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Too Much Calculus – Gilbert Strang (2001) [pdf]

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Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#111

Post by mcguire »

ams6110 wrote: post #59
ekidd wrote: post #7

As a programmer, I've found far more uses for linear algebra than for calculus. It shows up in robotics, in machine learning, in game development, and in many other cool subfields. I didn't get very much out of my university linear algebra course. (It was pure theory with nearly zero applications.) I don't think I really developed a good intuition for linear algebra—and why it's so useful—until I read Jim Hefferon's…

As a programmer, I've used calculus and linear algebra exactly never in my professional (20+ years) career. It just doesn't really come up doing web apps, sysadmin or dba work.

In my 20-odd years, I'd agree---uses of calc or linear algebra have been vanishingly rare, although my current project is changing that a bit. Not because I didn't have opportunities to use it, but because I didn't seek out those opportunities.

On the other hand, I'm prepared to defend to the death the statement that programming is applied formal logic. (Or possibly abstract algebra; that is likely more appropriate for some more modern things. On the other hand, I grew up in the land of Dijkstra, so....)

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#112

Post by peterfirefly »

jmilloy wrote: post #56
jeffreyrogers wrote: post #48

Earlier quoted context omitted.

I agree completely, but calculus touches so much of advanced math that at some point you have to learn it in depth (assuming you actually want to learn advanced mathematics). I'd argue that linear algebra is a more interesting subject for most people, but calculus contains more important ideas that are crucial if you intend to continue studying mathematics.

I have a major in Mathematics, and I truly didn't need calculus after the multivariable/vector calculus course I had to take my Freshman fall. Why did I have to do so much if it wasn't going to be necessary until graduate school at the earliest, or perhaps never depending on specialization? And why was it a prerequisite for my other math courses (in algebra, topology, real analysis, number theory, and combinatorics)?

How was topology motivated without talking about generalizations of limits?

(Honest question, no snark.)

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#113

Post by msandford »

cabinpark wrote: post #36
davidwihl wrote: post #13

Earlier quoted context omitted.

I'm amazed that you say that calculus has had a much greater impact on society when virtually every database, graphics manipulation or economic analysis uses principles of linear algebra. Can you substantiate "millions of transfer students between thousands of universities every quarter"? That seems like outlandish hyperbole. Irrespective of the precise number, a system with momentum is not sufficient justification f…

>I'm amazed that you say that calculus has had a much greater impact on society when virtually every database, graphics manipulation or economic analysis uses principles of linear algebra. I'm not. Consider all of physics. Physics is written down in terms of differential equations. You need calculus to get these equations. In terms of graphics, for example, are you magically creating only static images that never cha…

When I hear someone say "We don't need calculus!"

I think:

Okay, then go back to pre-calculus times and somehow still invent the modern world without it and then get back to me on how irrelevant it is.

Also, you don't get to wear this shirt: http://www.topatoco.com/merchant.mvc?Screen=PROD&Store_Code=...

My feeling is that it would take a few hundred years extra (or maybe more!) to get to the point where you had a computer without calculus. Might even take a thousand. You'd only be able to do discrete stuff, so while Babbage's difference engine might be up for grabs (and mechanical calculators definitely would be) things like the Norden bomb sight or naval firing computers wouldn't be, since they're continuous not discrete systems.

You might not EVER get to computers because things like Nyquist's theorem and stuff like that doesn't even exist in a discrete-only world. Things just sometimes work and sometimes don't and it's sorta related to the frequency, but how is unclear. Aliasing is just something that randomly bites you in the ass and there's no way to develop a rigorous understanding of it and to engineer yourself out of the morass.

I would not volunteer for that mission. I like the modern world.

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#114

Post by varelse »

peterfirefly wrote: post #107
Dudelander wrote: post #106

Earlier quoted context omitted.

Totally agree, calculus is a prerequisite for understanding statistics, unless you're just plugging in numbers for an A-B test - and you don't really need a class to do that. Linear Algebra is very useful too, but it's not really fair to say one is more valuable than another. It just depends on what kinds of applications you end up working on. Universities should just teach both.

Many interesting things require a combination of more than one area of math. I don't understand how you can do much within robotics or vision without using all three of calculus, linear algebra, and statistics :)

I'll go even further and state that without an understanding of calculus, linear algebra, and statistics, an engineer or even a data scientist is little more than a glorified script kiddy.

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#115

Post by romaniv »

peterfirefly wrote: post #84
vowelless wrote: post #67

Earlier quoted context omitted.

I agree although I work in robotics / computer vision. What I wish I had done a lot more of is statistics. Being able to reason about probabilities clearly, getting an idea of how a system performs given various variables, and most importantly, being able to communicate this to others has been very useful to me. I think anyone (front end, backend, etc) can benefit from learning about statistics.

But statistics and probability is really just integration. It's either that or stamp collection. And optimization is really just differentiation (+ some geometry). I'd say the problem is too little calculus and not too much.

Statistic teaches you application and helps to develop correct intuition. Can's speak for everyone, but my college algebra courses were 100% abstract stuff detached from reality. All professors had the attitude that once you learn the theory you could apply it to anything. Problem is, it's hard to remember theory if you don't map to something tangible, and it's hard to apply it to reality when you're just used to crunching equations.

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#116

Post by jimhefferon »

jseliger wrote: post #43
ekidd wrote: post #7

As a programmer, I've found far more uses for linear algebra than for calculus. It shows up in robotics, in machine learning, in game development, and in many other cool subfields. I didn't get very much out of my university linear algebra course. (It was pure theory with nearly zero applications.) I don't think I really developed a good intuition for linear algebra—and why it's so useful—until I read Jim Hefferon's…

until I read Jim Hefferon's free textbook I still prefer reading textbooks in paper, and one interesting note: it looks like a couple of companies and individuals are offering paper copies on Amazon, but none of them appear to be Henderson himself. And the companies or individual offering paper copies still charge $20 – $30!

> And the companies or individual offering paper copies still charge $20 – $30!

$20 for the official version (see the book's web page for a link). That is, essentially, the price of printing and binding.

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#117

Post by jmilloy »

peterfirefly wrote: post #112
jmilloy wrote: post #56

Earlier quoted context omitted.

I have a major in Mathematics, and I truly didn't need calculus after the multivariable/vector calculus course I had to take my Freshman fall. Why did I have to do so much if it wasn't going to be necessary until graduate school at the earliest, or perhaps never depending on specialization? And why was it a prerequisite for my other math courses (in algebra, topology, real analysis, number theory, and combinatorics)?

How was topology motivated without talking about generalizations of limits? (Honest question, no snark.)

It's a good question.

I think my first topology course, like in many intro analysis courses, shared concepts from calculus without requiring any previous expertise from our calculus courses. A rigorous approach to real analysis or point-set topology will be very set-theoretic, and you build the general definitions of continuity, compactness, etc from the ground up. So, I would have used concepts from my analysis and topology courses in calculus, if only it had been taught in the other order.

I guess it feels like limits are a rigorous foundation for calculus, but they are not calculus. In any case, I didn't need anything from differential or integral calculus (or that entire required course developing the theoretical basis for Stokes...), which is what I mean to say.

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#118

Post by joeyyang »

SixSigma wrote: post #3

I watched Prof. Strang's Linear Algebra courseware, great stuff. He is a wonderful teacher. http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-... As someone who hadn't studied it for 20 years, it was invaluable to revisit the subject as an adult with the ability to reflect "ah, I can use that for... " instead of "Ok but when am I ever going to use this"

This OCW was my lifeline during my college Linear Algebra course. Learned almost all the course material from 18.06 videos. Awesome, awesome stuff.

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#119

Post by mahyarm »

jmilloy wrote: post #33

> We need to present the mathematics that is most useful to the most students. I wish it weren't this way! Mathematics is a wide and fascinating field, and I think more students could find interest if our schooling reflected that. But moving some of the detail and drudgery that is the calculus sequence to subject matter that requires it, in order to make way for linear algebra, is a great start.

Mathematics is not taught well at all. Read a calculus textbook and you find out most of the words are undefined to most students, which are then taught to them 2-3 calculus classes later in 'discrete math'.

I think starting with discrete math first, where you teach what sets are, how proofs work and so on would be a good start. Or make your textbooks usable and not just containers of problem sets with unreadable extra paper inserted to make your back hurt more. Or make the target students and not their professors.

Re: Too Much Calculus – Gilbert Strang (2001) [pdf]

#120

Post by YAYERKA »

tptacek wrote: post #38
SixSigma wrote: post #3

I watched Prof. Strang's Linear Algebra courseware, great stuff. He is a wonderful teacher. http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-... As someone who hadn't studied it for 20 years, it was invaluable to revisit the subject as an adult with the ability to reflect "ah, I can use that for... " instead of "Ok but when am I ever going to use this"

Those lectures are fantastic. Pro-tip: watch them at 2.5x speed (you'll keep up just fine, and you can always pause to play with the concepts in Sage or whatever). Bonus: when you switch him back to normal speed, he sounds stoned out of his mind.

This is a great tip. I've been using it for a few years now to watch various lectures on youtube--especially mathematics. Professors lecturing mathematics tend to speak slowly since they don't want to say anything erroneous. Speeding them up is usually amazing. For youtube; open the javascript console in your browser and type '$('video').playbackRate = 2.5;'. I've found that each lecturer usually has their own magic number regarding speed--after watching someone for a few hours and varying the speed you can usually find it.
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