I discover these concepts in my math class, and I want to understand the intuition behind the Laplacian, the curl and the divergence operators, how do you explain this to a student who is discovering this for the first time? and especially how to make the link between formulas and intuitions?
For me, the laplacian can be seen as an average over a neighbors, while the divergence quantifies for each point in the space whether it is a vector sender or receiver.
Divergence is something that tells you how strong something diverges or converges to a point. Imagine a hole out of which something comes out, it will have a positive divergence, vice-versa for the other case. Curl you can imagine as a rate of how much are particles rotating, imagine a stream of water or something, it can tell you how much is water going straight down and how much is resisting due to some force. I mean it can all go down but at some point it can turn due to strong force at that spot making it turn.