Is it possible to compute the curl (or divergence) when \nabla and X have different number of "variables"?

No, not really.

You can measure similar quantities (say, the circulation of a vector field in 3D along a 1D curve), but this is not the same thing...

What are curl and divergence mainly used for?

Is it possible to compute the curl (or divergence) when \nabla and X have different number of "variables"?

No, not really.

You can measure similar quantities (say, the circulation of a vector field in 3D along a 1D curve), but this is not the same thing...

What are curl and divergence mainly used for?