I have a fairly solid background in Mathematics.  I can comfortably do
most problems from simple algebra through multivariable calculus. 
However, I want to learn more.  Younger students have a natural math
progression--algebra, geometry, trigonometry, math analysis,
calculus...this isn't so for advanced mathematics.

I would like to know two things.  

First, what would be considered a natural mathematics progression
*beyond* basic calculus?  I would like to self-teach myself, or take
online courses, in order to be able to understand & follow complex
mathematics literature, and to have meaningful discussions with those
in fields (such as physics) that require the language of mathematics. 
In other words, I am looking for a mathematics program, i.e. take
Calculus II, then XXXX course, then XXXX course...etc

Second, please point me toward online resources that could instruct me
in such courses or toward books that would help enable me in this
endeavour.

The more swift and helpful the response, the more I will be inclined to tip.

Thank you in advance.

Regards,

Herkdrvr

When I was an undergraduate Computer Science major, the continuous
mathematics tracks went like this:

1. Single Variable Calculus
2. Multivariate Calculus
3. Differential Equations
4. Linear Algebra
5. Probability

One could interchange the order of Linear Algebra and Probability as
those two courses aren't really dependent on each other.  Technically,
we could take Linear Algebra and/or Probability as soon as we finished
multi variable calculus.

I also had a discrete (non-continuous) mathematics track that I
followed.  After multivariate calculus, I took a two courses in
Discrete Mathematics (essentially we covered the entirety of this
book: http://www.mhhe.com/math/advmath/rosenindex.mhtml), and then
course in Algorithms.

The only class I felt that I missed out on was Statistics (calculus-based).

In my opinion, a lot of schools are really intensive in their
continuous mathematics, but lack on the discrete mathematics.  Where I
work in the real world, I *rarely* use calculus if ever, but the stuff
that comes in handy all the time is the probability, statistics,
linear algebra, and discrete mathematics.  They tend to be more
practical for a computer scientist/systems engineer, as I'm sure my
colleagues in physics and mechanical/electrical engineering use their
continuous mathematics much more than their discrete (although they
rely a lot on probability too).

My suggestion, if you are familiar with single variable calculus
already, give discrete mathematics a try and see if you like it.  It's
so different from the math you've seen already, which makes it really
interesting, especially if you like doing things like solving puzzles.
 You'll also get a chance to practice your proof skills, look at
methods of proof, set theory, graph theory, logic, formal languages,
basic algorithms, relations, combinatorics/counting/number theory
among other topics.  It's sort of ecclectic, but I think it's a great
way to sample things other than calculus.

I was a mathematics major in college, so I might as well shoot out what we took...

Calculus 1,2,3, 4(Calc 4 was called Differnetial Equations)
Linear Algebra
Modern Geometry
Mathematical Statistics (as opposed to the Prob & Stats offered to everyone else)
Real Analysis 1&2 (the "back bone" of calculus)
Abstract Algebra 1&2 (basically an elaborate treatment of sets)
Topology

This is my personal collection from 
http://www.jimmyr.com/education.php


Should add the wiki and dmoz though too:
http://en.wikipedia.org/wiki/Mathematics
http://dmoz.org/Science/Math/


Calculus
http://www.calculus.org/ -- Calculus.org
http://www.calculus.net/ci2/practice/ -- Calc Probs
http://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom7/deluxe-content.html
-- Calc Probs 1
http://www.jtaylor1142001.net/ -- Calc Probs 2
http://archives.math.utk.edu/visual.calculus/ -- Visual Calculus
http://www.ies.co.jp/math/products/calc/menu.html -- Calc Problems
http://www.webmath.com/index6.html -- Webmath Calculus
http://mathworld.wolfram.com/topics/Calculus.html -- Wolfram Calculus
http://www.geocities.com/calculusisnumber1/VisCalc.html -- Encyclo
http://www.sciencedaily.com/encyclopedia/list_of_calculus_topics -- Encyclo2

Math
http://www.webmath.com/ -- Webmath
http://mathforum.org/mathtools/cell.html?tp=9.5&new_tp=9.3 -- Mathforums tools
http://amath.colorado.edu/computing/mmm/ -- Mathematical, maple
http://mathforum.org/ -- Mathforums
http://www.onlineconversion.com/ -- Convertions
http://www.gomath.com/ -- Go Math


GoMath has thousands of math questions an answers. Calculus.org is
just phenomenal. They have more calculus and links to really hard
questions. I thought calculus was pretty easy before I went to that
site.



The local library is always a great resource for advanced mathematics topics.


Check out wolfram's book too. I have no course outline, but I hope
this resources help. I too enjoy having math as a hobby.

I suppose that "advanced mathematics" can be meant several ways.  Good
comments have already laid out what additional topics might be covered
in an undergraduate curriculum typically after calculus, and the
distinction between upper division undergraduate courses and
first-year graduate courses in mathematics is often an emphasis on
rigor that may vary with the academic environment.

In terms of practical difficulty, I suggest that calculus is already
quite advanced.  The follow-ons to differential equations, linear
algebra, and probability/statistics are largely developments of themes
(differentiation, integration, changes of variables) introduced with
single- or multi-variable calculus.

In my frame of reference "advanced" is then mostly the degree of rigor
with which mathematical topics can be dealt with, leading ultimately
to a study of logic and the foundations of mathematics.

regards, mathtalk-ga

Aww, I left out the best link:
http://ocw.mit.edu/OcwWeb/Mathematics/index.htm

If you'd like to learn more about how to write mathematical proofs,
including basic logic and proof by induction, I recommend this book:
Foundations of Higher Mathematics by Fletcher and Patty
Perhaps you can find it used on www.abebooks.com
My college used this book in a course that mainly taught how to write
proofs.  It was the transition from the lower level problem-solving
courses (e.g., calculus) to the higher level courses in which all you
do is prove theorems (e.g., abstract algebra).

If you are more interested in solving problems than proving theorems,
you might next study differential equations.  Any intro diff eq
textbook should be fine.  You'll also want to learn some linear
analysis (enough to compute eigenvectors and eigenvalues).  If you are
in the US, you might be able to study both of those at your local
community college if you're interested.  After that, you can read
about chaos theory.  "Nonlinear dynamics and chaos" by Strogatz is
very readable.

Congratulations on deciding on a mathematics field.  Unfortunately
there is no linear progression past first year caclulus... but if your
interested in sticking with calculus the next most natural progression
is Analaysis of complex variables.  Basically you take caclulus to the
imaginary realm.

I would suggest this book.

Function Theory of One Complex Variable 

by Robert E. Greene and Steven G. Krantz.

I used it for my 3rd and 4th year Complex Analysis courses.

As well i would recommend emailing someone in the department of
mathematics at your local university perhaps they could get you
involved or provide further guidance.