Paul's Online Math Notes
tutorial.math.lamar.edu
tutorial.math.lamar.edu
https://www.youtube.com/user/professorleonard57
I must also confess that his ability in helping me understand and fall in love with Calculus two years ago was the main impetus for me to select Mathematics as my major. I'm now focusing my attention on Number Theory with high hopes of one day becoming a Theoretical Mathematician.
I should also add that Princeton Companion book to Mathematics is a valuable resource for learning what is out there in Pure Mathematics.
what does that even mean ?
As an example, I wasn't very good at solving differential equations, a very important part of math used to solve many real-life problems. Instead I studied things like the Lebesgue Integration which extends the notion of integration to a larger set of functions for which the more familiar Riemann integral wouldn't be defined. That was taught in my second semester real analysis course that had these prerequisites: Real Analysis I, Complex Analysis I, Differential Equations, Calc I and Calc II. A lot of work to get to an interesting subject, but a subject of interest to Mathematicians not engineers using math. An undergraduate degree in Applied math, in contrast to theoretical math, would probably have involved learning more about, say, differential equations.
On the other hand, applied mathematics might also involve some pretty technical "theoretical maths", which though being applied, can be studied for its own interest without the application.
So in some sense, "applied maths" sometimes means maths that is applied, rather than maths that can be applied. For that reason it perhaps makes sense to instead use the term "theoretical maths", though I am not sure if that is standard terminology or not.
All homework problems were really realistic with messy answers and the test questions were easy. Only a few problems for both.
Once, when talking about higher dimensional math, a student asked if string theory said there are 13 dimensions. He quickly replied "yeah" and I'm sure he knew it was more complicated than that but it was an irrelevant question. Without thinking I blurted out it was consistent with up to 13 dimensions and the whole class turned around to look at this kid who corrected The Machine. He quickly admitted that was true and moved on. Highlight of my education.
(includes some linear algebra, despite the URL)
Another resource is https://www.youtube.com/user/patrickJMT/
Khan Academy is good if you don't know anything but I like PatrickJMT's videos better if you're reviewing.
Highly recommended!
Really great resource here.
a^m a^n = a^(m+n)
y = mx + b
For those who used these and found them useful: did you really use them as cheat sheets, that is, to look up things while working on a problem, or did you use them more as a checklist before entering an exam, to check that you remembered most of them?
The review of complex numbers, on the other hand (not a cheat sheet but a condensed review) I assign to some masters' students who have not used complex numbers for 2 years and need to recall what they once learned. There it's just a concise but reasonably comprehensive refresher list.
If you notice an error be sure to correct it though. I did once while studying for my high school finals while staying up to 3AM. He responded quickly and fixed it immediately. Great to see someone who pays attention.
Just checked my email account. Apparently, I've emailed him corrections three times over the period of 4 years. He always emailed back within 24 hours.