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There's also a book on the subject called "Div, Grad, Curl and all that"

https://www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161

which I would highly recommend to physics-minded folks, but would not recommend at all to maths-minded folks.

⬐ trendia

> would not recommend at all to maths-minded folks.

Why not?

⬐ ur-whale

I bought the book (a long time ago) thinking I was going to get something exactly like what the OP video is: a mathematical explanation of what the tools do, with some intuitive link between - say - the formula for div and why it measures how much a vector field does indeed "diverge" locally.

Instead, the whole book tries to explain the 3 tools using electrostatics as an intuitive justification for how they behave. Ugh.

To me, the way electromagnetic fields behave is no particularly intuitive or natural, and the way I do - sort of - manage to understand Maxwell's equations is because I have an intuitive feel for what grad, div and curl do to vector fields.

Where I think your argument falls flat is when you compare good teachers to bad ones.

Good teachers explain things so that students can understand - and explain the rakes.

Bad teachers put up walls of academic nonsense-speak - point at it with a laser - and expect you to know what they are teaching already.

Not all Academic Speech is "nonsense speak". For example - many students really learn the material from supplementary material (good academic speak) vs textbook (nonsense speak).

https://www.amazon.com/Signals-Systems-Made-Ridiculously-Sim...

https://www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161/

These examples show that you explain things well or poorly - explaining things poorly has a direct effect on humanity in the long run.

I learned just enough vector calculus in college to know I didn't know enough to actually understand Maxwell's equations & electromagnetic fields, or most of the other fun mathematical techniques introduced in mid-level physics and engineering classes for modeling interesting "real-world" systems.

Later, when I realized what I was missing out on, I tried to teach myself the missing concepts. I failed, until I found H.M. Schey's "Div, Grad, Curl, and All That: An Informal Text on Vector Calculus." It's a pragmatic, friendly, slim little math book that reads more like lecture notes than a classic textbook, and I can fairly say it's taught me everything I know about those operators (which isn't much).

So if you are like me, and got to calc III, vector math, and/or liner algebra without learning div, grad, curl and partial differential equations... check out the book. it's great: https://www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161

Also: can we get three cheers for HYPERPHYSICS?? http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html

When I was an undergrad and graduate EE student, the "go to" book to help with electromagnetics was Div, Grad, Curl, and all that (http://www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161). Back in my era it was the 1st edition of the book, not the 4th.

I see another book with good recommendations, A Student's Guide to Maxwell's Equations (http://www.amazon.com/Students-Guide-Maxwells-Equations/dp/0...) but I have not read that one.

Div, Grad, Curl helped a bit, but what really made it click for me was an excellent professor some other EE math-class-in-disguise that explained those vector calc operations in terms of divergence (source density) and flux (change in time/space).

As far as understanding the linked paper, I can't follow the proof either. Equations 1-4 I've never seen, 5-8 are Maxwell's Equations which are familiar but we wrote them with different notation, 9-18 are again equations I've never seen. The meat of the proof in 19-21 is built on 14 mystery equations and 4 that I recognize.

As a former EE I guess we didn't prove equations as much as take their existence as given and then figured out what that implied for the real world. ;) Other posters aroberge and wraithm have mentioned also needing QM from physics to follow the proof, which must be where the other equations are from!

⬐ aroberge

Surely you have seen equation 1 (but perhaps with a slightly different notation). Each dot above an x represent a time derivative. So, the left-hand side is the second derivative of the position with respect to time aka the acceleration, times the mass. So, the first equation is simply F = ma (but, yeah... the notation can easily throw one off if one has not see it before.)

⬐ thoth

You're correct, I didn't recognize it. Thanks for pointing that out!