Comment by dannymi | Hacker News Reader

Comment by dannymi | Hacker News Reader

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>But then came the physics class, why we could go from dy/dx = f(x) to dy = f(x).dx and integrate both sides remain a mystery. "dx" suddenly became an infinitely small things, when it used to be a notation for a linear form OR a notation for f'(x) === df/dx.

It's the mathematicians who changed it.

Infinitesimals used to be how differential equations were defined as in the first place (by Newton and Leibniz). Physicists still think like that because we are end users of mathematics. So the interface has to stay the same. Old text have to keep working etc. Mathematics for us is not so different to software as an end user.

In the quest for formalization, infinitesimals have been removed and replaced by limits by mathematicians.

And now they have found a way to have a rigorous notation that has infinitesimals again:

https://people.math.wisc.edu/~keisler/foundations.pdf

Your example is explained on page 34.

On the other hand, a notation of mathematics that is incompatible with infinitesimals will not be a success in physics or engineering, for obvious reasons (compatibility; and also by using infinitesimals one almost always gets to the correct result anyway--even in very complicated cases--, and they are way more intuitive than limits).

So there's no problem--it's just that life is short and there's too much to learn.

If you want to prove theorems, by all means, use the most rigorous tools--but to use the result it's nice to have a simple interface.

Mathematics, for us, is a tool to communicate ideas (about the physical world). Overcomplicating it will obscure the ideas.

So this should be explained to the students instead of letting them feeling around in the dark confused.

I just (barely) passed what is known as my university's hardest first year physics course: electricity and magnetism. We didn't use differential equations, but we had the same hand wave "just think about it" relationship with line integrals. The way my professor explained it was "we're engineers, not mathematicians, so we actually know what we're talking about".

What he meant by this is that as engineers, we need to have intuition as to what an integral actually is rather than knowing how to prove things with them. To an extent, this is a good way to go about it. Really it's not at all useful to understand things like delta-epsilon as an engineer because those concepts really have no application. It's much more useful to just gain an intuition as to what you're actually doing when you're taking a line integral, and applying that to, for example, calculating an electric field strength as a result of an object.

Physics teaches physics, not math.

Did this professor think conservative fields and independence of path were for schmucks too?

My engineering and physics intuition came from understanding the math behind it. For example, I've seen a lot of handwavy explanations of how jet engines worked, but understanding eluded me until I went through the math in jet engine class.

Can you elaborate? Im and Aero and while there is a defo a bunch of math involved suck, squeeze, bang, blow is pretty much the gist of how jet engines work and I don't find it really all that hand-wavy, its just a funny way of describing gas pressure and velocity graphs.

The hand-wavy explanations never explained why the exhaust went out the back rather than the front and the back.

Do elaborate. The most funky differential I know comes from EE's Maxwell eqns (how capacitors work, and weird ideas about "ground").

The compressor up front produces enough pressure to prevent the exhaust going out that way.

Another way to see it is you cannot start a turbine by dumping fuel in and igniting it. The turbine has to be spun up first, usually with an electric motor. The Me262 engines had an ingenious tiny gas engine in the nacelle which served that purpose, complete with a little handle to start it!

Next time you're on a jet, watch them do the engine start. You can see it slowly gaining speed, then suddenly it dramatically speeds up - that's when the fuel gets squirted in and ignited.

Anyhow, jet engines look deceptively simple. But they are real masterpieces of engineering.

Different strokes for different folks.

Michael Faraday was brilliant and driven, but his intuition did not come from understanding the math as he was famously math-illiterate.

On a personal note, the subject I had the most trouble with was classic electromagnetism (calc-based freshman physics): It was the first time I hit a wall/limit on what I could grasp.

The wall was because I could never couldn’t develop an intuition for the physics behind it.

>The wall was because I could never couldn’t develop an intuition for the physics behind it.

On the off chance that you can understand German, check out chapter 9.1.1 Fluidvolumen und Divergenz (until and including chapter 11 Erhaltung der Energie), in the book Einführung Theoretische Meteorologie by Michael Hantel (ISBN 978-3-8274-3055-7). It even uses a car analogy ;)

You will understand fluid dynamics, and by extension electrodynamics, like you never did before. And it will teach you the physical intuition first.

This book was such a lucky find for me ten years ago.

Equation (11.3) is the fluid analogon to the continuity equation (in electrodynamics that is: ∂/∂t ρ + ∇⃗⋅J⃗ = 0)--and you can see how, physically, you get to it, and what the parts and the whole mean. Lots of diagrams and geometry and simple analogies make that take about 30 pages.

I think you are just reiterating what the person I was replying to said. Makes sense, what I was saying is that it should be explained that there are differences with pure mathematics and that it's meant to be a pragmatic tool, and maybe even some of the reasoning as to why the differences are the way they are.

It was explained in my physics classes over and over again

It's not possible for a teacher to guess all the possible ways a student might be confused. When the student is confused, the onus to ask questions is on the student.

Everyone in this class was uniquely confused compared to every other class I took. Had a 80% drop-out rate. I don't think it was fully a problem of the students.

As you pointed out infintessimal manipulation in physics produces the correct results most of the time. In fact, it's almost always correct because Taylor Series say so. In the physical world we deal with differentials that are always real. Technically physicists are rejecting the later terms dealing with convergence because for all practical purposes they don't matter.

This frustrates math people like me to no end but I can see why they do that. To a mathematician dx is certainly not a number. It's an infintessimal and therefore not a real number. But physicists can get away with it...their math is simpler.

Thanks for this link. A bit hard to digest more than a decade after I last did formal math but very approachable non the less

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