} \cdot {\boldsymbol {\nabla }})\mathbf {A} -(\mathbf {A} \cdot {\boldsymbol {\nabla }})\mathbf {B} +\mathbf {A} ({\boldsymbol {\nabla }}\cdot \mathbf {B} )-\mathbf {B} ({\boldsymbol {\nabla }}\cdot \mathbf {A} )\\\mathbf {A} \times ({\boldsymbol {\nabla }}\times \mathbf {B} )&=({\boldsymbol {\nabla }}\otimes \mathbf {B} )\mathbf {A} -(\mathbf {A} \cdot {\boldsymbol {\nabla }})\mathbf {B} ,\end{aligned}}$

where the matrix has the following components:

${\big (}{\boldsymbol {\nabla }}\otimes \mathbf {B} {\big )}_{\alpha \beta }\equiv \nabla _{\alpha }B_{\beta }\equiv {\frac {\partial B_{\beta }}{\partial r_{\alpha }}},\quad {\hbox{with}}\quad \alpha ,\beta =1,2,3\leftrightarrow x,y,z,\quad {\hbox{and}}\quad (r_{1},\,r_{2},\,r_{3})\equiv (x,\,y,\,z).$

External link

MathWorld curl