curl is defined as del X F where f is a vector field. The physical interpretation of curl is that it measures the “circulation” of a vector field around a circular path at a point. thus the length of the curl vector correlates to how much the vector field “follows” an infinitesimal path around a point. thus in a conservative field with continuous partial derivatives are defined on all of R3 the curl is equal to 0. the reason being that at a point no path would be “preferred” which would be untrue if there was some curl (imagine a whirlpool).
The divergence of a vector field F is defined as del * F. interpretation of this is as follows: flux (which will come up later) roughly speaking is the net amount of vectors coming in and out of a surface. Divergence is then the amount of flux per unit volume of a closed surface. so i tells you how much something “diverges” from a point
