We could study General Relativity, when we consider d spatial dimensions (instead of 3). Then the computations are much more involved and there are many open issues in perturbation theory. Of course, such an extended theory is not realized in Nature, because we don’t observe extended extra dimensions. But people are still trying to search for small curved extra dimensions. Here is the presentation (pictures are taken from web).
Sometimes it looks mysterious, what is it the variational derivative and what is it the physical term variation. I try to give a simple explanation, how is the connection with the weak derivative.
Cross-product has interesting properties and exists only in dimension 3. I try to explain why.