I just stumbled upon Infinite Powers in B&N, and after reading a bit, decided to buy it. Loving it so far. I’ve been curious about calculus for a long time now, since I’ve never taken a formal class on it and am now in a graduate CS program, I feel like I’m missing the deeper understanding of many of the formulas that are presented.
My plan is to get through it to get some background on the main ideas of calculus, then work through khan academy and/or read through Aleksandrov’s Mathematics Contents/Meaning.
If anyone knows of active forums/q&a/online practice for self-learning calculus, it’d be a huge help if you could share.
The concepts of calculus (the mathematics of motion and change) are absolutely fundamental across science, but being able to get closed-form solutions to tricky indefinite integrals (while an enjoyable puzzle) is only marginally useful per se.
> work through khan academy
YMMV, but I don’t find KA to be especially pedagogically enlightened. I think of it as roughly an average-quality high school lecture videotaped, plus a big pile of mindless exercises. It’s still nice that it exists: it makes a quality floor for students with below-average teachers, and is free for everyone in the world, without any forced schedule.
You’ll get roughly the same result from just reading a standard introductory textbook and working the problems.
For better results, if you can afford the time, hire / find a private tutor to meet with regularly face to face.
I think KA fulfills a critical niche in the ecosystem of online study materials out there: he actually works through the math in real time on his 'blackboard'. That's one thing you can't get from watching 3Blue1Brown videos, or reading through textbooks.
Personally, I find I learn best when I can work my way through at least three/four different forms of understanding: initial intuition (ideally geometric) of the problem (i.e 3B1B, BetterExplained), the theoretical, proof-based solution you get from textbooks where you can see the derivation of the math concept, and finally, working through problems with a pen and paper. The fourth one is being able to code it from scratch or with the help from a basic library like numpy. KA helps immensely with that third form.
Downloaded the Calculus in Context when I read the your comment. That is exactly the kind of approach to math I appreciate and understand. I've been reading it and enjoying making simulations during the weekend. Thanks!
This is a fantastic thread https://twitter.com/nntaleb/status/1163192701472428032?lang=... from Taleb. As always, a little bit over the top, but mostly on point. I've seen the exact phenomenon he describes play out so many times now its not even funny. I'm doing a PhD in mathematical statistics, which is sort of like "calculus on steroids". I basically do calculus day in day out. Many of the problems we attack are simply not of much interest to mathematicians. I used to run my problems by math profs & they would say something like yeah it can be done, have you tried mathematica etc. instead of buckling down to do it. For instance, yesterday I had to prove that Fisher information of Cauchy is half. Now that's entirely calculus. You take a function f(x,t) = (pi*(1+(x-t)^2))^-1. You then take the log of that. Then you differentiate that w.r.t t. What you get is called the Fisher score function S(t). So you take the score function and differentiate that again. Lets call that g(x,t). Stick a negative sign in front of that. So now you have a complicated looking new function -g(x,t). You multiply the -g(x,t) with your original f(x,t) and integrate that product over the reals. The result is half.
Most mathematicians usually get stuck on that last step. But for (mathematical) statisticians, this os sort of our bread and butter integral. So we know a bunch of tricks. Here's one such trick - https://stats.stackexchange.com/questions/145017/cauchy-dist...
Its like riding a bicycle. Those who ride the most know how to ride. But there are some who want to know how do bikes actually work...which is not going to help you much with riding the bike.
Oh, hey. Are you me? I am wrapping my time in CS grad program, and I also, never took calculus as a formal class.
Now, I will say, save machine learning/AI, calculus isn't really necessary; the world is completely discrete.
That being said, that doesn't mean that knowing calculus wouldn't _enhance_ your ability to understand and digest some of the more difficult reductions and proofs in, say, a theory of computation course.
I relied on "The Calculus Tutoring Handbook"[0]. I wanted a book that had answers to _all_ the exercises for confidence building purposes. The book goes slow and provides a great amount of detail -- the authors are pretty good at not hand-waving.
I also found \r\learnmath useful as a "I have a problem and can't ask anyone" site. They are really friendly.
Am I wrong to say that even if the universe of your concern is discrete, calculus can at least describe the behavior of recursive discrete processes, among other things?
Depends on the process and the exact form of the discreteness.
Discreteness introduces discontinuities and errors, and it's usually possible to describe the errors analytically. But there are situations where discrete systems become numerically unstable and blow up while the smooth analytic equivalent has no problems.
Apparently I'm beyond the edit window of my original post. I mean the "world [in CS] is completely discrete[, in the context of mathematical modeling and abstraction.]"
I did not intend to imply that the world is discrete in the strictest sense. Just that, except for AI/ML, discrete math will prove much more helpful to understanding the concepts and material presented in a graduate CS curriculum.
The benefit studying continuous maths provides in the context of CS is the rigor and modeling skills one gains.
All of my thesis is rooted in Programming Languages, Compilers, and Type Theory. Continuous math is utterly useless in this context. It's all SAT/SMT, set theory, and graphs -- all of which are topics in discrete math.
I recommend "The Mechanical Universe" TV program, produced by Caltech with the Annenberg foundation.
Although it's not about calculus per se, it shows how it is used with physics. Newton having invented calculus in order to describe physics. However, they use the more standard Leibniz's notation on the program.
You probably won't be able to sit down and solve integrals after the show, but the program helps to take a practical and beautiful mathematics branch and gives viewers an intuition to its application that I didn't find in an actual math course.
If you have the chance, try to go through a formal proof and analysis based course that requires convergence proofs and all that (mathematical "analysis" isn't what you might think; it's a specific subject). For calculus, it is what drove home the point and the magic for me.
I loved calculus so much more once the professor walked us through the proofs and I was willing and able to understand them. Proving convergence is like being a cheeky kid saying, "Well, if you pick that small number, I'll just find a smaller one!"
The point of Spivak’s book is to be rigorous and offer hard problems, not teach the basics of how calculus is used.
Doing epsilon–delta proofs can be fun, but it’s mostly useful for aspiring pure mathematicians, and not really relevant per se for the grandparent poster.
If someone has never taken a calculus course, and many formulas are presented in other types of technical books which were developed using calculus, then those formulas will seem somewhat mysterious/foreign.
Doing delta–epsilon proofs isn’t necessary to clear that up though. Just a regular introductory calculus curriculum is likely sufficient.
My plan is to get through it to get some background on the main ideas of calculus, then work through khan academy and/or read through Aleksandrov’s Mathematics Contents/Meaning.
If anyone knows of active forums/q&a/online practice for self-learning calculus, it’d be a huge help if you could share.