[HN Gopher] Richard Feynman's Integral Trick (2018)
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Richard Feynman's Integral Trick (2018)
Author : TheTrotters
Score : 216 points
Date : 2021-06-15 16:23 UTC (6 hours ago)
(HTM) web link (www.cantorsparadise.com)
(TXT) w3m dump (www.cantorsparadise.com)
| ChuckMcM wrote:
| Funny story, I was a huge fan boi of Feynman's and read
| everything I could about him. I used this technique to integrate
| the "extra credit" problem on my freshman final, which I turned
| in with about 20 minutes to spare, and the professor accused me
| of cheating by knowing the answer ahead of time. When I showed
| him the steps and explained the origins he allowed that perhaps I
| hadn't cheated, but was disappointed I had used a technique that
| wasn't taught in class so that was somewhat unfair to the other
| students.
|
| Not my best professor.
| aidenn0 wrote:
| I've only ever been accused of cheating once[1]; which is
| perhaps surprising given my poor study habits and high test
| scores. It probably helps that I tend towards a fairly
| deliberate pace on my tests so I was rarely done super-early.
|
| 1: That one time was not even in school; when I graduated I
| applied to OCS, and the recruiter told me point-blank that his
| superiors thought I cheated on the test. It was apparently the
| highest score anyone at this particular recruiting site had
| ever seen, and my grades were mediocre (see above about poor
| study habits).
| bongoman37 wrote:
| How is knowing the answer 'cheating' by any stretch of that
| word?
| ChuckMcM wrote:
| The assertion was that I had somehow acquired the answers to
| the test before I took the test. The evidence for that,
| flimsy as it was, was that I finished the test "too quickly
| to have done the work."
|
| Getting the answers before the test was the alleged cheating.
| regularfry wrote:
| What was frustrating to me, in the days before this sort of
| thing was all over YouTube, was reading that Feynman had a
| magical technique in biographies and so on, but not being able
| to find any reference to what the damn trick was.
| acchow wrote:
| > When I showed him the steps and explained the origins
|
| Did you not put the steps on the paper you submitted?
| ChuckMcM wrote:
| Not enough apparently.
|
| But to be fair, as a freshman in a college calculus class
| that was essentially reviewing what I had already done in
| high school I was kind of an asshole. I skipped a lot of his
| classes and often did the formulaic questions in my head and
| just wrote down the answer.
|
| I don't recommend this approach.
| sillysaurusx wrote:
| One thing I've often wondered: Is there a reason to learn all of
| these methods in the modern era, when Wolfram Alpha or
| Mathematica can apply hundreds of methods automatically?
|
| It's good to understand things conceptually. But once I got the
| concept of integrals as the area under a curve, it felt like a
| lot of grunt work to learn so many tactics for solving them. But
| most of my focus has been on computers rather than pure
| mathematics. For pure math, it probably makes more sense to learn
| as many different methods as possible.
| 6gvONxR4sf7o wrote:
| If you just want the value of an integral, maybe not. If you
| want to understand the integral, or manipulate it in different
| ways, yeah there's value. Frequently, a computational solver
| will spit out some giant expression, while if you did it
| manually, you'd end up with something more compact, perhaps
| making new intuitive definitions along the way (generally
| manual common subexpression elimination and factoring).
| qsort wrote:
| Speaking as someone who is completely useless at integrals
| except for the basic undergrad-level tricks, yes, there is.
|
| Rewriting formulas in different forms can allow you to see
| analogies between them, allowing you to prove "mini-theorems",
| which you can use to make computations more efficient or to
| adapt slightly different mathematical tools to your problem.
|
| Those things happen frequently even if you are a "just" a
| developer, (not necessarily with integrals/real analysis, for
| example combinatorics tricks are extremely common), but it's
| definitely a nice tool to have.
| reedjosh wrote:
| If you have any interest in master's level engineering, you
| cannot get by without a strong understanding.
|
| I took a single master's course as a deal to get my
| bachelor's and that was Random Signals and Stochastic
| Processes. Wow, you cannot get these concepts without a super
| strong mathematics background.
|
| To this date I think it was both the hardest and most
| fulfilling course I took.
| neffy wrote:
| Absolutely, because after a while this helps you internalise
| and see errors in other people's work, that everybody else is
| just letting flow past them.
|
| Even being able to do simple arithmetic in your head to check
| for errors in slides/talks is a major step up. It will only
| take one moment of realising that the person talking has made a
| mistake in their calculation, and is basing their argument on
| that mistake, to realise the power of this - and that's just
| the simple stuff.
| reasonabl_human wrote:
| This is the same line of thinking a lot of 'coding boot camps'
| seem to take- why learn 4 years of computer science and
| engineering fundamentals when you'll just be using node.js in
| the workplace?
|
| A dangerous trap IMO- both are valuable but incredibly
| different. The former is teaching a narrow-scoped trade as
| opposed to learning a full-fledged engineering discipline. The
| latter is much more generalizable and equips you to understand
| / build / use most tools going forward, rather than overfitting
| use only to the current fad of high-level tooling.
| chalst wrote:
| I don't think the two are so analogous. A problem with
| placing a high value on finding exact closed forms for
| integrations is that it encourages behaviour like the drunk
| looking across the street from where he lost his key because
| that is where the streetlight is. Most integrations just
| don't have closed solutions and we have to live with that
| fact.
|
| Bayesian statistics has been liberated by the ability to
| perform many dimensional integrations on the kind of
| likelihood functions appropriate for the problem, where
| before the advent of modern computational techniques Bayesian
| statistics had a reputation for concentrating on artificial
| problems that we happened to know how to solve.
| runawaybottle wrote:
| The counter point to that I can offer is that you can kill
| whatever spirit exists to learn it deeply if you throw
| someone into the deep first, and then deeper. The will to
| learn gets lost, whereas being effective can fuel wanting to
| be more effective. Human energy has to be paced, and will
| cultivated.
| phkahler wrote:
| >> being effective can fuel wanting to be more effective.
|
| I rather like that statement. Going to use it.
| brennanpeterson wrote:
| Mostly agreed, but I did like the last sentence: sometimes it
| is easier to solve the general problem than the specific one.
|
| That is a useful trick to keep in mind all the time. The point
| of course, is not to learn to solve integrals, but to learn a
| transferrable piece off mental jiujitsu.
| abdullahkhalids wrote:
| One of the goals of science is to explain the world.
|
| * While computational tools will symbolically solve a lot of
| integrals, they won't solve them all. Resorting to numerics
| often means you have lose some understanding along the way,
| because you no longer have a closed form expression to analyze.
|
| * One general strategy in Physics is to take a complicated
| expression and make different sets of simplifying assumptions
| to reduce it to simpler forms. This adds explanation to your
| model because you understand how the system is said to behave
| under different limitating scenarios. But if you are not adept
| at manipulating complicated expressions, you won't be able to
| use the strategy fully. Computer solvers are really bad at
| writing mathematical expressions in the nicest way possible so
| that the simplifying assumptions pop out naturally.
|
| Full disclosure: I am a physicist, who uses Mathematica quite a
| lot to solve various expressions (but I know the limitations of
| the tool).
| sillysaurusx wrote:
| Your second point is very interesting! Can you point me to
| any problems of that type? I'd be interested in learning how
| to make a mathematical model of something, and then simplify
| parts of it with various assumptions.
|
| The one I can think of offhand is the pendulum problem, where
| sin(theta) is approximately equal to theta for small values.
| But you made it sound like there are problems with multiple
| parts, and many different simplifications.
| petschge wrote:
| There definitely are. The simple pendulum (or rather
| harmonic oscillators) are the problem we want to transform
| harder stuff to, because we know how to solve that one. In
| other words, you might have to make a bunch of assumptions
| till you have reduced your problem to simple harmonic
| oscillations around a steady state that you have found
| separately.
| bernulli wrote:
| Check out Prandtl's Boundary Layer Theory
| https://en.wikipedia.org/wiki/Boundary_layer
|
| It's a set of smart physical observations/assumptions that
| allow you to find closed form solutions for the Navier
| Stokes equations in fluid mechanics
| https://en.wikipedia.org/wiki/Navier-Stokes_equations for
| the important case of flow close to some body, such as an
| airfoil.
| boxed wrote:
| The article says wolfram alpha times out on this problem.
| Kranar wrote:
| The free version of it does but the Pro version will
| calculate it.
| muench wrote:
| It seems yes there is a reason if you need integrals. From TFA:
| "You can also try having Wolfram Alpha compute it, and it will
| time out. We will need to be more creative."
| whatshisface wrote:
| Mathematica/Maple/Sagemath don't have a freemium timeout
| mechanic and can solve a lot more. Truth be told I think that
| integration techniques are much less broadly crucial for
| everyone to learn than they used to be, although you need to
| have some clue of what's going on because you need to be able
| to guide yourself towards posing problems in such a way that
| the integrals that can be solved.
| dragonwriter wrote:
| > Mathematica/Maple/Sagemath don't have a freemium timeout
| mechanic and can solve a lot more.
|
| Mathematica/Maple don't have freemium timeouts because they
| are not free; Sagemath, OTOH, is a good point.
| [deleted]
| kaba0 wrote:
| But there are integrals that you can easily solve by hand
| but both WolframAlpha and sage will (effectively) timeout
| on them. And I'm not even talking about something made
| deliberately hard for computers to symbolically analyze.
| dasudasu wrote:
| This is when 99.99% of the population would just whip out a
| numerical solver.
| mr_mitm wrote:
| Sometimes you are looking for deeper insights in some
| equations that can only be achieved by finding symbolic
| solutions.
|
| You would never figure out that black holes are a solution
| to the Einstein field equations of you just threw a
| numerical solver at it, for example. (Bad example because
| that's arguably the easiest solution to them but I hope you
| get my point.)
| hyperbovine wrote:
| It's going to take you a while to numerically solve that
| integral for the uncountably many values of \alpha that you
| are being asked to...
| benlivengood wrote:
| It's probably best to return a function that takes alpha
| as a parameter and numerically integrates for whichever
| finite set of alphas are required by the caller.
| fooker wrote:
| The usual method is to computer a reasonable number of
| solutions and do curve fitting.
| Kranar wrote:
| It's the algorithm in and of itself that serves as the
| solution to the integral.
| jjgreen wrote:
| 0.01% would whip out a numerical solver; 99.98% would go
| "huh?"
| ShinyRice wrote:
| Of course there is. It's vital to check your calculations in
| some way or another, and cross checking with other humans that
| know what they're doing ought to yield the correct answer
| eventually. I suppose this is mostly useful if you find nobody
| that knows how to use library X, and everyone uses Y, so the
| only other practical option to cross check are other humans.
| zwieback wrote:
| For me it's in-between: I like to understand the basics but
| happy to use numerical algorithms for everyday use.
|
| However, when I'm working on something that heavily uses a
| specific mathematical method I like to dig in and deeply
| understand that aspect, otherwise you can become dependent on
| other peoples implementations that may not be optimal for your
| use case.
| jordan_curve wrote:
| I'll disagree with everyone else here. There's not really a
| good reason to understand them, no. It's good to have a broad
| understanding of what tricks are out there and to get a general
| sense of what techniques might work where, but no reason to
| know them in depth or learn all the tricks.
|
| Computer Algebra is actually not great at solving integrals
| (and the problem is unsolvable in general). But it's not
| extremely common that one needs to symbolically integrate
| gnarly expressions and you can look up the tricks when it comes
| up.
|
| Much in the same way that I believe introductory linear algebra
| is bogged down by endless matrix computation without a
| computer, I think forcing students to compute a million
| different gross integrals quickly has diminishing returns.
| _jal wrote:
| If you don't know how a calculation is performed, depending on
| what you're doing, there's between a chance and a good chance
| you'll look at an error and not know it. Everything from bugs
| to typos to just using the wrong method for the job can cause
| you grief, and having an idea of what's going on makes it far
| easier to spot.
|
| I have been on both sides of that, and far prefer to know what
| I'm doing.
| bordercases wrote:
| If you can understand all the nuances and special cases of a
| concept through one-shot learning, go ahead.
| _0ffh wrote:
| Not sure, but I've had computer generated symbolic
| differentiations that the sw was absolutely not able to boil
| down to the compactness of the result I was able to come up
| with manually. It was really only useful as a test to verify my
| own result.
| rtomanek wrote:
| https://outline.com/XZ93MY
| prof-dr-ir wrote:
| How do we see that f(1) = 0 in the example? That claim is
| equivalent to:
|
| int_0^pi ln[1 - cos(x)] dx = - pi ln[2]
|
| Is this easy?
| kinkrtyavimoodh wrote:
| I think it's not obvious but it reduces to a relatively
| 'common' integral.
|
| 1 - cos(x) can be written as 2*cos^2(x/2)
|
| Take the logarithm and you get 2ln (cos(x/2)), which is
| relatively common and the solution is
| https://www.quora.com/How-do-I-integrate-log-cos-x-from-0-to...
| cryvate1284 wrote:
| Because the integral is then over ln(1) and the log of 1 is 0.
| prof-dr-ir wrote:
| No, that is saying that f(0) = 0 which is correct but not
| what I asked about (nor what was stated in the article).
|
| Incidentally, your correct observation is in contradiction
| with f(a) = 2 pi log(|a|) since the latter only holds for |a|
| >= 1. This is because f(a) is not a smooth function.
| contravariant wrote:
| Yeah I can't immediately spot something that would make it
| easy.
|
| It's much easier to prove that the difference between the
| integral of log(a^2 - 2 a cos(x) + 1) and 2 pi log(a) goes to 0
| when a goes to infinity.
| evouga wrote:
| As far as I can tell, it's not trivial, but can be derived by
| repeatedly exploiting symmetries of the trig functions and
| properties of logarithms.
|
| One observation is that by symmetry
|
| int_0^pi ln[1 - cos(x)] dx = int 0^pi ln[1 + cos(x)] dx
|
| and so the calculation is equivalent to
|
| 1/2 int_0^pi ln[ sin(x)^2 ] dx = 2 int_0^pi/2 ln[ sin(x) ] dx.
|
| Now we can use a similar trick again:
|
| int_0^pi/2 ln[ sin(x) ] dx = int_0^pi/2 ln[ cos(x) ] dx
|
| so
|
| 4 int_0^pi/2 ln[ sin(x) ] dx = 2 int_0^pi/2 ln[ sin(x) cos(x) ]
|
| = 2 int_0^pi/2 ln[1/2] dx + 2 int_0^pi/2 ln[sin(2x)] dx
|
| = - pi ln[2] + int_0^pi ln[sin(u)] du
|
| (using substitution u = 2x)
|
| = -pi ln[2] + 2 int_0^pi/2 ln[ sin(x) ] dx
|
| and so the original integral is -pi ln[2].
| Agingcoder wrote:
| While the technique is powerful, it may also not work : there are
| conditions to check to be allowed to differentiate under the
| integral sign.
| motohagiography wrote:
| If you don't have the background to get this, here's a quick
| tutorial on integrals https://cognicull.com/en/1dc797za , and it
| would be cool if cognicull included this Feynman's method in
| their ontology.
| wly_cdgr wrote:
| Really cool site, thanks
| SamBam wrote:
| A very interesting way to show information and to aid learning.
| I haven't explored it much, but I do feel like this could be
| great, but isn't quite fleshed out yet. I think the tree is
| very difficult to navigate without being able to see what each
| bubble represents before clicking on it. Also, since the site
| contains so much content, it would be nice if it could remember
| what you have understood (and maybe still show it grayed out,
| instead of the permanent-seeming deletion of nodes).
| motohagiography wrote:
| Interestingly, they have precisely that functionality, where
| you can "prune" nodes as you learn them, and scrolling down
| the page gives very good articles on each topic. It was an HN
| post a while ago and I revisit it to look up refreshers on
| math concepts in articles like this Feynman one.
| SamBam wrote:
| Right, I saw that, I was just nit-picking the "prune" UX.
| I'd far prefer to gray things out, not delete them for what
| seems to be for good.
| foolfoolz wrote:
| i'm so relieved i don't have to do math like this anymore
| renewiltord wrote:
| I recall learning this for an entrance exam and it was a right
| proper nightmare.
|
| Fifteen years ago and still terrifies me.
| strictnein wrote:
| Same. I was honestly getting anxious looking through all of
| that.
| fjert wrote:
| I've forgotten how to do most of it and when I look at it now
| it's hard to believe I was ever able to.
| Synaesthesia wrote:
| I think that when you're doing it for recreation or to practise
| problem solving it's probably a lot nicer than at school.
| paulpauper wrote:
| It worked for this one because we knew the answer beforehand and
| the best approach. Its not like we can generalize this. Change
| some of the terms and poof unsolvable
| kccqzy wrote:
| That's true in general for all integrals. Method A solves this
| formidable looking integral nicely and simply. Make one small
| change, method A completely fails and now you'd need method B
| to solve it, which is not at all related to method A.
| siraben wrote:
| A lot of undergraduate math programs in the US start with
| unnecessarily hard calculus classes "weed-outs" which is
| unfortunate, since it discourages students who might have pursued
| mathematics otherwise. I can say from personal experience that
| Calculus II was my worst math grade, I fared much better in
| rigorous and challenging classes like real analysis or
| differential topology. To do well in elementary calculus one has
| to seemingly practice integration techniques in various
| permutations for hours, and honestly for what future purpose I
| cannot say.
|
| EDIT: I finally understood calculus after taking introduction to
| real analysis, and it was amazing because for the first time all
| the hand-waving disappeared and could be replaced with rock-solid
| arguments and increasing levels of abstraction (starting from the
| very definition of what the real numbers are). This is also
| important because functions can get very pathological[0][1][2]
|
| [0] https://en.wikipedia.org/wiki/Weierstrass_function
| (continuous everywhere but differentiable nowhere)
|
| [1] https://en.wikipedia.org/wiki/Cantor_function (derivative is
| zero almost everywhere but f(x) goes from 0 to 1)
|
| [2] https://en.wikipedia.org/wiki/Thomae's_function (continuous
| at irrationals but discontinuous on rationals)
| acchow wrote:
| Integral problems is to math as Leetcode is to software
| engineering.
| ska wrote:
| > I finally understood calculus after taking introduction to
| real analysis,
|
| Real analysis is the version of these things taught to math
| students, rather than the (often mostly service) version that
| is taught for other programs (engineering, physics, etc.) that
| need calculus. It is unfortunate that many programs are
| structured so you can't even see this before surviving the
| standard 1st year calc progression, especially at large
| universities.
|
| How math-oriented or not a particular program is varies
| obviously, but it's pretty common to see this distinction. When
| I was an undergraduate, entry to the honors math program
| ignored all calculus classes and results entirely (if I recall
| correctly, it was based on having a 1st class standing in
| linear and modern algebra courses). I think was entirely
| possible to complete a math major with little or no calculus at
| all.
| gip wrote:
| Very similar experience for a lot of math programs in France.
| For about a year we did a lot of repetitive and uninteresting
| stuff (integration by parts, compute Taylor series on the
| whiteboard, ...).
|
| The only concept from that time I used in my day job is the
| binomial coefficients. Yet I don't regret taking the class, in
| some inexplicable way I feel it has made be better (at what I
| have no idea).
| commandlinefan wrote:
| > some inexplicable way I feel it has made be better
|
| The only thing I really "learned" from studying calculus that
| I actually apply to the real-world is the ability to _slow
| down_ and take each part individually. When I first took
| calculus, I tried to rush through the problems and inevitably
| dropped something important. It wasn 't until I started
| forcing myself to write down each step (even if I thought I
| could do it in my head) that I started actually getting the
| right answers, and knowing that I actually had.
| cameldrv wrote:
| My college roommate called this "Blue Collar Mathematics."
| ptmcc wrote:
| I quite like math, but I hated the college-level calc series. I
| had to struggle to just pull of a B average, even though I
| thoroughly understood the concepts. I went on to apply calculus
| and diff eqs in higher level classes quite successfully, where
| the rote memorization isn't the point.
|
| But those weed-out classes test a bunch of arcane mechanics and
| memorized formulas and transformations in the most
| intentionally obtuse exam questions possible. If you don't know
| the "one weird trick" you're kind of screwed.
|
| Math weed-out classes are a lot like the tech interview
| problem, but for STEM majors.
| kccqzy wrote:
| And the worst is that when you apply the "one weird trick"
| unexpectedly, you thoroughly confuse the graders. (In this
| particular case, it was an integration problem in which I
| used a substitution that wasn't taught by the professor.)
| paulpauper wrote:
| I am guessing that people with an aptitude for math ,such as
| top physicists and mathematicians, just breeze through calc
| 2. They don't study more than everyone else but rather it
| just clicks faster.
| concreteblock wrote:
| Yes, I think you're right. As a mediocre mathematician, I
| don't know of anyone in my good-but-not-top PhD program
| that struggled with computational math. This is the feeling
| that I got from their relaxed attitude towards TAing those
| classes. My classmates also had close to 4.0 gpas in
| undergrad.
| analog31 wrote:
| It's interesting that these classes have a particular
| character to them, and they are also coincidentally the core
| math "service" courses for engineering, the hard sciences,
| and pre-med students. The fun isn't allowed to begin until
| those kids are gone. Also, the accreditation requirements for
| those disciplines makes it very hard to change the lower
| level math curriculum.
|
| To help my kids stay interested in math, I offered them the
| following promise: "Suspend your judgement until you get a
| chance to do proofs, because proofs are when math comes
| alive." One of my kids became a math major.
| hansvm wrote:
| Ha, I became a math major accidentally when I happened to
| take a proof based course and math finally came alive for
| me.
| Afton wrote:
| It's funny because while I don't consider myself "good at
| math", I'd always learned math by learning the fundamentals,
| and deriving what I need at exam time, since I find
| memorizing leads me to (a) go insane with boredom and (b) if
| I don't understand what I'm memorizing, I might apply it
| incorrectly without it "looking wrong".
|
| Calc II was a class where it just wasn't possible because
| there was _too much_ to derive on any given exam. It took me
| 1 /2 way through the course to course-correct and make
| flashcards and such nonsense. Unfortunately that was the last
| (non-discrete) math class I took, so I never discovered what
| happens next.
| paulpauper wrote:
| Calculus 3 is probably what you mean. calculus 1 does not cover
| integrals that much beyond some of the basic techniques. The
| handwaving typically makes the class easier instead of harder.
| I find it hard to belive that someone who struggles with intro
| calculus will underderstand it by starting with elliptic
| functions.
| thebooktocome wrote:
| There are different course names in common use for the
| various divisions of the calculus curriculum. There's no
| standard, so quibbling over course titles is kind of empty.
| nwallin wrote:
| Not sure if the author's edit changed which calculus he's
| talking about, but at the time of this posting, they say
| Calculus 2, which meshes with my experience.
|
| Lots of schools break up calculus in different ways, and
| that's fine. My school (and the schools of lots of people I
| know) break calculus up into calc 1, which is limits,
| differentials, and a toe dipping into integrals. Calc 2 is
| the 8 or so different tools for integrating progressively
| more difficult integrals. Calc 3 is multivariate calculus.
|
| Lots of people have a very difficult time with calc 2. It
| feels very plodding- calc 1 and calc 3 (and diff eq and lin
| alg...) felt like I was learning new insight every week, calc
| 2 just felt like memorizing new vocabulary words. It wasn't
| just that it was hard, it was that it was hard and boring.
|
| (obviously if a school breaks calc up differently your
| experiences will probably be different)
| Retric wrote:
| AP classes have caused most US collages to split the
| material in similar ways between in Calc I, II, and III.
| Koshkin wrote:
| > _the very definition of what the real numbers are_
|
| I wonder what that was. (In my world it was just a bunch of
| axioms.)
| knicholes wrote:
| What finally made calculus click for me was my class in
| Numerical Mathematics where I actually wrote programs to take
| derivatives. Also because of that course, functions became
| familiar, fun, and easy to rearrange. It was life changing.
| a-dub wrote:
| going between discrete and continuous really helped me too. i
| think such a thing, using computers even, could do wonders
| for ug math education.
|
| some kind of awesome integrated class where kids work with
| robot toy cars comes to mind as an interesting way it could
| be presented. (start by measuring their behavior and
| collecting data, computing crude integrals on the computer,
| moving into analysis using the reals)
| rsj_hn wrote:
| At least when I went to undergrad, calculus was a university
| requirement, not a major requirement. Like English 101, except
| it is of course taught by math department faculty just as
| English 101 is taught by English faculty. Everyone had to take
| it or test out of it.
|
| For actual math requirements, you started with real and complex
| analysis, abstract algebra, geometry and topology, and then
| some applied math classes such as partial differential
| equations, or numerical methods. There was also a requirement
| for probability and statistics.
|
| One way you can tell the difference between a general
| requirement class and a major class is the size of the
| classroom and the majors taking the course. If you are in an
| auditorium with 300 freshmen taught by a TA and almost no one
| else in that course is a math major, then you are looking at a
| university requirement rather than a college requirement.
|
| University requirements are not intended to weed anyone out,
| that would be contrary to the goals of the university. They
| should be doable by all who are admitted. When I went to grad
| school, I had the pleasure of teaching some of these calculus
| classes, and no one considered this to be a weed out class or a
| math major class. All the math majors we had tested out of
| calculus in high school, and most of our students had
| humanities majors (as the STEM students also tended to test out
| of it). Giving those humanity majors lots of tricky problems in
| order to try to weed them out from their own majors wouldn't
| make any sense.
|
| Moreover the key skill in being a math major is the ability to
| do proofs. So the weed out classes tend to be real analysis or
| abstract algebra, as these are the classes where students first
| do proofs. As there are traditionally no proofs in calculus
| classes (the books may provide proofs, but you are not tested
| in your ability to prove theorems, but in your ability to
| calculate). Thus it wouldn't be a good weed out class for math
| majors even if it wasn't a general requirement class taken by
| all majors.
| acchow wrote:
| What university is this where a humanities major is required
| to learn and apply integration rules?
| rsj_hn wrote:
| undergrad was Arizona State. Yes, having a college
| education requires knowing basic stuff like how to write a
| college essay or how to find the area under a curve. At
| that time, it was grouped by Numeracy or Literacy
| Requirements, so calculus met the N1 requirement and you
| could satisfy your L1 with English 101. Of course you could
| take more advanced classes as well if you wanted, but there
| was no credit for taking high school math classes to
| satisfy the university Numeracy requirements. So no trig or
| pre-calc would cut it. There were also social studies
| requirements, etc. The idea is that a "liberal arts
| education" requires these. And of course virtually all math
| majors would already have tested out of them, even in big
| state schools.
| hammock wrote:
| _raises hand_
|
| I started freshman year intending to major in math, started
| with Calc 3. When the average grade among my classmates on the
| first test was a 56/100, curved of course, I knew something had
| to give. This was not the fun math I knew from before. A+
| student up until this moment.
| a-dub wrote:
| i always found it funny how much time was spent drilling
| techniques for different integral types or doing transforms for
| derivatives, yet the most important idea: the continuous nature
| of the reals and why this is the rug that pulls it all
| together, and the delta-epsilon definition of the limit, only
| saw about a grand total of 3 minutes of hand waving and an
| optional problem on one homework.
|
| ug calculus was about having algebraic/trig manipulations
| memorized along with a table of transforms and a handful of
| tricks; where the actually beautiful ideas that if we use an
| infinitely "elastic" representation of numbers, we can solve
| hard approximation problems both correctly and easily- get
| totally glossed over.
|
| physics has the same problem. basic physics without calculus is
| just a bunch of rote memorization. the idea that such a thing
| is taught and that is somehow "easier" is nuts. they should be
| taught together, as they were developed, as many of the
| expressions given to undergrads in physics are simply
| definitions of integrals and derivatives applied a few times.
|
| no student in calculus should ever be wondering what the
| constant is for in a computed integral and no student in
| physics should wonder where the constants come from in
| equations of motion. (there should be no equations of motion,
| just definitions in integral/derivative form and definitions of
| integrals and derivatives)
| concreteblock wrote:
| As someone who has taught intro calculus a few times, one
| reason for the emphasis on computational techniques is simply
| that 90% of the students in such a class do not care/are not
| capable of grasping the epsilon-delta definition of the
| limit.
|
| Solution: spend more time on epsilon-delta so that students
| have time to wrap their minds around the idea. But I think
| the engineering departments would complain that the students
| who we send on to them cannot do basic computations. Also
| students would complain that we spend too much time on theory
| and not enough on application. There are probably other
| reasons that someone more experienced would know about.
| Trung0246 wrote:
| I don't really understand the part from how did the author jumps
| from -pi*(1+a^2)/(1-a^2) to df/da = 2pi/a. Anyone knows how the
| author did it?
| gglon wrote:
| In Mathematica 12.3: Integrate[Log[1 - 2 a Cos[x] + a^2], {x, 0,
| Pi}, Assumptions -> Abs[a] >= 1 && a \\[Element] Reals] gives the
| correct answer -\\[Pi] Log[1/a^2]
| marosgrego wrote:
| Actually, this method was already used by Leibniz, although it
| was not that common at Feynman's time.
| https://en.wikipedia.org/wiki/Leibniz_integral_rule
| alisonkisk wrote:
| Yes, that's the first sentence of the main article body.
|
| > Today's article is going to discuss an obscure but powerful
| integration technique most commonly known as differentiation
| under the integral sign, but occasionally referred to as
| "Feynman's technique" due to his popularization of this
| technique in his book, and properly known as the Leibniz
| Integral Rule.
| maest wrote:
| A think I've been wondering is why is integration harder than
| differentiation. The latter can be done almost mechanically, as
| long as your primitive functions are "nice", but the former often
| requires cleverness like what's show in the article.
|
| I mean, sure, we have simpler rules for dfferentiation, but
| _why_?
|
| I sometimes wonder if it's differentiation is P and integration
| is NP (for the restricted case of functions where the primitives
| are "nice")
| jacobwilliamroy wrote:
| What's the difference between "an arbitrary constant" and "a
| variable"?
| Koshkin wrote:
| In programmer's terms, an (arbitrary) constant is a
| _parameter_. (A variable is, well, a variable.)
| analog31 wrote:
| The definition is arbitary. And it varies. ;-)
|
| Both are symbols. The difference is kind of subjective, and has
| to do with how you treat the symbol. Do you just carry it
| through your derivation, or are you interested in what happens
| when you feed it specific values?
|
| But I believe your objection is valid. And to be honest I got
| all the way through a college math major by just treating
| everything as symbol manipulation.
|
| Caring about numerical values was for my other major, physics.
| mywittyname wrote:
| This reminds me of how much I struggled with integral calc in
| college. My textbook (Stewart) had a table of integrals
| containing 120 forms that you'd need to solve the problems in the
| book, and looking through them, the calculations seem so
| insurmountable.
|
| Like,
| https://www.wolframalpha.com/input/?i=integrate+u%5En+sqrt%2...
|
| I looked at that and realize that I'd have no future as a
| physicist and switched to CS.
| siraben wrote:
| We also used Stewart, and I think struggling (or excelling for
| that matter) at integral calculus is a poor indicator of one's
| competency in higher-level math, which is much more than
| memorizing 120 rewrite rules :)
| bntyhntr wrote:
| My AP calc teacher was big into Leithold, specifically TC7
| (https://www.amazon.com/Calculus-7-Louis-
| Leithold/dp/06734691...) when I had him. Claimed Stewart was
| useless, but he had a lot of strong opinions. I don't really
| have anything to say except that I have this feeling that I'm
| supposed to preach the gospel of TC7 anytime it comes up, so
| may as well :). Haven't touched calc in 10 years but sometimes
| still find myself trying to remember the chain rule (usually
| when miserable on a run)
| marcosdumay wrote:
| > but sometimes still find myself trying to remember the
| chain rule
|
| There's an algebra of differentials that was formalized quite
| late (I think at the 19th century) but accepts all of the
| operations you can use for scalars. The chain rule is just
| fraction simplification.
| klyrs wrote:
| This paper is an old favorite of mine. It shows how to
| transform a program that computes the value of a multivariate
| function into a program that computes that value and all of
| its first derivatives. The resulting program requires at most
| 7x more instructions as the original. Spoiler: it's just the
| rules of differentiation, and the constant 7 comes from the
| quotient rule.
|
| https://courses.cs.washington.edu/courses/cse446/18wi/slides.
| ..
| runawaybottle wrote:
| Same textbook. The amount auxiliary material I needed to watch
| off YouTube to grok it was very real. I had one of those
| impenetrable profs that only explained calc in the most
| theoretical terms.
| BearOso wrote:
| I think the teachers who choose Stewart books don't
| necessarily know how to teach down very well, so that would
| probably be reflected in their lectures, too. In multi-
| variable calculus I had a Stewart book and the professor put
| me to sleep. I spent a lot more time figuring it out on my
| own than I should have.
|
| I was too naive to do so, but if anyone out there is in a
| class and the official suggested book doesn't help you, ask
| the Internet for a respected alternative.
| joshka wrote:
| I had good teachers and Stewart and did pretty well
| (2001-ish, not sure how that translates, just one counter
| point - sorta)
| scrozart wrote:
| IME, Stewart is pretty opaque. The intro section is a great
| refresher of things you'll need, but the remaining text is
| pretty muddy. I found an old Thomas Finney book that was much
| clearer, as was Richard Delaware's YouTube series.
| ofrzeta wrote:
| In Germany there's a proverb: "Differenzieren ist Handwerk,
| Integrieren ist Kunst" - differentiation is craft, integration is
| an art.
| lapetitejort wrote:
| Best exemplified by x^x. Differentiation is tricky but doable.
| Integration is impossible.
| AnimalMuppet wrote:
| Depends on your definition of "impossible" ;-)
|
| In high school, I transformed x^x into e^(x log x), expanded
| that into the Taylor series, and integrated term by term. I
| got a "solution", but it wasn't closed form - it was an
| infinite series. And, for a given error limit, it probably
| converged more slowly than a decent numerical integration.
| So, not worth much. But I "solved" it...
| dhosek wrote:
| In English too. I remember my high school calc teacher saying
| this (except IIRC it was differentiation is science, not
| craft).
| maxnoe wrote:
| Relevant xkcd: https://xkcd.com/2117/
| arcadi7 wrote:
| the funny thing is that the calculation in this article misses
| the point, especially in the Feynman context. First, beyond all
| trickery, the log(alpha) answer might suggest that something bad
| happens at alpha=0 . What makes this integral interesting is that
| it is equal to zero identically for alpha<1 .
|
| The reason, of course, is that this integral is not randomly
| chosen -- it represents the two-dimensional coulomb potential
| (log(r)) of the sphere (circle) of radius 1 at distance alpha
| from the center. By when point alpha is inside the circle , the
| potential is constant (or zero -- no force) . When alpha is
| outside, the potential is log(r) as if all the mass of a circle
| is at its center. The expression under the log in the integral is
| just (square of ) the distance between the point alpha and point
| on a unit circle.
|
| beyond tricks -- the physical reason for the singular behavior of
| this integral is gauss theorem for coulomb potential . so no
| magic.
| gobrewers14 wrote:
| The first integral can be solved replacing the integrand with a
| series of sort. Notice that the expression inside the logarithm
| has zeros at $\alpha = e^{\pm ix}$
|
| So we can rewrite the function we're integrating as
| $log((\alpha - e^{ix})(\alpha - e^{-ix}))$
|
| which is just $2log(\alpha) + log(1 -
| \frac{e^{ix}}{\alpha}) + log(1 - \frac{e^{-ix}}{\alpha})$
|
| Using $log(1 - x) = -\sum_{n=1}^{\infty}
| \frac{x^n}{n}$
|
| We get $2log(\alpha) -
| \sum\frac{e^{inx}}{n\alpha^{n}}-\sum\frac{e^{-inx}}{n\alpha^{n}}$
|
| which is just $2log(\alpha) -2\sum
| \frac{cos(nx)}{n\alpha^n}$
|
| The integral of the second half of this involves a $sin(nx)$ term
| which will evaluate to zero for all values of \alpha at 0 and
| \pi.
|
| Leaving just the integral of $2log(\alpha)$ which is just $2\pi
| log(\alpha)$
| dang wrote:
| Past related threads:
|
| _Differentiation Under Integral Sign (2015) [pdf]_ -
| https://news.ycombinator.com/item?id=26123750 - Feb 2021 (59
| comments)
|
| _Feynman 's Integral Trick_ -
| https://news.ycombinator.com/item?id=26040353 - Feb 2021 (6
| comments)
|
| _Richard Feynman 's Integral Trick_ -
| https://news.ycombinator.com/item?id=21055728 - Sept 2019 (8
| comments)
|
| _Richard Feynman 's Integral Trick_ -
| https://news.ycombinator.com/item?id=17558752 - July 2018 (35
| comments)
| dynm wrote:
| This is fantastic. I've tried several times to understand this
| idea over the years, with no success. This clearly expressed the
| idea in only a few minutes.
|
| One question: It mentions that Wolfram alpha will fail on
| integrals that this trick can work for. Is that just because it
| will time out (we need more compute) or is the trick difficult to
| automate?
| nimish wrote:
| A little bit of both. I don't think WA uses the full Risch
| algo, but even then, enough pattern matching rules
| https://rulebasedintegration.org/ beats it in perf.
|
| Differentiation under the integral sign only works for certain
| well behaved functions and isn't easy to automate since you now
| need to figure out where to parametrize and you don't have good
| structure theorems to help you.
|
| IMO contour integration is a more powerful and easier to intuit
| technique.
| kzrdude wrote:
| Amazing website. I love that I could just click ahead and
| find the .pdf transcripts of the tests ran.
| Kranar wrote:
| Unlike differentiation, indefinite integration is undecidable
| even for elementary functions so there will always be some
| limit to what an algorithm can compute.
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