Intuitive interpretation of some vector-dif-calc identities

  • #1
Dear All,

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
 

Answers and Replies

  • #2
robphy
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There are some interesting ideas on Bill Burke's page
http://www.ucolick.org/~burke/home.html

He was an advocate of using differential forms instead of vector calculus.

Along these lines, one might argue that
viewing or seeking patterns in "fields of vectors" might not be the right structure to use for visualizing your expressions.

Following Burke's approach (which likely came from Misner-Thorne-Wheeler's approach, which in turn likely came from Schouten's approach),
I can visualize vectors and covectors, bivectors and two-forms, and their algebraic operations...
but I don't yet have a good feel for visualizing the exterior derivative (which are related to div, grad, and curl).
 
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  • #3
Dear robphy,

thank you for the provided hint. I will examine the suggested web page with great interest.

Best regards
 

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