13 ms·

Math notes to take you from one year of college calculus to grad student level

217 points by crntaylor 14y ago

deleted 14y ago
[deleted]
psykotic 14y ago
This looks great based on my quick perusal. I'd be very surprised if the notes could teach these subjects to anyone who didn't have significant prior exposure. The notes seem better suited for reviewing and contextualizing material you already know rather well. My favorite book of this type is Shafarevich's Basic Notions of Algebra. The stated prerequisites are also more advanced than the submission title implies. At my university we didn't have a dedicated course in complex analysis until our third semester, and that was in Denmark, where students will study nothing but mathematics from day one. In the American system where even mathematics majors have a mixed course of study for their first several years, it's not unusual for rigorous complex analysis to be a final year subject. Even Harvard's infamous Math 55b second-semester honors course only treats complex analysis very superficially.
- crntaylor 14y ago
  I don't think it's implied that complex analysis is required for reading these notes. The author writes that he expects students will take a course on complex analysis "at some point" but as far as I can tell he doesn't do anything requiring complex analysis in these notes (e.g. every search for "complex" turns up something unrelated to complex analysis, and the word "contour" (as in contour integral) doesn't appear in the notes at all).
  - psykotic 14y ago
    I'll take your word for it. In that case, the notes can't very well be said to take you to the graduate level. There's no point in skating over arguably the most beautiful and interconnected area of mathematics (by which I mean complex analysis) in an effort to get to some nominal level of mathematical advancedness.
    - crntaylor 14y ago
      That's a fair point.
- sillysaurus 14y ago
  I'd be very surprised if the notes could teach these subjects to anyone who didn't have significant prior exposure. I'm self-taught, and these notes are probably the most useful resource I've yet come across. It's hard not having anyone to work through physics problems with. Learning in-person is much higher bandwidth. But thus far OCW has done a fair job in supplementing this. The problem is that there isn't a unifying thread across courses. Each course is isolated from every other course. That's a good way to build a toolkit, but it makes it rather difficult to understand how and why certain knowledge will be useful later on, and how to apply that knowledge. So these notes are the unifying thread I've wanted. But it's true that notes aren't a substitute for courses. Perhaps books are, though. These have served me well so far: http://dl.dropbox.com/u/315/books/list.html http://dl.dropbox.com/u/315/books/list.html and recommendations would be great.
  - ianfernz 14y ago
    I think "high bandwidth" is a good way to describe it.
xxpor 14y ago
I found the writing style to be incredibly obnoxious and hand wavy, and really make me stop reading.
- crntaylor 14y ago
  That's interesting. I guess 'obnoxious' is pretty subjective, but which part did you find 'hand-wavy'? The parts I've read have been quite rigorous.
  - xxpor 14y ago
    I don't think hand wavy was exactly the right way to describe it, more like flippant.
  - xxpor 14y ago
    Also, some examples of what I found objectionable: * Abstract algebra, topology (local and global) folds into a useful, intuitive toolset for ordinary differential equations and partial differential equations, be they linear or nonlinear. (This annoying to read be they construction is used multiple times) * Abstract algebra, topology, and algebraic topology aren’t vacuous constructs built for useless mental masturbation by “pure” mathematicians; the study of differential equations is more than a study of botany.
- dmcdougall_ 14y ago
  Admittedly, I stopped before then when I saw the poor mathematics rendering. For some reason unbeknownst to me, the equations appear cramped and difficult to read.
  - drivebyacct2 14y ago
    Yes, I know that I sound whiny but I'd love to see this rendered via LaTeX.
  - verroq 14y ago
    When I saw that this was a word document exported as a pdf I knew this wasn't a serious text.
kombinatorics 14y ago
this is gold. thanks!
msutherl 14y ago
It would be wonderful to have the privilege and dedication to learn all of this. Do you think it would be possible to construct a high level treatment that would impart a rough idea of to the layman? One that omitted all the business about finding solutions and stuck to merely tracing the structures? I have seen that lower-level concepts like the fundamental theorem of calculus and the Fourier transform can be easily explained in a matter of minutes with the help of diagrams. It is my hunch, but I lack proof, that the same could be done for all of mathematics. Of course I have been told a few times that it would be impossible.
- crntaylor 14y ago
  I think that the book "Q.E.D" by Richard Feynman does about as good a job as is possible at explaining quantum electrodynamics to the layman (I first read it when I was 17, and found it very understandable, with the possible exception of the final chapter). As to whether you could do this for all mathematics - I'm not sure. It's quite easy to 'visualise' the FTC or the fourier transform, and they have immediate applications to things that non-mathematicians care about. I'm not quite sure how one would go about explaining e.g. representation theory of lie algebras, since all of the motivating examples would only be of interest to mathematicians. It's a bit like the wall I hit when I tried to study category theory. It's perfectly possible for someone with very little math background to learn the basics, but until you've seen a lot of mathematics you won't understand what the point of it all is.
  - cdwhite 14y ago
    About representation theory of Lie algebras: physicists actually care about that quite a bit, as the the theory of spin is intimately tied up with the subject. Your point stands, though. I don't know of any major applications outside quantum mechanics, and other parts of mathematics have even fewer applications. (I have always found integration theory kind of tedious for exactly that reason---some results turn out to be handy, but it's more like you're laying the groundwork for background material for stuff that'll be useful to physicists.) Also, the way in which physicists care about the math is very different from the way mathematicians do: we want the moral reasoning---we want to have some intuition for why the result is true---but (to grossly stereotype) we don't really care about the detailed proof. (I must add that I heartily second your recommendation of /Q.E.D./)
jason_adleberg 14y ago
From Sentence 5 of Example 1.1: "The symmetry is a smooth (differentiable to all orders) invertible transformation mapping solutions of the ODE to solutions of the ^ODE^. Invertible means the Jacobian is nonzero: x'x y'y - x'y y'x != 0" Yeah, understood about 5% of that.
- yawgmoth 14y ago
  Here's how I broke it down, it has been a little while since I've been in a math class * Differentiable to all orders means that for each derivation, no cusp will appear in the curve. A cusp means that the next order of derivation will not be defined at that point on the curve. * 'The symmetry is a smooth invertible transformation mapping solutions of the X to solutions of the Y'. - I now understand that the stuff I just paraphrased means that it's just a mapping, and that it's invertible. - ODE = Ordinary Differential Equation. Cool. Rings a bell. It looks like ^ODE^ is just the next order of derivation? And this mapping, the symmetry, is just describing how the next order of derivation relates to the first (I think, that is not exactly clear in the time I spent). * Invertible means the Jacobian is nonzero... Describing to a sophomore that a mapping is invertible in these terms is pretty vague (this section is supposed to be accesible to sophomores). The Jacobian is the determinant of a particular form of matrix, http://mathworld.wolfram.com/Jacobian.html http://mathworld.wolfram.com/Jacobian.html So aside from that last bit it came apart okay. I have noticed that when you have completed a certain amount of math (or any topic) it is hard to exclude certain bits or to describe things in a simpler fashion
experiment0 14y ago
I guess this is a good a time as any to show the organic chemistry notes that I've been writing up. https://github.com/alexganose/chem1201 https://github.com/alexganose/chem1201 So far I've done my first year notes. They aren't particularly organised, they are literally just latex versions of my handwritten notes so they won't be good to learn from, however as a summary they are quite useful. I'm doing it for purely selfish means as I can revise from these notes better, but I thought it would be good to open source them so people can use them if they want.
sambomillo 14y ago
I'm sorry, grad student level on just notes? I don't think so. Maybe grad degress aren't worth what they used to be, but certainly more than notes.
- atondwal 14y ago
  Well, there's a difference between holding-a-grad-degree level, and being a grad student. On the other hand if the implication was that undergrad degrees teach your relatively little in comparison to what they should, then yes.
mrcactu5 14y ago
john baez has been blogging for years on math and physics * http://math.ucr.edu/home/baez/TWF.html http://math.ucr.edu/home/baez/TWF.html math is separated from the other disciplines in a very artificial way. but I am also skeptical of any one book who makes as bold claims this. Math (even freshman calculus) is very deep and takes years to master these notes rough around the edges, but great for self-teaching. Harvard's Math 55 tries to accomplish similar goals. Not as user friendly, but more traditional: * http://www.math.harvard.edu/~ctm/home/text/class/harvard/55a/08/html/index.html http://www.math.harvard.edu/~ctm/home/text/class/harvard/55a... * http://www.math.harvard.edu/~ctm/home/text/class/harvard/55b/09/html/index.html http://www.math.harvard.edu/~ctm/home/text/class/harvard/55b...
QuantumGuy 14y ago
This plus MIT OpenCourseware & Coursera could really teach someone physics. And I mean real physics not pop culture physics. IE breaking the fundamental laws of thermodynamics and having a negative temperature(the conclusion drawn by the website doesn't fit the actual paper).
SandB0x 14y ago
I'm surprised by the excitement at every piece of teaching material or set of course notes posted here. Free textbooks and PDF notes have been around for years, especially for mathematical topics. This PDF won't do the work for you, and you can't skim-read this kind of material. To properly understand an area of mathematics then you need to put a significant amount of time and effort into working through the text, and a set of condensed notes is probably not as good as a well written textbook with careful examples and exercises (and with fewer errors). I'm not making a judgement about the quality of this document, I guess I'm saying that if you really wanted to learn this material you would have started already.
- doktrin 14y ago
  >> if you really wanted to learn this material, then you would have started already This is pretty much the only statement i have an objection or reaction to. I simply don't understand the basis for assertions along these lines. These sorts of fatalistic proclamations are made not-infrequently in the context of programming and development as well. Its as if we feel that the only ones worthy of pursuing a given discipline are those who realized their passion and interest early in life. Why the exclusivity? This is just knowledge, after all.
  - SandB0x 14y ago
    I agree with your sentiment. Sometimes we get articles on (say) the Fourier transform, with its own intuitive take on how and why it works, some visualisation and some maths. I think these articles are great. I can understand how they would spark the interest of someone who is not familiar with the maths, whatever their age. That's not the case here. I don't think that anyone who has upvoted this has read any significant part of the document, simply because it would take months if not years to go through. It's like me posting a several-hundred page set of homemade notes on cell biology and saying "Notes to take you to medical school level biology".
    - doktrin 14y ago
      Fair. I happen to agree that the document is probably not going to be of much use to anyone who hasn't already studied the material. I do, however, firmly believe that anyone - no matter their age - may find interest and cause to learn math, even if starting with high school calculus.
j2kun 14y ago
This is a very one sided treatment of "all mathematics" between college calculus and graduate level mathematics. Sounds like typical mathematical physics, which is a far cry from all mathematics, and the treatment of things like, say, topoological spaces is quite shallow. You couldn't survive a minute in a graduate level mathematics class with this treatment of topology alone.
- verroq 14y ago
  I'm not quite sure why he always relates abstract algebra examples to ODES etc. Surely there are more motiving examples when discussing groups, esp from geometry.
j2kun 14y ago
This is a very one sided treatment of "all mathematics" between college calculus and graduate level mathematics. Sounds like typical mathematical physics, which is a far cry from all mathematics, and the treatment of things like, say, topoological spaces is quite shallow. You couldn't survive a minute in a graduate level mathematics class with this treatment of topology alone.
j2kun 14y ago
This is a very one sided treatment of "all mathematics" between college calculus and graduate level mathematics. Sounds like typical mathematical physics, which is a far cry from all mathematics, and the treatment of things like, say, topoological spaces is quite shallow. You couldn't survive a minute in a graduate level mathematics class with this treatment of topology alone.
ajtulloch 14y ago
If you're interested in this material, you may also like my LaTeX'ed lecture notes covering the last few years of my mathematics degree - mostly pure mathematics with some statistics and financial mathematics. http://tullo.ch/2011/mathematics-lecture-notes/ http://tullo.ch/2011/mathematics-lecture-notes/ for the PDFs, and https://github.com/ajtulloch/SydneyUniversityMathematicsNotes/ https://github.com/ajtulloch/SydneyUniversityMathematicsNote... for the LaTeX source.
- crntaylor 14y ago
  This looks really neat. Nice work!
jfarmer 14y ago
I can't stand the sans serif typesetting and cramped mathematical formulas. The tone is kind of obnoxious, too. When he introduces group theory: Group theory basics. It is time to note that our one-parameter symmetries are groups in the sense of modern algebra. Why? To masturbate with nomenclature as you do in an abstract algebra class? No. Because, as you will soon see, studying the group structure of a symmetry of a differential equation will have direct relevance to reducing its order to lower order, and will have direct relevance to finding some, possibly all of the solutions to the given differential equation—ordinary, partial, linear, or nonlinear. So what is a group? I don't get the pedagogical purpose of calling what one does in an abstract algebra class "masturbating with nomenclature." I think every word in a textbook should be crafted with a pedagogical goal in mind. Making the material more light-hearted and less daunting is a valid purpose, but this tone just seems sour. In fact, I count three uses of the word "masturbate" in the notes. I prefer something like Richard Feynman's style, where he makes a subject accessible while still respecting the subject. Here's a fantastic example of Feynman explaining how a computer works, using an analogy of an ever-faster filing clerk: http://www.youtube.com/watch?v=EKWGGDXe5MA http://www.youtube.com/watch?v=EKWGGDXe5MA
- crntaylor 14y ago
  What really struck me about that lecture is that he only uses one blackboard throughout the entire talk, and he doesn't start writing on it until 20 minutes in. I wish more lectures and talks were like that.
zemanel 14y ago
Oh just what i was looking for. I was actually meaning to post an Ask question for this a couple of days ago, as i've been wondering about giving a [long] shot at MIT next year and i was looking for reference material for SAT's and stuff.
atondwal 14y ago
This is extrordanarily good. For a similar, but more in depth covering of the same material I reccomend [Osborne --- Advanced Mathematical Techniques: for Scientists and Engineers](http://www.amazon.com/Advanced-Mathematical-Techniques-Scientists-Engineers/dp/1453798765 http://www.amazon.com/Advanced-Mathematical-Techniques-Scien...) and for a much more indepth, but less pedagogically useful (more of a reference) [Arfken --- Mathematical Methods for Physicists, Seventh Edition: A Comprehensive Guide](http://www.amazon.com/Mathematical-Methods-Physicists-Seventh-Comprehensive/dp/0123846544 http://www.amazon.com/Mathematical-Methods-Physicists-Sevent...) In addition anything by Penrose tends to target a lay audience, but quickly build up formalism and cover concepts interesting to even practicing physicists.
hcarvalhoalves 14y ago
You know what I would like? Math material that is less concise. Some math concepts are too dense to grasp without first understanding the reasoning behind it, the axioms it's based on, real-world applications, metaphors, diagrams... heck, even the history behind the mathematician helps sometimes (e.g., knowing Newton was a theologist is relevant to understand some things about classic physics [1]). In fact, I love how earlier mathematicians were mostly multi-disciplinary scientists, and almost always philosophers. We need a new Renaissance. [1] http://en.wikipedia.org/wiki/Isaac_Newton#Religious_views http://en.wikipedia.org/wiki/Isaac_Newton#Religious_views
- gnu8 14y ago
  That's what university is for. ~There is no royal road to mathematics~
  - hcarvalhoalves 14y ago
    Let me guess, you have a major in math.
- baaats 14y ago
  I think what you're looking for are textbooks.
  - hcarvalhoalves 14y ago
    You will have my eternal gratitude if you are kind enough to name one that doesn't suck.
    - MrVitaliy 14y ago
      Elementary Analysis: The Theory of Calculus by Ross IMHO very easy, introductory book on proofs and basic analysis. It's not a novel, you will need some effort to think through exercises. Getting through this book is like unlocking GODMODE on the first 2 years of college math courses (Calculus mostly).
    - 14y ago
deleted 14y ago
[deleted]
ivan_ah 14y ago
Looks like a very good summary of mathematical physics -- from ODEs all the way to Lie algebras / symmetries which are very important in quantum field theory and other advanced physics subjects. Would it be possible to have a version in the computer moder font and without so much space between the lines. I would print this and try to read it. I never liked/respected differential equations much, but this looks like a tutorial (300+ pages!!!) which could turn around my opinion.

217 points by crntaylor 14y ago

deleted 14y ago

[deleted]

This looks great based on my quick perusal. I'd be very surprised if the notes could teach these subjects to anyone who didn't have significant prior exposure. The notes seem better suited for reviewing and contextualizing material you already know rather well. My favorite book of this type is Shafarevich's Basic Notions of Algebra. The stated prerequisites are also more advanced than the submission title implies. At my university we didn't have a dedicated course in complex analysis until our third semester, and that was in Denmark, where students will study nothing but mathematics from day one. In the American system where even mathematics majors have a mixed course of study for their first several years, it's not unusual for rigorous complex analysis to be a final year subject. Even Harvard's infamous Math 55b second-semester honors course only treats complex analysis very superficially.

crntaylor 14y ago
I don't think it's implied that complex analysis is required for reading these notes. The author writes that he expects students will take a course on complex analysis "at some point" but as far as I can tell he doesn't do anything requiring complex analysis in these notes (e.g. every search for "complex" turns up something unrelated to complex analysis, and the word "contour" (as in contour integral) doesn't appear in the notes at all).
- psykotic 14y ago
  I'll take your word for it. In that case, the notes can't very well be said to take you to the graduate level. There's no point in skating over arguably the most beautiful and interconnected area of mathematics (by which I mean complex analysis) in an effort to get to some nominal level of mathematical advancedness.
  - crntaylor 14y ago
    That's a fair point.
sillysaurus 14y ago
I'd be very surprised if the notes could teach these subjects to anyone who didn't have significant prior exposure. I'm self-taught, and these notes are probably the most useful resource I've yet come across. It's hard not having anyone to work through physics problems with. Learning in-person is much higher bandwidth. But thus far OCW has done a fair job in supplementing this. The problem is that there isn't a unifying thread across courses. Each course is isolated from every other course. That's a good way to build a toolkit, but it makes it rather difficult to understand how and why certain knowledge will be useful later on, and how to apply that knowledge. So these notes are the unifying thread I've wanted. But it's true that notes aren't a substitute for courses. Perhaps books are, though. These have served me well so far: http://dl.dropbox.com/u/315/books/list.html http://dl.dropbox.com/u/315/books/list.html and recommendations would be great.
- ianfernz 14y ago
  I think "high bandwidth" is a good way to describe it.

xxpor 14y ago

I found the writing style to be incredibly obnoxious and hand wavy, and really make me stop reading.

crntaylor 14y ago
That's interesting. I guess 'obnoxious' is pretty subjective, but which part did you find 'hand-wavy'? The parts I've read have been quite rigorous.
- xxpor 14y ago
  I don't think hand wavy was exactly the right way to describe it, more like flippant.
- xxpor 14y ago
  Also, some examples of what I found objectionable: * Abstract algebra, topology (local and global) folds into a useful, intuitive toolset for ordinary differential equations and partial differential equations, be they linear or nonlinear. (This annoying to read be they construction is used multiple times) * Abstract algebra, topology, and algebraic topology aren’t vacuous constructs built for useless mental masturbation by “pure” mathematicians; the study of differential equations is more than a study of botany.
dmcdougall_ 14y ago
Admittedly, I stopped before then when I saw the poor mathematics rendering. For some reason unbeknownst to me, the equations appear cramped and difficult to read.
- drivebyacct2 14y ago
  Yes, I know that I sound whiny but I'd love to see this rendered via LaTeX.
- verroq 14y ago
  When I saw that this was a word document exported as a pdf I knew this wasn't a serious text.

kombinatorics 14y ago

this is gold. thanks!

msutherl 14y ago

It would be wonderful to have the privilege and dedication to learn all of this. Do you think it would be possible to construct a high level treatment that would impart a rough idea of to the layman? One that omitted all the business about finding solutions and stuck to merely tracing the structures? I have seen that lower-level concepts like the fundamental theorem of calculus and the Fourier transform can be easily explained in a matter of minutes with the help of diagrams. It is my hunch, but I lack proof, that the same could be done for all of mathematics. Of course I have been told a few times that it would be impossible.

crntaylor 14y ago
I think that the book "Q.E.D" by Richard Feynman does about as good a job as is possible at explaining quantum electrodynamics to the layman (I first read it when I was 17, and found it very understandable, with the possible exception of the final chapter). As to whether you could do this for all mathematics - I'm not sure. It's quite easy to 'visualise' the FTC or the fourier transform, and they have immediate applications to things that non-mathematicians care about. I'm not quite sure how one would go about explaining e.g. representation theory of lie algebras, since all of the motivating examples would only be of interest to mathematicians. It's a bit like the wall I hit when I tried to study category theory. It's perfectly possible for someone with very little math background to learn the basics, but until you've seen a lot of mathematics you won't understand what the point of it all is.
- cdwhite 14y ago
  About representation theory of Lie algebras: physicists actually care about that quite a bit, as the the theory of spin is intimately tied up with the subject. Your point stands, though. I don't know of any major applications outside quantum mechanics, and other parts of mathematics have even fewer applications. (I have always found integration theory kind of tedious for exactly that reason---some results turn out to be handy, but it's more like you're laying the groundwork for background material for stuff that'll be useful to physicists.) Also, the way in which physicists care about the math is very different from the way mathematicians do: we want the moral reasoning---we want to have some intuition for why the result is true---but (to grossly stereotype) we don't really care about the detailed proof. (I must add that I heartily second your recommendation of /Q.E.D./)

jason_adleberg 14y ago

From Sentence 5 of Example 1.1: "The symmetry is a smooth (differentiable to all orders) invertible transformation mapping solutions of the ODE to solutions of the ^ODE^. Invertible means the Jacobian is nonzero: x'x y'y - x'y y'x != 0" Yeah, understood about 5% of that.

yawgmoth 14y ago
Here's how I broke it down, it has been a little while since I've been in a math class * Differentiable to all orders means that for each derivation, no cusp will appear in the curve. A cusp means that the next order of derivation will not be defined at that point on the curve. * 'The symmetry is a smooth invertible transformation mapping solutions of the X to solutions of the Y'. - I now understand that the stuff I just paraphrased means that it's just a mapping, and that it's invertible. - ODE = Ordinary Differential Equation. Cool. Rings a bell. It looks like ^ODE^ is just the next order of derivation? And this mapping, the symmetry, is just describing how the next order of derivation relates to the first (I think, that is not exactly clear in the time I spent). * Invertible means the Jacobian is nonzero... Describing to a sophomore that a mapping is invertible in these terms is pretty vague (this section is supposed to be accesible to sophomores). The Jacobian is the determinant of a particular form of matrix, http://mathworld.wolfram.com/Jacobian.html http://mathworld.wolfram.com/Jacobian.html So aside from that last bit it came apart okay. I have noticed that when you have completed a certain amount of math (or any topic) it is hard to exclude certain bits or to describe things in a simpler fashion

experiment0 14y ago

I guess this is a good a time as any to show the organic chemistry notes that I've been writing up. https://github.com/alexganose/chem1201 https://github.com/alexganose/chem1201 So far I've done my first year notes. They aren't particularly organised, they are literally just latex versions of my handwritten notes so they won't be good to learn from, however as a summary they are quite useful. I'm doing it for purely selfish means as I can revise from these notes better, but I thought it would be good to open source them so people can use them if they want.

sambomillo 14y ago

I'm sorry, grad student level on just notes? I don't think so. Maybe grad degress aren't worth what they used to be, but certainly more than notes.

atondwal 14y ago
Well, there's a difference between holding-a-grad-degree level, and being a grad student. On the other hand if the implication was that undergrad degrees teach your relatively little in comparison to what they should, then yes.

mrcactu5 14y ago

john baez has been blogging for years on math and physics * http://math.ucr.edu/home/baez/TWF.html http://math.ucr.edu/home/baez/TWF.html math is separated from the other disciplines in a very artificial way. but I am also skeptical of any one book who makes as bold claims this. Math (even freshman calculus) is very deep and takes years to master these notes rough around the edges, but great for self-teaching. Harvard's Math 55 tries to accomplish similar goals. Not as user friendly, but more traditional: * http://www.math.harvard.edu/~ctm/home/text/class/harvard/55a/08/html/index.html http://www.math.harvard.edu/~ctm/home/text/class/harvard/55a... * http://www.math.harvard.edu/~ctm/home/text/class/harvard/55b/09/html/index.html http://www.math.harvard.edu/~ctm/home/text/class/harvard/55b...

QuantumGuy 14y ago

This plus MIT OpenCourseware & Coursera could really teach someone physics. And I mean real physics not pop culture physics. IE breaking the fundamental laws of thermodynamics and having a negative temperature(the conclusion drawn by the website doesn't fit the actual paper).

SandB0x 14y ago

I'm surprised by the excitement at every piece of teaching material or set of course notes posted here. Free textbooks and PDF notes have been around for years, especially for mathematical topics. This PDF won't do the work for you, and you can't skim-read this kind of material. To properly understand an area of mathematics then you need to put a significant amount of time and effort into working through the text, and a set of condensed notes is probably not as good as a well written textbook with careful examples and exercises (and with fewer errors). I'm not making a judgement about the quality of this document, I guess I'm saying that if you really wanted to learn this material you would have started already.

doktrin 14y ago
>> if you really wanted to learn this material, then you would have started already This is pretty much the only statement i have an objection or reaction to. I simply don't understand the basis for assertions along these lines. These sorts of fatalistic proclamations are made not-infrequently in the context of programming and development as well. Its as if we feel that the only ones worthy of pursuing a given discipline are those who realized their passion and interest early in life. Why the exclusivity? This is just knowledge, after all.
- SandB0x 14y ago
  I agree with your sentiment. Sometimes we get articles on (say) the Fourier transform, with its own intuitive take on how and why it works, some visualisation and some maths. I think these articles are great. I can understand how they would spark the interest of someone who is not familiar with the maths, whatever their age. That's not the case here. I don't think that anyone who has upvoted this has read any significant part of the document, simply because it would take months if not years to go through. It's like me posting a several-hundred page set of homemade notes on cell biology and saying "Notes to take you to medical school level biology".
  - doktrin 14y ago
    Fair. I happen to agree that the document is probably not going to be of much use to anyone who hasn't already studied the material. I do, however, firmly believe that anyone - no matter their age - may find interest and cause to learn math, even if starting with high school calculus.

j2kun 14y ago

This is a very one sided treatment of "all mathematics" between college calculus and graduate level mathematics. Sounds like typical mathematical physics, which is a far cry from all mathematics, and the treatment of things like, say, topoological spaces is quite shallow. You couldn't survive a minute in a graduate level mathematics class with this treatment of topology alone.

verroq 14y ago
I'm not quite sure why he always relates abstract algebra examples to ODES etc. Surely there are more motiving examples when discussing groups, esp from geometry.

j2kun 14y ago

ajtulloch 14y ago

If you're interested in this material, you may also like my LaTeX'ed lecture notes covering the last few years of my mathematics degree - mostly pure mathematics with some statistics and financial mathematics. http://tullo.ch/2011/mathematics-lecture-notes/ http://tullo.ch/2011/mathematics-lecture-notes/ for the PDFs, and https://github.com/ajtulloch/SydneyUniversityMathematicsNotes/ https://github.com/ajtulloch/SydneyUniversityMathematicsNote... for the LaTeX source.

crntaylor 14y ago
This looks really neat. Nice work!

jfarmer 14y ago

I can't stand the sans serif typesetting and cramped mathematical formulas. The tone is kind of obnoxious, too. When he introduces group theory: Group theory basics. It is time to note that our one-parameter symmetries are groups in the sense of modern algebra. Why? To masturbate with nomenclature as you do in an abstract algebra class? No. Because, as you will soon see, studying the group structure of a symmetry of a differential equation will have direct relevance to reducing its order to lower order, and will have direct relevance to finding some, possibly all of the solutions to the given differential equation—ordinary, partial, linear, or nonlinear. So what is a group? I don't get the pedagogical purpose of calling what one does in an abstract algebra class "masturbating with nomenclature." I think every word in a textbook should be crafted with a pedagogical goal in mind. Making the material more light-hearted and less daunting is a valid purpose, but this tone just seems sour. In fact, I count three uses of the word "masturbate" in the notes. I prefer something like Richard Feynman's style, where he makes a subject accessible while still respecting the subject. Here's a fantastic example of Feynman explaining how a computer works, using an analogy of an ever-faster filing clerk: http://www.youtube.com/watch?v=EKWGGDXe5MA http://www.youtube.com/watch?v=EKWGGDXe5MA

crntaylor 14y ago
What really struck me about that lecture is that he only uses one blackboard throughout the entire talk, and he doesn't start writing on it until 20 minutes in. I wish more lectures and talks were like that.

zemanel 14y ago

Oh just what i was looking for. I was actually meaning to post an Ask question for this a couple of days ago, as i've been wondering about giving a [long] shot at MIT next year and i was looking for reference material for SAT's and stuff.

atondwal 14y ago

This is extrordanarily good. For a similar, but more in depth covering of the same material I reccomend [Osborne --- Advanced Mathematical Techniques: for Scientists and Engineers](http://www.amazon.com/Advanced-Mathematical-Techniques-Scientists-Engineers/dp/1453798765 http://www.amazon.com/Advanced-Mathematical-Techniques-Scien...) and for a much more indepth, but less pedagogically useful (more of a reference) [Arfken --- Mathematical Methods for Physicists, Seventh Edition: A Comprehensive Guide](http://www.amazon.com/Mathematical-Methods-Physicists-Seventh-Comprehensive/dp/0123846544 http://www.amazon.com/Mathematical-Methods-Physicists-Sevent...) In addition anything by Penrose tends to target a lay audience, but quickly build up formalism and cover concepts interesting to even practicing physicists.

hcarvalhoalves 14y ago

You know what I would like? Math material that is less concise. Some math concepts are too dense to grasp without first understanding the reasoning behind it, the axioms it's based on, real-world applications, metaphors, diagrams... heck, even the history behind the mathematician helps sometimes (e.g., knowing Newton was a theologist is relevant to understand some things about classic physics [1]). In fact, I love how earlier mathematicians were mostly multi-disciplinary scientists, and almost always philosophers. We need a new Renaissance. [1] http://en.wikipedia.org/wiki/Isaac_Newton#Religious_views http://en.wikipedia.org/wiki/Isaac_Newton#Religious_views

gnu8 14y ago
That's what university is for. ~There is no royal road to mathematics~
- hcarvalhoalves 14y ago
  Let me guess, you have a major in math.
baaats 14y ago
I think what you're looking for are textbooks.
- hcarvalhoalves 14y ago
  You will have my eternal gratitude if you are kind enough to name one that doesn't suck.
  - MrVitaliy 14y ago
    Elementary Analysis: The Theory of Calculus by Ross IMHO very easy, introductory book on proofs and basic analysis. It's not a novel, you will need some effort to think through exercises. Getting through this book is like unlocking GODMODE on the first 2 years of college math courses (Calculus mostly).
  - 14y ago

deleted 14y ago

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ivan_ah 14y ago

Looks like a very good summary of mathematical physics -- from ODEs all the way to Lie algebras / symmetries which are very important in quantum field theory and other advanced physics subjects. Would it be possible to have a version in the computer moder font and without so much space between the lines. I would print this and try to read it. I never liked/respected differential equations much, but this looks like a tutorial (300+ pages!!!) which could turn around my opinion.