Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

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

math.ucr.edu

Post reply on HN
Print view View original

11–20 of 70 posts

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

#11

Post by crntaylor »

msutherl wrote: post #9

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…

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.

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

#12

Post by crntaylor »

psykotic wrote: post #10
crntaylor wrote: post #8

Earlier quoted context omitted.

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…

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.

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

#13

Post by sillysaurus »

psykotic wrote: post #3

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…

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.

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

#14

Post by cdwhite »

crntaylor wrote: post #11
msutherl wrote: post #9

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…

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 appli…

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./)

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

#15

Post by 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.

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

#16

Post by drivebyacct2 »

dmcdougall_ wrote: post #6
xxpor wrote: post #4

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

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.

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

#18

Post by randv »

msutherl wrote: post #9

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…

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.

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

#19

Post by ianfernz »

sillysaurus wrote: post #13
psykotic wrote: post #3

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…

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 unify…

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

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

#20

Post by 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/55b...

Post reply on HN