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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 method
by sillysaurusx 5y ago
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.
- brennanpeterson 5y agoMostly 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.
- muench 5y agoIt 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 5y agoMathematica/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 5y ago> 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 5y ago[deleted]
- kaba0 5y agoBut 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.
- whatshisface 5y agoReally? I've never seen one, could you give some examples?
- bzbarsky 5y agoFor Wolfram Alpha, I just tried the integral from https://en.wikipedia.org/wiki/Contour_integration#Example_2_%E2%80%93_Cauchy_distribution https://en.wikipedia.org/wiki/Contour_integration#Example_2_... and it times out. The by-hand solution is pretty straightforward, if you know contour integration.
- dasudasu 5y agoThis is when 99.99% of the population would just whip out a numerical solver.
- hyperbovine 5y agoIt'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 5y agoIt'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 5y agoThe usual method is to computer a reasonable number of solutions and do curve fitting.
- Kranar 5y agoIt's the algorithm in and of itself that serves as the solution to the integral.
- mr_mitm 5y agoSometimes 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.)
- jjgreen 5y ago0.01% would whip out a numerical solver; 99.98% would go "huh?"
- ShinyRice 5y agoOf 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.
- _jal 5y agoIf 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.
- qsort 5y agoSpeaking 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 5y agoIf 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 5y agoAbsolutely, 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 5y agoThis 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.
- runawaybottle 5y agoThe 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 5y ago>> being effective can fuel wanting to be more effective. I rather like that statement. Going to use it.
- chalst 5y agoI 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.
- zwieback 5y agoFor 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.
- _0ffh 5y agoNot 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.
- 6gvONxR4sf7o 5y agoIf 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).
- abdullahkhalids 5y agoOne 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 5y agoYour 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 5y agoThere 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 5y agoCheck out Prandtl's Boundary Layer Theory https://en.wikipedia.org/wiki/Boundary_layer 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 https://en.wikipedia.org/wiki/Navier–Stokes_equations for the important case of flow close to some body, such as an airfoil.
- jordan_curve 5y agoI’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.
- boxed 5y agoThe article says wolfram alpha times out on this problem.
- Kranar 5y agoThe free version of it does but the Pro version will calculate it.
- bordercases 5y agoIf you can understand all the nuances and special cases of a concept through one-shot learning, go ahead.