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I am studying electrodynamics and I am trying hard to clearly understand each and every formula. By "understand" I mean that I can "truly see its meaning in front of my eyes". Generally, I am not satisfied only by being able to prove or derive certain formula algebraically; I want to "see through it".

With this regards, there are three formulas from vector differential calculus, which I can happily derive or prove (by expanding and comparing individual vector components, etc.), however, I am unable to truly understand them.

The three formulas are famous: (1) curl(AXB), (2) div(AxB) and grad(A dot B).

Expansions of these formula consist of several terms, whose meaning I cannot grasp intuitively. By this I mean that I cannot "visualize" how all these terms together coherently join to form e.g. curl of the cross product, etc..

Is there any book or other resource, which does not deal with these formulas purely algebraically, and which does not simply states "expand the expression in term of individual components and be happy". I am looking for a resource which graphically or in any other way gives an intuitive explanation of these formulas.

Any hints about such a reference would be highly appreciated.

Thank you in advance and best regards