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Curved Coordinates 012 — The Cross Product and the Curl
Building both from the Levi-Civita symbol
This is the twelfth article in my series on curvilinear coordinates. In the previous article I built the divergence and the Laplacian by analogy with their Cartesian forms. Here I do the same for the cross product and the curl, starting from their Cartesian forms.
I strongly recommend reading the earlier articles in this series before proceeding.
The cross product in Cartesian coordinates
Start with cross products of the Cartesian unit vectors, also briefly covered in the first article in this series:
Look at the pattern. A cross product of two different unit vectors gives the third, with a sign that flips when you swap the order. In the index notation below, I label the Cartesian directions 1, 2, and 3 instead of x, y, and z. The pattern is then captured compactly by a single object:
In Cartesian coordinates the dual basis vectors are the same as the basis vectors…
