I've always believed that the difference between those who know things and those who don't is access to information. And that on a level playing field, we could all pretty much know whatever we truly wanted to know. But while books are a wonderful resource, lectures on any given subject are even better.
Usually lectures require money (tuition) or at the very least time (rearranging your schedule to match a class's schedule)... but not any more!
I started out as a physics major at UCSD, but because I started falling in love with the geometric foundations of General Relativity I switched over to mathematics. So while I would love for this thread to be about how to reach some of my favorite areas of mathematics (specifically topics in differential geometry and differential topology), what I have found so far is actually best suited for those interested in physics (to about the equivalence of a bachelors in the subject).
Foundations...
I really want this to be inclusive, but in the end some background has to be assumed. So for this, I am assuming that you've had a first and second course in calculus (covering single variable calculus, analytic geometry and multivariable calculus). These courses are generally followed by a first course in linear algebra and differential equations, both of which are included in the lower division materials linked to below. I don't know of a place to get the calculus stuff online, but high school calculus (AP Calculus AB and BC) or even those offered at a community college level should cover everything one would need in this area.
If you have had any of this stuff before (and most science and engineering students have the same type of lower division course work as I recall), then skip what you don't need and start up at the first course where you feel the material is new for you.
The upper division classical mechanics is an important subject, and some people will assume that the term classical means it is stuff that can be skipped... believe me, it shouldn't be skipped! This is usually people's first introduction to both Lagrangian and Hamiltonian mechanics (which I don't recall being part of most lower division physics courses).
General Relativity is covered, but I've always felt that the subject can only be best understood by those who have taken the time to really learn the mathematics behind the subject. Unfortunately, I can't (currently) find any good lecture resources of the topics I would want to include. So the best I can do here is recommend some courses that you might be able to sit in on during the latter part of the upper division course work listed here. These courses would include classical differential geometry (upper division math), calculus on manifolds (upper division math), differentiable manifolds (graduate level math), Riemannian geometry (graduate level math) and integration on manifolds (graduate level math).
For other areas of physics I would recommend a course in modern algebra (upper division math, for an introduction to group theory) and complex analysis (upper division math). Between the differential geometry and group theory background one should have the foundations needed to start studying Lie Groups and Lie Algebras (graduate level math), all of which have applications in gauge theories.
If I come across any of the courses I think are missing, I'll add them into this thread.
The courses...
I think that this list of courses is fine if what you are looking for is increased knowledge. In fact, just watching the courses, with or without taking notes and watching some segments more than once (if you don't get a clear understanding the first time) would be more than enough if you aren't planning on doing research or the like in this area.
You can take these materials even further, of course, but at some point I would hope that someone investing a lot of their time into this would realize that they really might want to jump into a real university environment and work towards a degree in this area.
Lower Division Courses
Usually lectures require money (tuition) or at the very least time (rearranging your schedule to match a class's schedule)... but not any more!

I started out as a physics major at UCSD, but because I started falling in love with the geometric foundations of General Relativity I switched over to mathematics. So while I would love for this thread to be about how to reach some of my favorite areas of mathematics (specifically topics in differential geometry and differential topology), what I have found so far is actually best suited for those interested in physics (to about the equivalence of a bachelors in the subject).
Foundations...
I really want this to be inclusive, but in the end some background has to be assumed. So for this, I am assuming that you've had a first and second course in calculus (covering single variable calculus, analytic geometry and multivariable calculus). These courses are generally followed by a first course in linear algebra and differential equations, both of which are included in the lower division materials linked to below. I don't know of a place to get the calculus stuff online, but high school calculus (AP Calculus AB and BC) or even those offered at a community college level should cover everything one would need in this area.
If you have had any of this stuff before (and most science and engineering students have the same type of lower division course work as I recall), then skip what you don't need and start up at the first course where you feel the material is new for you.
The upper division classical mechanics is an important subject, and some people will assume that the term classical means it is stuff that can be skipped... believe me, it shouldn't be skipped! This is usually people's first introduction to both Lagrangian and Hamiltonian mechanics (which I don't recall being part of most lower division physics courses).
General Relativity is covered, but I've always felt that the subject can only be best understood by those who have taken the time to really learn the mathematics behind the subject. Unfortunately, I can't (currently) find any good lecture resources of the topics I would want to include. So the best I can do here is recommend some courses that you might be able to sit in on during the latter part of the upper division course work listed here. These courses would include classical differential geometry (upper division math), calculus on manifolds (upper division math), differentiable manifolds (graduate level math), Riemannian geometry (graduate level math) and integration on manifolds (graduate level math).
For other areas of physics I would recommend a course in modern algebra (upper division math, for an introduction to group theory) and complex analysis (upper division math). Between the differential geometry and group theory background one should have the foundations needed to start studying Lie Groups and Lie Algebras (graduate level math), all of which have applications in gauge theories.
If I come across any of the courses I think are missing, I'll add them into this thread.
The courses...
I think that this list of courses is fine if what you are looking for is increased knowledge. In fact, just watching the courses, with or without taking notes and watching some segments more than once (if you don't get a clear understanding the first time) would be more than enough if you aren't planning on doing research or the like in this area.
You can take these materials even further, of course, but at some point I would hope that someone investing a lot of their time into this would realize that they really might want to jump into a real university environment and work towards a degree in this area.
Lower Division Courses
Physics:Mathematics:
Upper Division CoursesGraduate Courses