Definition of CURL in $\mathbb{R}^n$.

Question

In calculus we saw that how to define curl in $\mathbb{R}^3$. My question is about the generalization; I mean is there any way that one can define curl in $\mathbb{R}^n$? If yes, how can I do it?

Answer 1

This is just an idea...

As others have alluded to in the comments it is not necessarily well defined. One way of extending might be working with the determinant idea $$ \nabla \times \mathbf{v} = \begin{vmatrix} i & j & k \\ \partial_x & \partial_y & \partial_z \\ v_x & v_y & v_z\end{vmatrix} $$ for the $4\times4$ case you have the choice of adding another row of $i,j,k,l$ symbols, or derivatives, or an extra vector $$ \nabla \times \mathbf{v} = \begin{vmatrix} e_1 & e_2 & e_3 & e_4 \\ \partial_{x_1} & \partial_{x_2} & \partial_{x_3} & \partial_{x_4}\\ v_1 & v_2 & v_3 & v_4 \\ ? & ? & ? & ? \end{vmatrix} $$ if you add extra unit vectors, then you have dyadic like quantities. If you add extra derivatives, you will have a second order derivative for each term, and if you add two vectors it will progress from a unary/binary operation to a ternary, or n-ary operation.

Would be fun to see how that plays out, but don't take this answer too seriously it's just an idea.

Answer 2

This is to give you the idea, but for the details you have to study some exterior calculus. The concept behind the usual curl in a three-dimensional space is that of exterior derivative $d$ of a vector field $v$ (namely the derivative, or "differential", $dv$ of an object $v$ called 1-form). Now, the exterior derivative of a 1-form gives you a 2-form $dv$ (you can think of a 2-form as a matrix, namely a "vector field with two indexes"). There is also a "duality operation" over forms called "Hodge duality". This duality is implemented by an operator called "Hodge star" $*$. This operator takes a 2-form and gives you a dual $(N-2)$-form, where $N$ is the dimension of the space you are considering. The curl of a vector field is $\nabla \times v = *(dv)$. Let's see what happens for $N=3$: $dv$ is a 2-form (the differential of your vector field $v$) and the Hodge star gives you the dual (3-2)-form. Therefore this dual form $*dv$ is a 1-form, like the initial 1-form $v$. This tells you that in $N=3$ the curl of a vector field is still a vector field. For $N=9$ you find that the generalized curl of a 1-form is a $7$-form (a sort of generalization of a vector field with 7 indexes).