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In terms of vector field analogies to closed and exact differential forms, conservative and incompressible vector fields (gradient and divergence) generalize to higher dimensions, but curl and irrotational fields do not. Why? Cross product doesn't generalize either but one can use exterior products and hodge duals to fullfill the need. In differential geometry, there is a duality between the boundary operator on chains and the exterior derivative as expressed by the general Stokes' theorem. By a theorem of De Rham, the exterior derivative is the dual of the boundary map on singular simplices. From this perspective, there should be a generalized curl. Perhaps, there is an explanation in terms of the Poincare Lemma where for n>3 (but perhaps not 7), curl fails for higher dimensions?

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What properties do you want for the generalized curl Do you want it to be a vector field? What should it do for you? – Deane Yang Jan 22 '11 at 15:14
The curl in $\mathbb{R}^3$ is the exterior derivative of a $1$-form under the identification: vector fields = $1$-forms=$*$2-forms given by the Euclidean metric. So in this sense, it surely does generalize. – Donu Arapura Jan 22 '11 at 16:39
Answers to this question might be relevant: – José Figueroa-O'Farrill Jan 22 '11 at 18:11
up vote 6 down vote accepted

Let ${\bf K}$ be a vector field in the neighbourhood of ${\bf p}\in{\mathbb R}^n$, and let ${\bf X}$ and ${\bf Y}$ be two tangent vectors at ${\bf p}$. These two vectors span a parallelogram $P$ with one vertex at ${\bf p}$. The "circulation" of ${\bf K}$ around $P$ computes to $$\int_{\partial P}{\bf K}\bullet d{\bf x}= (L.{\bf X})\bullet{\bf Y}- (L.{\bf Y})\bullet{\bf X} + o(|P|^2)$$ with $L:=d{\bf K}({\bf p})$ and $|P|$:= diam$(P)$. It follows that there is a certain skew bilinear function ${\rm Rot}\thinspace{\bf K}({\bf p}):T_{\bf p}\times T_{\bf p}\to{\mathbb R}$ with $$\int_{\partial P}{\bf K}\bullet d{\bf x}={\rm Rot}\thinspace{\bf K}({\bf p}).({\bf X},{\bf Y})+ o(|P|^2) \ \ \ (|P|\to 0).$$ In the case $n=3$ the bilinear form ${\rm Rot}$ can be represented by the vector ${\rm curl}\thinspace{\bf K}$ in the form $$ {\rm Rot}{\bf K}({\bf p}).({\bf X},{\bf Y}) = {\rm curl}\thinspace{\bf K}({\bf p})\bullet({\bf X}\times{\bf Y}).$$

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