The cross product from linear algebra is most often used in conjunction with 2-dimensional or 3-dimensional vectors. However, it is also said to work with 7-dimensional vectors, although this version appears to live a bit deeper in the sea of mathematical prerequisites.

In multivariable calculus, 2D-Curl and 3D-Curl are very much connected to the 2D cross product and 3D cross product, respectively.

Curvature, with respect to arclength, has a loosely similar geometry to Curl (both involve some measure of straightness or lack thereof), and also connects with the cross product $(k = \frac{\|S'(t)\ \times\ S''(t)\|}{\|S'(t)\|^3}$).

This raises the question, does the 7-dimensional version of the cross product lead to a 7D-Curl, 7D curvature, or 7-dimensional versions of any other topics in multivariable calculus tied to the cross product which don't exist in every dimension?

  • $\begingroup$ You should certainly be able to define a 7D curl in the analogous manner. $\endgroup$
    – user856
    Nov 4 '18 at 5:24

There is brief discussion of 7 dimensional curl, and electromagnetism, in The Mathematical Heritage of C F Gauss By George M. Rassias



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