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I totally agree that calculus is unlike "real math" (reasoning from first principles, proofs, abstractions). Perhaps learning linear algebra would be time bette

by ivan_ah 13y ago

I totally agree that calculus is unlike "real math" (reasoning from first principles, proofs, abstractions). Perhaps learning linear algebra would be time better spent[1]. Sill, calculus is the first applied math course and is very useful for many areas of science so I'm not altogether in agreement that learning calculus is bad for you. On the contrary---in combination with a mechanics class it can be very good in terms of learning how to model the real world. That being said, I think UGRADs spend waaaaaay too much time learning calculus: Calc I, Calc II, multivariable, vector calculus, etc. That's like 4 semesters of calculus! This signals to the student that calculus is somehow a big deal, when in fact it is not: it's just calculation techniques. I think this should be cut-down to 2 semesters: Calc I+II together, then multivariable+vector calc. Just learn it quickly and get it over with[2]. _________ [1] Gilbert Strang saying we should learn more LA and less Calc. http://www-math.mit.edu/~gs/papers/essay.pdf http://www-math.mit.edu/~gs/papers/essay.pdf [2] My short book on Calculus and Mechanics http://minireference.com/launch40 http://minireference.com/launch40 (disclaimer: self-plug, but on topic)

dalke 13y ago
A minor correction. Calculus has loads of proofs. The calculus taught to high school and non-math undergrads tends to avoid more than a handful of proofs because most students don't need to know, nor care, about the actual math involved. Indeed, it would likely drive more students away. I would say that calculus is the first "real math" that students study which is so complicated that they can't really learn the first principles without years of effort. That's why classes teach the most immediately useful parts instead of covering the details. Someone who only studies those basics might even come out thinking that calculus is just a bunch of techniques, without grasping how gorgeous the underlying concepts and abstractions are. Take geometry as something I think you regard as "real math." How come geometry classes never prove that it's impossible to trisect the angle? Or double the cube? It turns out that compass and straightedge can only produce quadratic constructions, and Wantzel showed that these problems require cubics. This shows just how limited geometry really is, which is partially why students can feel that they have a handle on the entire topic. (Also, few if any secondary schools cover non-Euclidean geometries, or Euclid's book X as it explores incommensurable magnitudes.) I would also say that trigonometry is the first applied math course that students learn, not calculus. Let's go to calculus. Take for example the chain rule, D(g∘f)(c) = Dg(f(c))∘Df(c). This is one of those rules everyone learns in Calculus I. Can you prove it? I once could. Here's the start of the proof from my text book: "The hypothesis implies the c is an interior point of the domain of h = g∘f. (Why?) Let e>0 and d(e, f) be as in Definition 39.2. It follows from Lemma 39.5 that there exists strictly positive numbers g, J such that if |x-c|<=g then f(x) is an element of B and |f(x)-f(c)|<=K |x-c|. For simplicity, we write L_f = D f(c) and L_g = D h(b). By Theorem 21.3 there is a constant M such that |L_s(u)|<=M|u| for all u in R^q. If |x-c|<infimum(g, (1/K) ..." And so on for another 10 lines of the book. Do you really want to subject all calculus students to that level of detail? I don't. Who other than a mathematician needs to learn about Lebesgue integration and measure theory, which are even more advanced topics in calculus? BTW, my undergrad had multivariable and vector calculus as a single Calculus III class, so there were only 3 semesters of calculus before getting into the foundations of calculus. Also, calculus books do have some proofs. For example, every chapter I looked at from http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/ http://ocw.mit.edu/resources/res-18-001-calculus-online-text... contains a few proofs.
- eli_gottlieb 13y ago
  >BTW, my undergrad had multivariable and vector calculus as a single Calculus III class, so there were only 3 semesters of calculus before getting into the foundations of calculus. So did mine, which is biting me on the ass now that I'm studying for my Machine Learning exam in graduate school. This course expected matrix calculus.
wwweston 13y ago
> Calc I, Calc II, multivariable, vector calculus, etc. That's like 4 semesters of calculus! I'm counting at least eight classes that were arguably Calculus in my undergrad program: * Calc I (basic differentiation/antidifferentiation) * Calc II (integration techniques/applications, series/sequences) * Calc III (vector calc) * Careful/Proven Calc (Real Analysis I) * Careful/Proven Calc II (Real Analysis II) * Differential Equations * Partial Differential Equations Five of which I took (unsurprising for a math major). It's a lot, and I could definitely wish there was more linear algebra (the intro LA course was a significant part of what convinced me to look at the major, and I took the grad level matrix analysis class as one of my electives because there wasn't anything else). At the same time, I can see why it's all there.

by ivan_ah 13y ago

dalke 13y ago

A minor correction. Calculus has loads of proofs. The calculus taught to high school and non-math undergrads tends to avoid more than a handful of proofs because most students don't need to know, nor care, about the actual math involved. Indeed, it would likely drive more students away. I would say that calculus is the first "real math" that students study which is so complicated that they can't really learn the first principles without years of effort. That's why classes teach the most immediately useful parts instead of covering the details. Someone who only studies those basics might even come out thinking that calculus is just a bunch of techniques, without grasping how gorgeous the underlying concepts and abstractions are. Take geometry as something I think you regard as "real math." How come geometry classes never prove that it's impossible to trisect the angle? Or double the cube? It turns out that compass and straightedge can only produce quadratic constructions, and Wantzel showed that these problems require cubics. This shows just how limited geometry really is, which is partially why students can feel that they have a handle on the entire topic. (Also, few if any secondary schools cover non-Euclidean geometries, or Euclid's book X as it explores incommensurable magnitudes.) I would also say that trigonometry is the first applied math course that students learn, not calculus. Let's go to calculus. Take for example the chain rule, D(g∘f)(c) = Dg(f(c))∘Df(c). This is one of those rules everyone learns in Calculus I. Can you prove it? I once could. Here's the start of the proof from my text book: "The hypothesis implies the c is an interior point of the domain of h = g∘f. (Why?) Let e>0 and d(e, f) be as in Definition 39.2. It follows from Lemma 39.5 that there exists strictly positive numbers g, J such that if |x-c|<=g then f(x) is an element of B and |f(x)-f(c)|<=K |x-c|. For simplicity, we write L_f = D f(c) and L_g = D h(b). By Theorem 21.3 there is a constant M such that |L_s(u)|<=M|u| for all u in R^q. If |x-c|<infimum(g, (1/K) ..." And so on for another 10 lines of the book. Do you really want to subject all calculus students to that level of detail? I don't. Who other than a mathematician needs to learn about Lebesgue integration and measure theory, which are even more advanced topics in calculus? BTW, my undergrad had multivariable and vector calculus as a single Calculus III class, so there were only 3 semesters of calculus before getting into the foundations of calculus. Also, calculus books do have some proofs. For example, every chapter I looked at from http://ocw.mit.edu/resources/res-18-001-calculus-online-textbook-spring-2005/textbook/ http://ocw.mit.edu/resources/res-18-001-calculus-online-text... contains a few proofs.

eli_gottlieb 13y ago
>BTW, my undergrad had multivariable and vector calculus as a single Calculus III class, so there were only 3 semesters of calculus before getting into the foundations of calculus. So did mine, which is biting me on the ass now that I'm studying for my Machine Learning exam in graduate school. This course expected matrix calculus.

wwweston 13y ago

> Calc I, Calc II, multivariable, vector calculus, etc. That's like 4 semesters of calculus! I'm counting at least eight classes that were arguably Calculus in my undergrad program: * Calc I (basic differentiation/antidifferentiation) * Calc II (integration techniques/applications, series/sequences) * Calc III (vector calc) * Careful/Proven Calc (Real Analysis I) * Careful/Proven Calc II (Real Analysis II) * Differential Equations * Partial Differential Equations Five of which I took (unsurprising for a math major). It's a lot, and I could definitely wish there was more linear algebra (the intro LA course was a significant part of what convinced me to look at the major, and I took the grad level matrix analysis class as one of my electives because there wasn't anything else). At the same time, I can see why it's all there.