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

www-math.mit.edu

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

#81

Post by superobserver »

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…

Thank you for the link. I had the same experience in uni. It's a general issue that plagues mathematics education, I think: theory should come after some basic understanding of applications, not before.

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

#82

Post by omaranto »

jmilloy wrote: post #40
omaranto wrote: post #32

Earlier quoted context omitted.

This I understand, but that sentence with "cliché" in it I still don't get.

It's pretty clear to me that the OP just used the word cliche where it didn't belong. Fortunately the idea still came across. Perhaps you were truly confused, but it sounds more like you're playing dumb to force the OP to admit the mistake.

If by "the idea still came across", you mean I was able to guess approximately what the OP meant, you're right. I was curious about exactly what he or she meant and that's why I asked.

And I wasn't playing dumb. For one thing, even if I assumed the word cliché was used incorrectly, I still couldn't easily guess exactly what the OP really meant. Second, I don't like assuming people are using words incorrectly, it is safer to ask what is meant.

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

#83

Post by pmalynin »

xamuel wrote: post #10

(Speaking from my experience as a calc teacher at a big U.S. state U.; mileage may vary elsewhere) Most non-STEM students won't see the 'Calculus I, Calculus II, Calculus III' that starts the essay so dramatically. Increasingly, liberal arts students are able to get by with no more than a "math appreciation" course and maybe what amounts to high school algebra. The "math appreciation" course might actually touch on v…

I must disagree. In my experience Linear Algebra has been so much easier than Calculus because even though at times it can be very abstract it is still more familiar as most of the time you're just solving equations. My math professors have done quite a good job in connecting the two subjects. For example, differentiation and integration of polynomials can be expressed as a linear algebra problem, etc. etc.

"The fundamental theorem of calculus is a legitimately earth-shattering breakthrough: none of the theorems in linear algebra come anywhere near it in cultural importance."

Linear Algebra has the Fundamental Theorem of Algebra (i.e. every polynomial has a solution in the complex numbers) and also the Isomorphism Theorems that say all vector spaces of n-dimensions are the same. For example polynomials of the form c+bx are equivalent to the complex numbers (over the reals) which are equivalent to two-vectors (x,y). Quantum physics is basically linear algebra, with quantum states defined as vectors in a quantum vector-space.

Don't dismiss the impact of Linear Algebra.

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

#84

Post by peterfirefly »

vowelless wrote: post #67
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…

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.

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

#85

Post by relaytheurgency »

cageface wrote: post #11

I really can't agree with this strongly enough. I had enough Calculus and Differential Equations courses to kill a horse in college (Chemistry major) and I've never used that knowledge once in my entire programming career. But I find uses for linear algebra and matrices in coding all the time. Thank god Strang's excellent course is available online for free.

"I learned more than enough welding when apprenticing as a metalsmith but now as a chef use none of it."

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

#86

Post by samatman »

andrepd wrote: post #26
cageface wrote: post #11

I really can't agree with this strongly enough. I had enough Calculus and Differential Equations courses to kill a horse in college (Chemistry major) and I've never used that knowledge once in my entire programming career. But I find uses for linear algebra and matrices in coding all the time. Thank god Strang's excellent course is available online for free.

If you were looking to become a programmer, maybe you shouldn't have majored in chemstry, where strong Calculus foundations make perfect sense :)

Chemist here, who also programs. Inorganic chemists do calculus all the time. Organic chemists do arithmetic (stoichiometry) and some discrete analysis (instrumentation, the computers do the heavy lifting), orgo is primarily a linguistic discipline, i.e. we are expected to parse sentences like "benzophenone in THF was refluxed in the presence of Pd black for 2 hours" and to know, without being told, that there's a workup before the next step in the reaction.

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

#87

Post by krussell »

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!

I just noticed under the "Download Linear Algebra" section on http://joshua.smcvt.edu/linearalgebra/ , he gives a link to a paperback version of Linear Algebra on Amazon (ISBN-13: 978-0989897563).

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

#88

Post by cageface »

samatman wrote: post #86
andrepd wrote: post #26

Earlier quoted context omitted.

If you were looking to become a programmer, maybe you shouldn't have majored in chemstry, where strong Calculus foundations make perfect sense :)

Chemist here, who also programs. Inorganic chemists do calculus all the time. Organic chemists do arithmetic (stoichiometry) and some discrete analysis (instrumentation, the computers do the heavy lifting), orgo is primarily a linguistic discipline, i.e. we are expected to parse sentences like "benzophenone in THF was refluxed in the presence of Pd black for 2 hours" and to know, without being told, that there's a wo…

Exactly. I was an organic chemist. Never used a lick of Calculus. I didn't get into programming until I'd already been working as a chemist for about five years.

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

#89

Post by pakled_engineer »

ludicast wrote: post #20

I agree. As far as calculus goes, I am more enamored with books like Spivak's ( http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098... ) that take a proof-centric approach to teach calculus from first principles. Incidentally, for those who want to learn linear algebra for CS in a mooc setting there are 3 classes running at this very moment: https://www.edx.org/course/linear-algebra-foundations-fronti... (fr…

My calc I course in university was applied calculus without a text. I had to go back and redo single variable by reading Spivak (and Polya's How to Solve It) to figure out the proofs in Concrete Math by D. Knuth, et. al.

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

#90

Post by RogerL »

xamuel wrote: post #10

(Speaking from my experience as a calc teacher at a big U.S. state U.; mileage may vary elsewhere) Most non-STEM students won't see the 'Calculus I, Calculus II, Calculus III' that starts the essay so dramatically. Increasingly, liberal arts students are able to get by with no more than a "math appreciation" course and maybe what amounts to high school algebra. The "math appreciation" course might actually touch on v…

"Consider a spherical cow in a vacuum on a frictionless plane..."

That joke comes from the analytic calculus tradition of physics. I took a lot of physics while an undergrad, and I came out unable to do any useful physics. This is because as soon as you try even very simple problems the partial differential equations that result have either no, or extremely difficult to find solutions, and it is usually the former.

I spent years learning analytical tricks that I just don't apply.

There has been a lot of push back on this in education recently. There are Giordano's and Sherwood&Chabay's textbooks in physics now which emphasize computation over analytics. The reality is that if you want to do the vast majority of work in physics you will be doing things like reducing your problem to Ax=b, and throwing the linear algebra machinery against it.

I type this while taking a break from Strang's "Linear Algebra, Geodesy, and GPS" textbook, which I need for work right now. There is an endless need for this sort of math.

Sure, it is a matter of balance. But I flip through my diff eq books and I don't see a lot in there that I actually need. There is nothing in Strang that I don't know, and ditto for a good numerical methods book.

Even the work that Gauss did with orbits did was linear algebra and computation. He did not analytically derive the motion of Ceres, for example, but used linear algebra to compute it.

I, along with many other posters on this thread, and everyone that I have talked to, wish that there STEM education had been much more about Ax=b, the only problem we really solve, and a lot less about using huge tables of analytic integrations to try to find answers to equations that we will never encounter in the real world because they are so specially crafted as to have an analytic solution.

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