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Calculus for the Curious

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Re: Calculus for the Curious

#31

Post by vharuck »

6gvONxR4sf7o wrote: post #5
ncfausti wrote: post #2

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, th…

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. We used this book ( https://smile.amazon.com/Introduction-Calculus-Analysis-Clas... ) but I expect it may be a bad fit for self study. Try to fi…

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!"

Re: Calculus for the Curious

#32

Post by ngcc_hk »

It is an odd review mixing negative comment and positive appreciation. The last bit did me as I am not historian but what to know the latest development of the field. And the Zeno etc kind of philosophical discussion lately.

Re: Calculus for the Curious

#33

Post by jacobolus »

vostok wrote: post #23
jacobolus wrote: post #6

Earlier quoted context omitted.

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.

I took this to mean that they're interested in a more rigorous treatment. > [I] 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

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.

Re: Calculus for the Curious

#34

Post by jacobolus »

yessenia1 wrote: post #17

I used to always associate 'math smarts' with 'code smarts'. Spent most of my life telling myself that since I was bad at math I had no hope learning how to code. Now I'm 2 weeks into a coding bootcamp after losing my job and am realizing they are complete different parts of the brain. I believe in myself more with every push to heroku. maybe I will do a Calculus class next and prove to myself I can learn anything. A…

> Spent most of my life telling myself that since I was bad at math

Note that very few people are actually “bad at math” in any kind of inherent way. The problem is usually a combination of psychologically damaging (and technically poor) teaching, parental/peer pressure, etc. leading to a phobia/mental block, which eventually leads people to construct an identity as “not a math person” (which has been tragically normalized in our society – in some places this doesn’t happen).

Plenty of the folks who say they are “bad at math” try again later under more relaxed and encouraging circumstances and are plenty successful.

So good luck!

If you are serious about it, my recommendation is to try to find a private tutor to meet with face to face.

Re: Calculus for the Curious

#35

Post by lilott8 »

lilott8 wrote: post #7
ncfausti wrote: post #2

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, th…

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…

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.

Re: Calculus for the Curious

#36

Post by chrisdhal »

lilott8 wrote: post #7
ncfausti wrote: post #2

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, th…

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…

Maybe I'm getting old, but I can't imagine a CS grad student not having taken a formal calculus class.

When I started my undergrad CS program in 1989 it required 4 semesters (2 full years) of calculus. This was in addition to 4 semesters of physics.

Maybe I'm just not up to date on what "Computer Science" is today.

Re: Calculus for the Curious

#37

Post by [deleted] »

chrisdhal wrote: post #36
lilott8 wrote: post #7

Earlier quoted context omitted.

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…

Maybe I'm getting old, but I can't imagine a CS grad student not having taken a formal calculus class. When I started my undergrad CS program in 1989 it required 4 semesters (2 full years) of calculus. This was in addition to 4 semesters of physics. Maybe I'm just not up to date on what "Computer Science" is today.

[deleted]

Re: Calculus for the Curious

#38

Post by rainbowzootsuit »

ncfausti wrote: post #2

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, th…

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.

https://www.youtube.com/playlist?list=PL8_xPU5epJddRABXqJ5h5...

Re: Calculus for the Curious

#39

Post by saeranv »

jacobolus wrote: post #3
ncfausti wrote: post #2

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, th…

Try http://www.math.smith.edu/~callahan/intromine.html which is a very conceptual and simulation-focused calculus curriculum, without so much symbol twiddling. 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 k…

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.

Re: Calculus for the Curious

#40

Post by [deleted] »

soVeryTired wrote: post #19
braindeath wrote: post #16

Earlier quoted context omitted.

Euler lived in the 1700s. Not sure how that would have affected your training unless you’re much older than average. Cauchy and Taylor both formalized many concepts, again in the 18th and 19th century. What are you thinking is ad box?

In high school mathematics, you're not really given the definition of a limit. Consider the definition of the derivative limit as h -> 0 of (f(x+h) - f(x)) / h. That's well-defined on (0, inf) but not on [0, inf). So you can't just evaluate at h=0 and be done with it. The intuition is 'as h gets smaller and smaller, the ratio gets closer and closer to a new function of x'. But many high-school students aren't given a…

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