Intuitive Guide to Maxwell's Equations • Hacker News Board

Re: Intuitive Guide to Maxwell's Equations

#41

Post by uoaei » Wed, Jul 01, 2020, 4:45 PM UTC

fsckboy wrote: post #34

chronolitus wrote: post #3

These kinds of visual, well thought out explanations of topics which are often taught with terse, obscure and uninviting methods are a gift. On a tangent: I remember asking my Calculus 101 professor what the "intuitive meaning" of divergence and curl was, outside of the formal math and equations. He was shocked that one could ask to sully these perfect mathematical concepts with dirty intuitive reductions. A guide li…

just a quibble, if your Calculus 101 covers div and curl, you are going to an intense STEM school and really could be expected to understand the math.

Understanding the math and understanding the importance of the mathematical concept are two different things.

Personally, div and curl didn't quite click for me until I took fluid dynamics in my final year. I could do the homework but didn't really get why they were useful until it made sense why "div = 0 always for incompressible fluids".

Re: Intuitive Guide to Maxwell's Equations

#42

Post by soVeryTired » Wed, Jul 01, 2020, 4:57 PM UTC

martincmartin wrote: post #22

chronolitus wrote: post #3

These kinds of visual, well thought out explanations of topics which are often taught with terse, obscure and uninviting methods are a gift. On a tangent: I remember asking my Calculus 101 professor what the "intuitive meaning" of divergence and curl was, outside of the formal math and equations. He was shocked that one could ask to sully these perfect mathematical concepts with dirty intuitive reductions. A guide li…

The book Geometrical Vectors does exactly that. In fact, it considers the gradient vector to be a different kind of vector than a normal distance vector. If I have a vector between two points, and I compress space so that the points move closer together, then the distance vector gets smaller. However, the gradient vector gets bigger. The gradient is basically a density, it's units are units-of-whatever-your-taking-th…

That sounds like the beginnings of intuition for exterior algebra. While the gradient is a vector (i.e. it lives in a vector space), it's better thought of as being different to a standard 'displacement' vector.

Look at a topographical map of a landscape and note the contour lines. As you zoom into the contour lines (if they're detailed enough), they'll start to look more and more like parallel lines, densely spaced for a steep slope, or sparsely spaced for a mild slope. These parallel lines are the gradient.

A gradient and a vector 'fit together' to give a real number. The more parallel lines the vector pierces, the bigger the number [0]. So a gradient 'eats' a vector, spitting out a real number, and vice-versa. Just like a row vector 'eats' a column vector and spits out a real number.

I'm saying 'gradient' here, but really what I mean is 'one-form'. Language deliberately imprecise for all y'all mathematicians out there.

[0] https://neutronstars.utk.edu/class/p616_s19/lect3/index.html...

Re: Intuitive Guide to Maxwell's Equations

#43

Post by 6gvONxR4sf7o » Wed, Jul 01, 2020, 5:07 PM UTC

C4stor wrote: post #28

ghaff wrote: post #7

Earlier quoted context omitted.

Wikipedia is probably the most obvious modern exemplar of this. The editors of most mathematics articles clearly are much fonder of playing with the equations editor than they are of actually explaining things.

Well, it's not Wikipedia's fault if people producing intuitive content are not contributing to it. It's not like editors go out of their way to remove nice explanations.

I think it is wikipedia's fault, but that it's not far from how it should be. Wikipedia is a secondary source, designed have be a reference taken from primary sources, rather than be a primary source teaching information. It's a weird blurry distinction, but seems to work well. It'd be great to have a pedagogy section that either explicitly linked out to pedagogical primary sources, or a section explicitly designed to teach.

Re: Intuitive Guide to Maxwell's Equations

#44

Post by titanomachy » Wed, Jul 01, 2020, 5:11 PM UTC

chronolitus wrote: post #3

These kinds of visual, well thought out explanations of topics which are often taught with terse, obscure and uninviting methods are a gift. On a tangent: I remember asking my Calculus 101 professor what the "intuitive meaning" of divergence and curl was, outside of the formal math and equations. He was shocked that one could ask to sully these perfect mathematical concepts with dirty intuitive reductions. A guide li…

This was one nice thing about studying physics, most of the math we learned was motivated by some application. I think my first exposure to div and curl was using the analogy of fluids flowing in the plane.

Intuitively, divergence is how much fluid is being created (or destroyed) at a point, curl is how much a waterwheel would spin if placed at a point.

Re: Intuitive Guide to Maxwell's Equations

#45

Post by sreevisakh » Wed, Jul 01, 2020, 5:13 PM UTC

C4stor wrote: post #28

ghaff wrote: post #7

Earlier quoted context omitted.

Wikipedia is probably the most obvious modern exemplar of this. The editors of most mathematics articles clearly are much fonder of playing with the equations editor than they are of actually explaining things.

Well, it's not Wikipedia's fault if people producing intuitive content are not contributing to it. It's not like editors go out of their way to remove nice explanations.

The problem is more due to a general culture that prefer rigour over clarity. Mathematics by definition is made of layers upon layers of intuitive conclusions. Then why do universities prescribe books with long dry derivations over clear explanations? Rigour is important to ensure correctness - and that should never be compromised. But that isn't the reason why humans learn mathematics. People learn it because it extends their imagination. Clarity and intuition are important for that and those who can't see it becomes disillusioned.

Rigour is better left to computers these days. They are better at following rules. We can have better proofs with software like metamath. Surprisingly, you can even gain deep insights by writing automated proofs (compared to manual proofs). Ideally, practitioners should consider simple and clear explanations as their primary goal and equations and proofs as necessary supplements. Think of this as literate programming for mathematics.

Re: Intuitive Guide to Maxwell's Equations

#46

Post by soVeryTired » Wed, Jul 01, 2020, 5:13 PM UTC

uoaei wrote: post #40

C4stor wrote: post #28

Earlier quoted context omitted.

Well, it's not Wikipedia's fault if people producing intuitive content are not contributing to it. It's not like editors go out of their way to remove nice explanations.

There are many anecdotal reports of tyrannical Wikipedia editors doing exactly this. They claim dominion over a subset of Wikipedia articles and then manicure them exactly to their own personal tastes. I'm sure the situation you describe happens often in various corners of the website.

I nearly based my Master's dissertation on a topic I discovered on a wikipedia math page when I was looking for something to cover. Turns out that subsection was maintained by the 'inventor' of topic and essentially served as a vanity page. The topic itself had no recognition in the community and if I had forged ahead on it, I would have failed my dissertation pretty hard.

Re: Intuitive Guide to Maxwell's Equations

#47

Post by potiuper » Wed, Jul 01, 2020, 5:14 PM UTC

ABeeSea wrote: post #11

chronolitus wrote: post #3

These kinds of visual, well thought out explanations of topics which are often taught with terse, obscure and uninviting methods are a gift. On a tangent: I remember asking my Calculus 101 professor what the "intuitive meaning" of divergence and curl was, outside of the formal math and equations. He was shocked that one could ask to sully these perfect mathematical concepts with dirty intuitive reductions. A guide li…

Most mathematicians consider the equations for div and curl to be the “dirty equations” while the “perfect mathematical concept” is the exterior derivative of a 1-form.

If the objective is to generalize, then differentiation is simply the adjoint of the boundary operation, which is a homomorphism; extending the notions of closed and exact forms to general differential forms along with that the differential operator is nilpotent allows setting up de Rham cohomology, which also has analogs by multiple authors.

Re: Intuitive Guide to Maxwell's Equations

#48

Post by signaru » Wed, Jul 01, 2020, 5:17 PM UTC

A chapter in Shen and Kong's Applied Electromagnetism helped me a lot back in college. It is graphically explaining the integrals and vector differential equations, something I didn't see in other textbooks. My intuition was then subsequently reinforced by coding a lot of FDTD (Finite Difference Time Domain) for research. The FDTD algorithm is so simple, that I wish physics teachers covered it in class.

Re: Intuitive Guide to Maxwell's Equations

#49

Post by imranq » Wed, Jul 01, 2020, 5:17 PM UTC

potiuper wrote: post #47

ABeeSea wrote: post #11

Earlier quoted context omitted.

Most mathematicians consider the equations for div and curl to be the “dirty equations” while the “perfect mathematical concept” is the exterior derivative of a 1-form.

If the objective is to generalize, then differentiation is simply the adjoint of the boundary operation, which is a homomorphism; extending the notions of closed and exact forms to general differential forms along with that the differential operator is nilpotent allows setting up de Rham cohomology, which also has analogs by multiple authors.

I guess there's also an aesthetic component to limit the extent of generalization

Re: Intuitive Guide to Maxwell's Equations

#50

Post by martincmartin » Wed, Jul 01, 2020, 5:40 PM UTC

billfruit wrote: post #33

martincmartin wrote: post #22

Earlier quoted context omitted.

The book Geometrical Vectors does exactly that. In fact, it considers the gradient vector to be a different kind of vector than a normal distance vector. If I have a vector between two points, and I compress space so that the points move closer together, then the distance vector gets smaller. However, the gradient vector gets bigger. The gradient is basically a density, it's units are units-of-whatever-your-taking-th…

Can you mention the name of the authours..

It's by Gabriel Weinreich.