Curl

The curl of a Tensor field is given by

$$(\nabla\times A)^\alpha = \epsilon^{\alpha\mu\nu}A_{\nu;\mu},$$ (1)

where $\epsilon_{ijk}$ is the Levi-Civita Tensor and ``;'' is the Covariant Derivative. For a Vector Field, the curl is denoted

$${\rm curl}({\bf F})\equiv\nabla\times{\bf F},$$ (2)

and $\nabla \times {\bf F}$ is normal to the Plane in which the ``circulation'' is Maximum. Its magnitude is the limiting value of circulation per unit Area,

$$(\nabla \times {\bf F})\cdot \hat {\bf n} \equiv \lim_{A\to 0} {\oint_C {\bf F}\cdot d{\bf s}\over A}.$$ (3)

Let

$${\bf F} \equiv F_1\hat {\bf u}_1+F_2\hat {\bf u}_2+F_3\hat {\bf u}_3$$ (4)

and

$$h_i \equiv \left\vert{\partial {\bf r}\over \partial u_i}\right\vert,$$ (5)

then

$$\nabla \times {\bf F} \equiv {1\over h_1h_2h_3}\left\vert\begin{array}{ccc}h_1\hat {\bf u}_1 &... ...& {\partial\over\partial u_3}\\ h_1F_1 & h_2F_2 & h_3F_3\end{array}\right\vert$$

$$= {1\over h_2h_3} \left[{{\partial\over \partial u_2}(h_3F_3)- {\partial\over\partial u_3}(h_2F_2)}\right]\hat {\bf u}_1$$

$$\phantom{=} +{1\over h_1h_3} \left[{{\partial\over \partial u_3}(h_1F_1)-{\partial\over \partial u_1}(h_3F_3)}\right]\hat {\bf u}_2$$

$$\phantom{=} +{1\over h_1h_2} \left[{{\partial\over \partial u_1}(h_2F_2)- {\partial\over \partial u_2}(h_1F_1)}\right]\hat {\bf u}_3.$$ (6)

Special cases of the curl formulas above can be given for Curvilinear Coordinates.

See also Curl Theorem, Divergence, Gradient, Vector Derivative

References

Arfken, G. ``Curl, $\nabla\times$.'' §1.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 42-47, 1985.