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Learning Feynman's Trick for Integrals

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Re: Learning Feynman's Trick for Integrals

#21

Post by zamadatix »

analog31 wrote: post #14

I don't know if this is exactly the same as what I learned in high school as "integration by substitution." A number of years after I finished school, I was in a new town without a job, and got hired to teach a freshman algebra course at the nearby Big Ten university. About halfway into teaching the class, I was struck by the realization that virtually every problem was solved in the same way, by recognizing the "for…

I think the difference is something like Feynman’s trick simplifies a hard integral by introducing a parameter and differentiating the whole integral, while substitution simplifies an integral by changing variables to undo the chain rule. But it has been so long since I've done integration manually I'm not 100% sure that's an accurate description/the full story.

The thing I hated about integration was which approach would work and the best option for each approach were much more "do a lot and see what's right" and I was too lazy :).

Re: Learning Feynman's Trick for Integrals

#22

Post by jmyeet »

Back in college I stopped doing maths in second year as a major because of the way it was taught. I just hated it. Numerical methodds in particular broke me. My main problem was we never really got told how things fit together. Resources like 3blue1brown just didn't exist at that time, sadly. We just had dusty and expensive and very dry textbooks to rely on. For example, we just got through into ODEs and were told "just use e^at". We started doing contour integrals without really telling us what was going on. Honestly, things like linearity were never really taught for basic stuff like derivatives and integrals.

But I had always loved maths and went back to it much later. After having done some computer science, some concepts just made it click more for me. Like sets were a big one. Seeing functions as just a mapping between sets. Seeing functions as set elements. Seeing derivatives and integrals as simply the mapping between sets of functions.

What fascinates me is that differentiation is solved, basically. Don't come at me about known closed form expressions. But integration is not. Now this makes a certain amount of sense. Differentiation is non-injective after all. But what's more fascinating (and possibly really good evidence of my own neurodivergence) is that integration isn't just an algorithm. It requires some techniques to find, of which the Feynman technique is just one. I think I was introduced to it with the Basel problem. I have to confess I end up watching daily Tiktok integration problems. It scratches an itch.

I kinda wish I'd made it to complex analysis at least in college. I mean I kinda did. I do remember doing something with contour integrals. But it just wasn't structured well. By that I mean Laplace transforms, poles of a function in the S-plane and analytic continuations.

I'm not particularly proficient at the Feynman technique. Like I can't generally spot the alpha substitution that should be made. Maybe one day.

Re: Learning Feynman's Trick for Integrals

#23

Post by CyberDildonics »

jmyeet wrote: post #22

Back in college I stopped doing maths in second year as a major because of the way it was taught. I just hated it. Numerical methodds in particular broke me. My main problem was we never really got told how things fit together. Resources like 3blue1brown just didn't exist at that time, sadly. We just had dusty and expensive and very dry textbooks to rely on. For example, we just got through into ODEs and were told "j…

This seems like a bizarre comment that has almost nothing to do with the title.

Re: Learning Feynman's Trick for Integrals

#24

Post by pvitz »

When I was a student of physics and came across this paragraph in Feynman's book, I was curious if he really meant the simple technique explained in the article, a more general one (also described in the article with the integral bounds as functions of a parameter) or something else. I don't know, but this led me to read the text "Advanced Calculus" by Edwin Bidwell Wilson (1912), which includes a lot of examples and gems. If there is some young student out there who wants to go well beyond the basic techniques of calculus taught in analysis or mathematical physics courses, have a look at [0].

[0] https://archive.org/details/advancedcalculus031579mbp/mode/1...

Re: Learning Feynman's Trick for Integrals

#25

Post by [deleted] »

CyberDildonics wrote: post #23
jmyeet wrote: post #22

Back in college I stopped doing maths in second year as a major because of the way it was taught. I just hated it. Numerical methodds in particular broke me. My main problem was we never really got told how things fit together. Resources like 3blue1brown just didn't exist at that time, sadly. We just had dusty and expensive and very dry textbooks to rely on. For example, we just got through into ODEs and were told "j…

This seems like a bizarre comment that has almost nothing to do with the title.

[deleted]

Re: Learning Feynman's Trick for Integrals

#26

Post by [deleted] »

CyberDildonics wrote: post #23
jmyeet wrote: post #22

Back in college I stopped doing maths in second year as a major because of the way it was taught. I just hated it. Numerical methodds in particular broke me. My main problem was we never really got told how things fit together. Resources like 3blue1brown just didn't exist at that time, sadly. We just had dusty and expensive and very dry textbooks to rely on. For example, we just got through into ODEs and were told "j…

This seems like a bizarre comment that has almost nothing to do with the title.

[deleted]

Re: Learning Feynman's Trick for Integrals

#27

Post by praptak »

tacitusarc wrote: post #6

I just finished Mathematica by David Bessis and I wish this information was presented in the way he talks about math: using words and imagery to explain what is happening, and only using the equations to prove the words are true. I just haven’t had to use integral calculus in so many years, I don’t recall what the symbols mean and I certainly don’t care about them. That doesn’t mean I wouldn’t find the problem domain…

My intuition for the Feynman's trick is that we construct a "morph" which produces the given function (the parameter t drives the morphing).

The key to the trick is that we construct the morph so that: a) we can tell the rate at which it increases the "area under curve" b) the rate is easier to integrate that the original function and c) the starting function has a known integral

a) is generally easier because differentiation under integral sign lets use use the standard differentiation rules.

b) this is where the difficulty in constructing the morph lies.

So we start from a known value of the integral (from c above) and then just add whatever the morph adds, which is the integral of the rate from a) over the interval of the morph.

Re: Learning Feynman's Trick for Integrals

#28

Post by eru »

charcircuit wrote: post #20
analog31 wrote: post #14

I don't know if this is exactly the same as what I learned in high school as "integration by substitution." A number of years after I finished school, I was in a new town without a job, and got hired to teach a freshman algebra course at the nearby Big Ten university. About halfway into teaching the class, I was struck by the realization that virtually every problem was solved in the same way, by recognizing the "for…

I think it's intuitive to assume what you are being tested on is what is being taught by the book or the teacher. It's unfair otherwise.

Depends on your sense of fairness. Math Olympiads don't test what's in the book, but they are also fair.

Re: Learning Feynman's Trick for Integrals

#29

Post by TrackerFF »

lordnacho wrote: post #4

My issue with both this and u-substitution is that you don't know what expression to use. There are a LOT of expressions that plausibly simplify the integral. But you have to do a bunch of algebra for each one (and not screw it up!), without really knowing whether it actually helps. OTOH, if I'm given the expression, it's just mechanical and unrewarding.

I see your point, but as it is stated in the article, it is one of those techniques that require practice, and time to mature. And like it mentions, it's a bit like chess...when you're presented with some troubling integral, you can parametrize it in a number of ways. Most will bring you back to the beginning (like with the standard integration by parts), but the right one will make your life much easier.

It can be frustrating when math does not have any clear single path, but that's just the nature of the beast. In the beginning you'll just have to explore all the paths, but do that a couple of hundred times, and you'll start to notice patterns and what will work / what will not. Kind of like chess, where a good chess player can think N moves ahead in time.

Re: Learning Feynman's Trick for Integrals

#30

Post by kqr »

To people who find this stuff useful in practise today (and not merely fascinating or useful 50 years ago): what is your line of work?

I have needed to know the values of a few integrals in my job, but I have always ended up with a close enough answer using computational methods. What am I missing by not solving analytically?

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