Math notes to take you from one year of college calculus to grad student level | Hacker News Reader

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

math.ucr.edu

217 pointscrntaylor63 comments

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.

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".

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.

This kind of mathematics is at a level that requires a certain amount of self-motivation, passion, desire, whatever you want to call it, to learn. It's probably something you're born with, which is why not many people pursue graduate studies in mathematics and those who do are mostly people with a real love for it. It's unlikely that at some point in your adult life you're going to suddenly develop the necessary burning desire to motivate such study on the basis of a set of notes someone posted on the internet.

>> It's probably something you're born with

What are you basing this claim on?

>> It's unlikely that at some point in your adult life you're going to suddenly develop the necessary burning desire to motivate such study

While it's true that adults are not likely to re-enroll in University to study mathematics, that doesn't necessarily mean the desire isn't there.

For instance, a "burning" desire may be sufficient to motivate a 22 year old to study mathematics, however that same desire in a 44 year old might not overcome the pressures, realities and obligations of life, work & family.

> For instance, a "burning" desire may be sufficient to motivate a 22 year old to study mathematics, however that same desire in a 44 year old might not overcome the pressures, realities and obligations of life, work & family.

You'd be surprised at how many there are.

I'm mid-30s and I've recently finished a maths degree via distance learning (Open University in the UK). During the time I've been studying I've moved house, got married, had increased work pressures, become a father and I am helping bootstrap a startup in what little free time is left from all of that.

At the various tutorials, revision day schools and exams during my studies I met lots of others who were in similar positions. It's more common than you think.

So you're actually proving the point: you weren't born with this burning desire, you didn't learn it in college or high school or (as burning desire would indicate) when you were twelve - you waited until now, and were perfectly capable of learning it once your motivations changed.

As they do, as time passes. Anybody out of their 20s will know that.

I had a desire to study it since I was young. I excelled at both Comp Sci and Maths during school[1]. Studied both all the way prior to University and then had to pick one (I didn't want to do a joint degree as I wanted to study one in depth rather than two partially).

A Comp Sci degree was going to be a bigger advantage for the kind of job I was looking for and so that won. The desire to study Maths has always been there, it's just taken a while since leaving University until I was in a position where I had the time/money to study it in my spare time.

[1] First computer was a ZX81 when I was just 5 years' old.

  if you really wanted to learn this material you would have 
  started already.

People change.

> People change.

People can change; most won't. While I, like you, take issue with the fatalistic statement of "you would have started already" (how young a person would the GP say this to? 30? 25? 18?), there is a grain of truth to it. Most people, even if they want to, won't get much out of this. Not that I'd like to discourage anyone.

I woke up one day 4 years ago and realised that I wanted to learn mathematics. I realised I had always wanted to learn Mathematics - only I had never before understood what it actually was. I now have a MSc in mathematics and I am getting to the point where I am able to bring that knowledge to bear on my satellite interests, mathematical physics and my own ideas for GAI.

It is documents like this which have helped me get to here and will take me to where I want to be.

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.

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.

I agree, math books tend to be on the terse side as they are - you wouldn't want the material to be even more terse, because you'll end up spending more time grasping it without the help the additional text may provide.

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.

Gilbert Strang's Linear Algebra textbook is designed to be useful to the self-learner, and I think the latest edition was revised after MIT's OCW was a thing, so actually references all the material online, IIRC.

I've read some of it and it's quite good. His lectures are great too.

Rudin's Analysis book is very good, so is Lang's Algebra. In college I found I learned much more from the textbook than from lectures

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 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).

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.

That's a fair point.

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 think "high bandwidth" is a good way to describe it.

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.

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

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.

john baez has been blogging for years on math and physics

* 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...

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.

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./)

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.

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

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.

http://tullo.ch/2011/mathematics-lecture-notes/ for the PDFs, and https://github.com/ajtulloch/SydneyUniversityMathematicsNote... for the LaTeX source.

This looks really neat. Nice work!

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.

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

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.

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.

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

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.

I don't think hand wavy was exactly the right way to describe it, more like flippant.

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.

Yes, I know that I sound whiny but I'd love to see this rendered via LaTeX.

When I saw that this was a word document exported as a pdf I knew this wasn't a serious text.

I'm pretty sure I know the guy that wrote this (seriously, how many physicists named Alex Alaniz are there?), and he can't have LaTeX on his work computer because of some really dumb rules. So I don't blame him for it being an exported Word document because a LaTeX version would have meant doing this all at home instead of his spare time at work.

I don't blame him. Instead, I just didn't read it.

iconjack, you are a dead user. Time to create a new account.

What do you mean, a dead user? I was just quoting from the document, as an example of writing some might find obnoxious. Personally I find it rather entertaining.

No worries. This happens when you 'double post' a comment (usually because HN dies in mid submission) and then delete the original. The duplicate then stays grayed out making people think that your account has been hellbanned.

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

That's what university is for.

~There is no royal road to mathematics~

Let me guess, you have a major in math.

I think what you're looking for are textbooks.

You will have my eternal gratitude if you are kind enough to name one that doesn't suck.

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).

Rudin, Apostol, Spivak, Lang, Munkres ..

I'd like to say that Apostol's "Introduction to Analytic Number Theory" is bloody awful and is in no way suitable to be an introduction to analytic number theory. A reference book of proofs of some common theorems, yes, but an introduction? No.

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-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.

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.

this is gold. thanks!

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