Is the curl of a vector field only defined on $\Bbb R^3$?
I was wondering if the criterion $$\nabla \times \vec{F}=\vec{0} \implies \vec{F} \space\text{is conservative}$$
only applies to three dimensional vector fields or if it also applies to $n$-dimensional vector fields?
From wikipedia
"Unlike the gradient and divergence, curl does not generalize as simply to other dimensions; some generalizations are possible, but only in three dimensions is the geometrically defined curl of a vector field again a vector field. This is a similar phenomenon as in the 3 dimensional cross product, and the connection is reflected in the notation ∇ × for the curl."
So you can define $\triangledown \times \vec{F} $ in higher dimensions but it does not have special geometrical properties there.