Conceptually Relating 2nd Partials to Electrodynamics

  • #1
Hey All,

In deriving applications of Maxwell's equations in advanced electromagnetics, it seems like one is exposed to almost all of the vector calculus identities in the book! (or at least on the current Wikipedia entry :smile:).

Personally, I'm a non-traditional learner, and grasp advanced physics concepts the best when I can relate to them in a visuospatial manner (hey, it got me through a B.S. in AP, albeit slowly!).
Generally, I was wondering if anyone has ever tried to conceptualize some of these more advanced mathematical statements in the context of what actually occurs physically. Sure, taking the curl of the divergence isn't a terribly advanced statement mathematically, but when attempting to conceptualize the behavior of a vector field without using pen and paper, or a computer, it can cause cranial pressures to asymptotically approach critical levels :smile:.

More specifically, it's been years since I took Calc. 3 and advanced E&M, and I'm having trouble with grasping how a vector field would appear when treated with each combination of the second partials of the del operator, curl and divergence.
For example, within Potential Theory (the electromagnetic application), the curl of the curl of a Magnetic potential field must be taken to forge the Maxwell-Ampere equation into a pretty nifty harmonic function. How would one visualize the curling of a curl at a point in R3? Is it akin to an acceleration of the magnitude of curling?
A vector identity equates this to the gradient of a divergence, minus the Laplacian, of a vector (or tensor) field. Unfortunately, this doesn't seem to help me grasp what's physically happening in this case!


Many Thanks,

Mike
 
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Answers and Replies

  • #2
vanhees71
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Well, the most intuitive picture for the vector-calculus operations you get from applying it to hydrodynamics. A very good summary is given in Sommerfeld, Lectures on Theoretical Physics, vol. 2 (fluid dynamics).
 

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