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> I really don't think that such topics can be self-studied Can you explain this? What makes something extremely difficult to be studied without a teacher? (I

> I really don't think that such topics can be self-studied Can you explain this? What makes something extremely difficult to be studied without a teacher? (I took calc in high school and never took diff eq, so my knowledge in this specific domain is basically zero)

sampo 8y ago
If you have some mathematical maturity (you can read notation, work to understand each equation, have the patience to pretty much understand each equation and each page, because later ideas are built on understanding the previous ideas first), you can self-study a lot of mathematics. But almost everyone (mathematicians, physicists, engineers) who has that maturity, has probably taken a first course in differential equations during their undergrad studies. So the number of people who have developed some mathematical maturity without ever taking a diff.eq. course is probably small. So we don't know much about these people and how the topic of differential equations appears to them. But yes, it would be interesting to hear about experiences from someone with e.g. a strong background in theoretical computer science, probably including calculus but not including diff.eqs., who has self-studied a book on differential equations.
- mclehman 8y ago
  Just chiming in to say we do indeed exist and your example is pretty spot on, but I unfortunately have not self-studied diff.eqs. if I eventually do, I'll make sure to write something up about it. I'd like to think I have developed a hint of mathematical maturity (CS + late declared Math major), but I went more the algebra route and ended up never taking a diff.eq. course. The closest I came was in a complex analysis course with some motivating examples that assumed we'd have picked up some tricks in a diff.eq. course.
- kenjackson 8y ago
  I had self-studied this and then later studied under a teacher as well. I think the thing that I found hardest in my self-study was (and unfortunately this is about 25 years ago so my recollection might be a bit off) that it seemed like there was a lot written on just two equations (the heat equation and the wave equation). I didn't get why is 50 pages dedicated to one equation. Up until that point it felt like Calculus was about techniques to solve equations, and then it suddenly became mostly about how to solve these two equations (there was a third, but I can't recall what it was now), which never really resonated with me.
  - sampo 8y ago
    Heat equation / diffusion equation (same thing): parabolic. Wave equation: hyperbolic. Laplace's equation: elliptic. Laplace's equation is the steady state (equilibrium) solution of heat equation when the boundary conditions (or anything) don't change in time.
    - kenjackson 8y ago
      Thanks. And yes, Laplace was the third one. I think even your simple categorization would have made things much simpler for me. Or maybe I was just an idiot as an undergrad. :-)
graycat 8y ago
With calculus, you are already well into ordinary differential equations -- partial differential equations are different, but with calculus you have a good start on the basics of those, too. A simple ordinary differential equation is below where of course just from calculus y'(t) = d/dt y(t) and the equation is y'(t) = k y(t) ( b - y(t)) So, for the context: t is time, say, in seconds. y is some real valued function of t, that is, y(t), b and k are constants. We are given the value of y at 0, that is y(t) at t = 0, that is, y(0). We want the value of y(t) for t > 0. Okay, with that, just need to use freshman calculus, for positive time s, integrate y'(t) from 0 to s. This is a simple exercise, without looking up the last dozen times I did that, maybe use integration by parts or some such. End up with a quotient with some exponentials. Differential equations pop up in motion, e.g., from Newton's second law, AC circuit theory, and some other areas of science and engineering. Boundary value problems, e.g., vibrating stings, parts of deterministic optimal control, are closely related but, still, significantly different. Long some of the pure mathematicians went, in a word, "nuts" studying differential equations. The best of the results are good, and some of those are nicely useful. In places the work has nice contact with linear algebra, matrix theory, and vector spaces of functions, functional analysis, e.g., Hilbert and Banach spaces. But hanging over the whole subject is a suspicion that, really, as nice as the general theories are, mostly the applications are just a few, standard differential equations. It's a little like learning everything about civil engineering when really are only going to do framing carpentry, hang drywall, and apply roof shingles. Once I bought Garrett Birkhoff and Gian-Carlo Rota, {\it Ordinary Differential Equations,\/} Ginn and Company, Boston, 1962.\ \ I looked through it, saw lots of intricate stuff, but wondered just why I should dig into that. Since then I read a story about Rota about how, apparently, he felt much the same about the material, got stuck teaching the differential equations course because he wrote that book, and wanted, essentially, to f'get about that book and its material! I had a full college course in ordinary differential equations. Okay: It left me wildly over educated for the differential equations in AC circuit theory. Otherwise I didn't much like the book, the teacher, or the course. On the advanced stuff, here is some more Earl A.\ Coddington and Norman Levinson, {\it Theory of Ordinary Differential Equations,\/} McGraw-Hill, New York, 1955.\ \ It has a nice result of Caratheodory, but in general could lose a lot of sleep working through that! I had a course from a Ph.D. from MIT from the book, apparently long a standard at MIT, Francis B.\ Hildebrand, {\it Advanced Calculus for Applications,\/} Prentice-Hall, Englewood Cliffs, NJ, 1962.\ \ So, yes, can find out about solutions via infinite series and boundary value problems. The book was very short on proofs, and to take such material seriously I wanted to see the proofs. Now that I know a lot more math, no doubt some of it originally motivated by material in that book, maybe I could fill in the proofs. When I was at FedEx, I wondered about the cheapest way to climb, cruise, and descend the airplanes, had heard about Michael Athans and Peter L.\ Falb, {\it Optimal Control:\ \ An Introduction to the Theory and Its Applications,\/} McGraw-Hill Book Company, New York, 1966.\ \ and flew up to MIT and met with Athans, got his course notes, etc. He explained that an application would be a "two point boundary value problem with mixed end conditions" -- okay, I'd had a course on numerical methods for that. But, in the early parts of the book will see something interesting -- fast, and well written coverage of the differential equations material needed for the book. This is an example of a general situation: Sometimes the best place to learn something is in an introduction or appendix written by a real expert who is also a good writer, intended as background for the rest of the book. So, such a source cuts out the tangential, maybe curious cruft can't much hope to use. At one point after college on my own I carefully read, not nearly new at the time (TeX markup): Earl A.\ Coddington, {\it An Introduction to Ordinary Differential Equations,\/} Prentice-Hall, Englewood Cliffs, NJ, 1961. Coddington was not just a grand expert in the field but also a good writer. I really liked his stuff on variation of parameters -- a bit amazing. Note: Can find mention of that in the famous movie The Day the Earth Stood Still -- apparently that math was hot stuff in applied math about when the movie was made. Can say some quite similar things about partial differential equations -- e.g., there are deep books, some connections with functional analysis (and even the theory of distributions) but the main interests are the partial differential equations of mathematical physics, especially, Maxwell's equations, the heat equation, the wave equation, Schrödinger equation, a wave equation, and the notorious Navier-Stokes equations -- which likely should attack only for limited goals and in somewhat special cases. Net, unless you have some significant reason for more, I suggest you learn what you need to know, just in time, when and if you need it. But, in that case, as elsewhere, a good pure math background in calculus, and advanced calculus with the proofs, etc. will be good to have.
spapas82 8y ago
Diff. eq. is a difficult topic. Not only you need to have a good understanding of a lot of other mathematic topics (derivatives, integrals etc) to properly understand it since it builds upon those but you also need somebody to guide you on how to think when solving such problems (i.e you'll need to solve diff. eq. together with your teacher).
- commandlinefan 8y ago
  Yeah - I never studied differential equations in college; I sort of know what they are, and that there's a famous problem that involves a snow-plow plowing a field while it's still snowing, but that's about it. I've always been curious, though, so I clicked through the link. Per the introduction, I jumped ahead to the "what do you need to know from calculus" appendix just to see how much I did remember. Derivatives, integrals, ok, sure I remember that. Trigonometric functions? Um, yeah, I think I remember that - adjacent over hypotenuse, tangent, theta, sounds familiar. Half-angle, double-angle... getting rustier, but I think I remember that stuff. Hyperbolic functions? Oh, boy... yeah, something to do with the way power lines hang. I remember remembering that once. Integration by parts? Oh, man, I don't know if I ever learned that. Geometric series? Power series? Binomial expansion? I read about those in TAOCP, but if I ever studied those in class, it went in one ear and out the other. And that's what I need to know to start reading... maybe I do need a teacher.

sampo 8y ago

If you have some mathematical maturity (you can read notation, work to understand each equation, have the patience to pretty much understand each equation and each page, because later ideas are built on understanding the previous ideas first), you can self-study a lot of mathematics. But almost everyone (mathematicians, physicists, engineers) who has that maturity, has probably taken a first course in differential equations during their undergrad studies. So the number of people who have developed some mathematical maturity without ever taking a diff.eq. course is probably small. So we don't know much about these people and how the topic of differential equations appears to them. But yes, it would be interesting to hear about experiences from someone with e.g. a strong background in theoretical computer science, probably including calculus but not including diff.eqs., who has self-studied a book on differential equations.

mclehman 8y ago
Just chiming in to say we do indeed exist and your example is pretty spot on, but I unfortunately have not self-studied diff.eqs. if I eventually do, I'll make sure to write something up about it. I'd like to think I have developed a hint of mathematical maturity (CS + late declared Math major), but I went more the algebra route and ended up never taking a diff.eq. course. The closest I came was in a complex analysis course with some motivating examples that assumed we'd have picked up some tricks in a diff.eq. course.
kenjackson 8y ago
I had self-studied this and then later studied under a teacher as well. I think the thing that I found hardest in my self-study was (and unfortunately this is about 25 years ago so my recollection might be a bit off) that it seemed like there was a lot written on just two equations (the heat equation and the wave equation). I didn't get why is 50 pages dedicated to one equation. Up until that point it felt like Calculus was about techniques to solve equations, and then it suddenly became mostly about how to solve these two equations (there was a third, but I can't recall what it was now), which never really resonated with me.
- sampo 8y ago
  Heat equation / diffusion equation (same thing): parabolic. Wave equation: hyperbolic. Laplace's equation: elliptic. Laplace's equation is the steady state (equilibrium) solution of heat equation when the boundary conditions (or anything) don't change in time.
  - kenjackson 8y ago
    Thanks. And yes, Laplace was the third one. I think even your simple categorization would have made things much simpler for me. Or maybe I was just an idiot as an undergrad. :-)

graycat 8y ago

With calculus, you are already well into ordinary differential equations -- partial differential equations are different, but with calculus you have a good start on the basics of those, too. A simple ordinary differential equation is below where of course just from calculus y'(t) = d/dt y(t) and the equation is y'(t) = k y(t) ( b - y(t)) So, for the context: t is time, say, in seconds. y is some real valued function of t, that is, y(t), b and k are constants. We are given the value of y at 0, that is y(t) at t = 0, that is, y(0). We want the value of y(t) for t > 0. Okay, with that, just need to use freshman calculus, for positive time s, integrate y'(t) from 0 to s. This is a simple exercise, without looking up the last dozen times I did that, maybe use integration by parts or some such. End up with a quotient with some exponentials. Differential equations pop up in motion, e.g., from Newton's second law, AC circuit theory, and some other areas of science and engineering. Boundary value problems, e.g., vibrating stings, parts of deterministic optimal control, are closely related but, still, significantly different. Long some of the pure mathematicians went, in a word, "nuts" studying differential equations. The best of the results are good, and some of those are nicely useful. In places the work has nice contact with linear algebra, matrix theory, and vector spaces of functions, functional analysis, e.g., Hilbert and Banach spaces. But hanging over the whole subject is a suspicion that, really, as nice as the general theories are, mostly the applications are just a few, standard differential equations. It's a little like learning everything about civil engineering when really are only going to do framing carpentry, hang drywall, and apply roof shingles. Once I bought Garrett Birkhoff and Gian-Carlo Rota, {\it Ordinary Differential Equations,\/} Ginn and Company, Boston, 1962.\ \ I looked through it, saw lots of intricate stuff, but wondered just why I should dig into that. Since then I read a story about Rota about how, apparently, he felt much the same about the material, got stuck teaching the differential equations course because he wrote that book, and wanted, essentially, to f'get about that book and its material! I had a full college course in ordinary differential equations. Okay: It left me wildly over educated for the differential equations in AC circuit theory. Otherwise I didn't much like the book, the teacher, or the course. On the advanced stuff, here is some more Earl A.\ Coddington and Norman Levinson, {\it Theory of Ordinary Differential Equations,\/} McGraw-Hill, New York, 1955.\ \ It has a nice result of Caratheodory, but in general could lose a lot of sleep working through that! I had a course from a Ph.D. from MIT from the book, apparently long a standard at MIT, Francis B.\ Hildebrand, {\it Advanced Calculus for Applications,\/} Prentice-Hall, Englewood Cliffs, NJ, 1962.\ \ So, yes, can find out about solutions via infinite series and boundary value problems. The book was very short on proofs, and to take such material seriously I wanted to see the proofs. Now that I know a lot more math, no doubt some of it originally motivated by material in that book, maybe I could fill in the proofs. When I was at FedEx, I wondered about the cheapest way to climb, cruise, and descend the airplanes, had heard about Michael Athans and Peter L.\ Falb, {\it Optimal Control:\ \ An Introduction to the Theory and Its Applications,\/} McGraw-Hill Book Company, New York, 1966.\ \ and flew up to MIT and met with Athans, got his course notes, etc. He explained that an application would be a "two point boundary value problem with mixed end conditions" -- okay, I'd had a course on numerical methods for that. But, in the early parts of the book will see something interesting -- fast, and well written coverage of the differential equations material needed for the book. This is an example of a general situation: Sometimes the best place to learn something is in an introduction or appendix written by a real expert who is also a good writer, intended as background for the rest of the book. So, such a source cuts out the tangential, maybe curious cruft can't much hope to use. At one point after college on my own I carefully read, not nearly new at the time (TeX markup): Earl A.\ Coddington, {\it An Introduction to Ordinary Differential Equations,\/} Prentice-Hall, Englewood Cliffs, NJ, 1961. Coddington was not just a grand expert in the field but also a good writer. I really liked his stuff on variation of parameters -- a bit amazing. Note: Can find mention of that in the famous movie The Day the Earth Stood Still -- apparently that math was hot stuff in applied math about when the movie was made. Can say some quite similar things about partial differential equations -- e.g., there are deep books, some connections with functional analysis (and even the theory of distributions) but the main interests are the partial differential equations of mathematical physics, especially, Maxwell's equations, the heat equation, the wave equation, Schrödinger equation, a wave equation, and the notorious Navier-Stokes equations -- which likely should attack only for limited goals and in somewhat special cases. Net, unless you have some significant reason for more, I suggest you learn what you need to know, just in time, when and if you need it. But, in that case, as elsewhere, a good pure math background in calculus, and advanced calculus with the proofs, etc. will be good to have.

spapas82 8y ago

Diff. eq. is a difficult topic. Not only you need to have a good understanding of a lot of other mathematic topics (derivatives, integrals etc) to properly understand it since it builds upon those but you also need somebody to guide you on how to think when solving such problems (i.e you'll need to solve diff. eq. together with your teacher).

commandlinefan 8y ago
Yeah - I never studied differential equations in college; I sort of know what they are, and that there's a famous problem that involves a snow-plow plowing a field while it's still snowing, but that's about it. I've always been curious, though, so I clicked through the link. Per the introduction, I jumped ahead to the "what do you need to know from calculus" appendix just to see how much I did remember. Derivatives, integrals, ok, sure I remember that. Trigonometric functions? Um, yeah, I think I remember that - adjacent over hypotenuse, tangent, theta, sounds familiar. Half-angle, double-angle... getting rustier, but I think I remember that stuff. Hyperbolic functions? Oh, boy... yeah, something to do with the way power lines hang. I remember remembering that once. Integration by parts? Oh, man, I don't know if I ever learned that. Geometric series? Power series? Binomial expansion? I read about those in TAOCP, but if I ever studied those in class, it went in one ear and out the other. And that's what I need to know to start reading... maybe I do need a teacher.