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.<p>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).<p>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.
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.<p>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.
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 guess this is a good a time as any to show the organic chemistry notes that I've been writing up.<p><a href="https://github.com/alexganose/chem1201" rel="nofollow">https://github.com/alexganose/chem1201</a><p>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.<p>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.
john baez has been blogging for years on math and physics<p>* <a href="http://math.ucr.edu/home/baez/TWF.html" rel="nofollow">http://math.ucr.edu/home/baez/TWF.html</a><p>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<p>these notes rough around the edges, but great for self-teaching.<p>Harvard's Math 55 tries to accomplish similar goals. Not as user friendly, but more traditional:<p>* <a href="http://www.math.harvard.edu/~ctm/home/text/class/harvard/55a/08/html/index.html" rel="nofollow">http://www.math.harvard.edu/~ctm/home/text/class/harvard/55a...</a><p>* <a href="http://www.math.harvard.edu/~ctm/home/text/class/harvard/55b/09/html/index.html" rel="nofollow">http://www.math.harvard.edu/~ctm/home/text/class/harvard/55b...</a>
It would be wonderful to have the privilege and dedication to learn all of this.<p>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?<p>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 can't stand the sans serif typesetting and cramped mathematical formulas. The tone is kind of obnoxious, too.<p>When he introduces group theory:<p>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?<p>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.<p>In fact, I count three uses of the word "masturbate" in the notes.<p>I prefer something like Richard Feynman's style, where he makes a subject accessible while still respecting the subject.<p>Here's a fantastic example of Feynman explaining how a computer works, using an analogy of an ever-faster filing clerk: <a href="http://www.youtube.com/watch?v=EKWGGDXe5MA" rel="nofollow">http://www.youtube.com/watch?v=EKWGGDXe5MA</a>
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.<p><a href="http://tullo.ch/2011/mathematics-lecture-notes/" rel="nofollow">http://tullo.ch/2011/mathematics-lecture-notes/</a> for the PDFs, and
<a href="https://github.com/ajtulloch/SydneyUniversityMathematicsNotes/" rel="nofollow">https://github.com/ajtulloch/SydneyUniversityMathematicsNote...</a> for the LaTeX source.
From Sentence 5 of Example 1.1:<p>"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"<p>Yeah, understood about 5% of that.
You know what I would like? Math material that is <i>less</i> concise.<p>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.<p>[1] <a href="http://en.wikipedia.org/wiki/Isaac_Newton#Religious_views" rel="nofollow">http://en.wikipedia.org/wiki/Isaac_Newton#Religious_views</a>
This is extrordanarily good. For a similar, but more in depth covering of the same material I reccomend<p>[Osborne --- Advanced Mathematical Techniques: for Scientists and Engineers](<a href="http://www.amazon.com/Advanced-Mathematical-Techniques-Scientists-Engineers/dp/1453798765" rel="nofollow">http://www.amazon.com/Advanced-Mathematical-Techniques-Scien...</a>)<p>and for a much more indepth, but less pedagogically useful (more of a reference) [Arfken --- Mathematical Methods for Physicists, Seventh Edition: A Comprehensive Guide](<a href="http://www.amazon.com/Mathematical-Methods-Physicists-Seventh-Comprehensive/dp/0123846544" rel="nofollow">http://www.amazon.com/Mathematical-Methods-Physicists-Sevent...</a>)<p>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.<p>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.<p>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.