
Math notes to take you from one year of college calculus to grad student level - crntaylor
http://math.ucr.edu/home/baez/Alex_Alaniz_Lie.pdf
======
SandB0x
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
>> 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
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
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.

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psykotic
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.

~~~
sillysaurus
_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.

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

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j2kun
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
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.

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experiment0
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.

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mrcactu5
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/55a/08/html/index.html)

* [http://www.math.harvard.edu/~ctm/home/text/class/harvard/55b...](http://www.math.harvard.edu/~ctm/home/text/class/harvard/55b/09/html/index.html)

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msutherl
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
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
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./)

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jfarmer
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>

~~~
crntaylor
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.

------
ajtulloch
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...](https://github.com/ajtulloch/SydneyUniversityMathematicsNotes/)
for the LaTeX source.

~~~
crntaylor
This looks really neat. Nice work!

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jason_adleberg
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
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

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sambomillo
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
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.

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

~~~
dmcdougall_
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.

~~~
olympus
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.

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

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hcarvalhoalves
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>

~~~
baaats
I think what you're looking for are textbooks.

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

~~~
mahmud
Rudin, Apostol, Spivak, Lang, Munkres ..

~~~
EliRivers
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.

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atondwal
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...](http://www.amazon.com/Advanced-Mathematical-Techniques-Scientists-
Engineers/dp/1453798765))

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...](http://www.amazon.com/Mathematical-Methods-Physicists-Seventh-
Comprehensive/dp/0123846544))

In addition anything by Penrose tends to target a lay audience, but quickly
build up formalism and cover concepts interesting to even practicing
physicists.

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ivan_ah
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.

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QuantumGuy
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).

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zemanel
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.

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kombinatorics
this is gold. thanks!

