Usage
In practice, the above definition is rarely used because in virtually all cases, the curl operator can be applied using some set of curvilinear coordinates, for which simpler representations have been derived.
The notation ∇ × F has its origins in the similarities to the 3 dimensional cross product, and it is useful as a mnemonic in Cartesian coordinates if we take ∇ as a vector differential operator del. Such notation involving operators is common in physics and algebra. If certain coordinate systems are used, for instance, polar-toroidal coordinates (common in plasma physics) using the notation ∇ × F will yield an incorrect result.
Expanded in Cartesian coordinates (see: Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations), ∇ × F is, for F composed of :
where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. This expands as follows:
Although expressed in terms of coordinates, the result is invariant under proper rotations of the coordinate axes but the result inverts under reflection.
In a general coordinate system, the curl is given by
where ε denotes the Levi-Civita symbol, the metric tensor is used to lower the index on F, and the Einstein summation convention implies that repeated indices are summed over. Equivalently,
where ek are the coordinate vector fields. Equivalently, using the exterior derivative, the curl can be expressed as:
Here and are the musical isomorphisms, and is the Hodge dual. This formula shows how to calculate the curl of F in any coordinate system, and how to extend the curl to any oriented three dimensional Riemannian manifold. Since this depends on a choice of orientation, curl is a chiral operation. In other words, if the orientation is reversed, then the direction of the curl is also reversed.
Read more about this topic: Curl (mathematics)
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