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.
>> 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.
Almost complete agreement here. There seems to be a 'shortcut mentality' among some hackers - the idea being that if you are faced with a difficult subject, begin with the assumption that standard learning material is padded with lots of useless filler material, and hence conclude you can save time by going for the more summarised stuff.
I think this is not generally a bad strategy in many subjects (business comes to mind), but mathematics is different. There really is no 'royal road' to any subset of it. There are shortcuts, sometimes, but every shortcut you take (with the exception of clever mathematical tricks, which count as solid learning here) deprives you of the opportunity to make a small but significant improvement to your logical problem-solving apparatus.
And that is probably a greater waste of time than anything else: shallow learning. Again, this is possibly not a bad strategy in many subjects, but again mathematics is not one of those.
probably not as good as a well written textbook with careful examples and exercises
Name one. While I agree that this document (probably) won't compensate for a more complete education, including textbooks, the truth of the matter is that most textbooks (even many highly regarded ones) are horrible for learning on one's own. Rigorous and sound, yes, but many are bad for pedagogy, and even worse for self-teaching. A good majority are muddled and unclear to the layman, with not a very good "big picture" or "here's why it's done this way" approach. Just look at the K&R C article from the other day.
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.
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 and recommendations would be great.
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).
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.
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.
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.
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:
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.
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.
I dont know your level, for someone like me who studied mathematics years ago but not applied it for years, I found "In pursuit of unknown, 17 equations that changed world" by Ian Stewart very helpful. While it is not a detailed math book, it gave me enough pointers to refresh and prepared me to dive deeper through other material.
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.
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.
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.
"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"
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
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
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.
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.
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.
In addition anything by Penrose tends to target a lay audience, but quickly build up formalism and cover concepts interesting to even practicing physicists.
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.
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).
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.
[+] [-] SandB0x|13 years ago|reply
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|13 years ago|reply
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.
[+] [-] hdivider|13 years ago|reply
I think this is not generally a bad strategy in many subjects (business comes to mind), but mathematics is different. There really is no 'royal road' to any subset of it. There are shortcuts, sometimes, but every shortcut you take (with the exception of clever mathematical tricks, which count as solid learning here) deprives you of the opportunity to make a small but significant improvement to your logical problem-solving apparatus.
And that is probably a greater waste of time than anything else: shallow learning. Again, this is possibly not a bad strategy in many subjects, but again mathematics is not one of those.
[+] [-] npsimons|13 years ago|reply
Name one. While I agree that this document (probably) won't compensate for a more complete education, including textbooks, the truth of the matter is that most textbooks (even many highly regarded ones) are horrible for learning on one's own. Rigorous and sound, yes, but many are bad for pedagogy, and even worse for self-teaching. A good majority are muddled and unclear to the layman, with not a very good "big picture" or "here's why it's done this way" approach. Just look at the K&R C article from the other day.
[+] [-] Confusion|13 years ago|reply
[+] [-] psykotic|13 years ago|reply
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.
[+] [-] sillysaurus|13 years ago|reply
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 and recommendations would be great.
[+] [-] crntaylor|13 years ago|reply
[+] [-] j2kun|13 years ago|reply
[+] [-] verroq|13 years ago|reply
[+] [-] experiment0|13 years ago|reply
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.
[+] [-] mrcactu5|13 years ago|reply
* 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...
* http://www.math.harvard.edu/~ctm/home/text/class/harvard/55b...
[+] [-] msutherl|13 years ago|reply
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|13 years ago|reply
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.
[+] [-] randv|13 years ago|reply
[+] [-] jfarmer|13 years ago|reply
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
[+] [-] crntaylor|13 years ago|reply
[+] [-] ajtulloch|13 years ago|reply
http://tullo.ch/2011/mathematics-lecture-notes/ for the PDFs, and https://github.com/ajtulloch/SydneyUniversityMathematicsNote... for the LaTeX source.
[+] [-] crntaylor|13 years ago|reply
[+] [-] jason_adleberg|13 years ago|reply
"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|13 years ago|reply
* '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
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
[+] [-] sambomillo|13 years ago|reply
[+] [-] atondwal|13 years ago|reply
[+] [-] xxpor|13 years ago|reply
[+] [-] crntaylor|13 years ago|reply
[+] [-] dmcdougall_|13 years ago|reply
[+] [-] snatch_backside|13 years ago|reply
[+] [-] iconjack|13 years ago|reply
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[+] [-] hcarvalhoalves|13 years ago|reply
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
[+] [-] baaats|13 years ago|reply
[+] [-] gnu8|13 years ago|reply
~There is no royal road to mathematics~
[+] [-] atondwal|13 years ago|reply
[Osborne --- Advanced Mathematical Techniques: for Scientists and Engineers](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-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.
[+] [-] ivansavz|13 years ago|reply
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.
[+] [-] QuantumGuy|13 years ago|reply
[+] [-] zemanel|13 years ago|reply
[+] [-] unknown|13 years ago|reply
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[+] [-] kombinatorics|13 years ago|reply
[+] [-] unknown|13 years ago|reply
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[+] [-] j2kun|13 years ago|reply
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