Real and Complex Analysis

Physics: Feynman's lectures on physics. Maths: Rudin's Real and Complex analysis. Software engineering: The Mythical Man-Month. Computer science: Knuth's TAOCP.

Books that are given nicknames by grad students. :) For a foundational understanding of Calculus, and just becoming a better mathematician, a good start would be -> Baby Rudin (Principles of Mathematical Analysis, Walter Rudin) Papa Rudin (Real and Complex Analysis, Walter Rudin) Grandpa Rudin (Functional Analysis, Walter Rudin)

Bluntly, with only occasional exceptions, have to learn from the books. Period. No biggie: Once are in grad school or a prof, have to learn from papers, and the good books are typically MUCH easier to read than the papers! For first calculus, read one of the well respected books. In my experience and opinion, don't need high school advanced placement calculus -- regard it as a waste of time. Also the last time I looked at Khan Academy, I concluded that it was a waste or worse and by people who didn't have a good background in calculus. Generally the Internet sources are too elementary. Solution: Just use some good books. Protter and Morrey is good (I taught from it), but it is a bit elementary. I learned from Johnson and Kiokmeister; I like it. At the time it was also being used at Harvard. The exercises are especially good. It's old, but calculus hasn't changed much since 1950 or so about when it gave up on teaching infinitesimals. But might check out the Maxima software for symbolic indefinite integration -- as I recall, its algorithms are so good they do any indefinite integration that can be done in closed form.…

Linear algebra, and more than one such book. IMHO long and still the best linear algebra book is Halmos, Finite Dimensional Vector Spaces (FDVS). It was written in 1942 when Halmos was an "assistant" to John von Neumann at the Institute for Advanced Study. It is intended to be finite dimensional vector spaces but done with the techniques of Hilbert space. The central result in the book, according to Halmos, is the spectral decomposition. One result at a time, the quality of von Neumann comes through. Commonly physicists have been given that book as their introduction to Hilbert space for quantum mechanics. But FDVS is a little too much for a first book on linear algebra, or maybe even a second book, should be maybe a third one. Also high quality is Nering, Linear Algebra and Matrix Theory. Again, the quality comes through: Nering was a student of Artin at Princeton. There Nering does most of linear algebra on just finite fields, not just the real and complex fields; finite fields in linear algebra are important in error correcting codes. So, that finite field work is a good introduction to abstract algebra.…

I think that those who are down-voting people suggesting Rudin are missing the forest for the trees. The level is not as low as Feynman's lectures, but if you want a text that has the hallmarks of a true master at work, Rudin fits the bill. I personally always felt that what made Feynman's lectures what they are were, was that the man actually understood the subjects on such a deep level that his mind was able to hone in on those little simple thought experiments and ways of looking at things that gave you huge insight into what he was lecturing on. Rudin comes at analysis with the same level of understanding and that's why Real & Complex and Baby Rudin are still the gold standard for analysis texts. It is impossible to convey the sense of elegance that mathematicians speak about until you've seen it and been blown away by it. Rudin will do that and leave you speechless, he's the master. I can tell you that my first encounter with Real & Complex leaving me thinking that Rudin was on a fundamentally higher plane than any author I had read before, it was like watching a magician. If you're looking for a real Feynman in mathematics, IMHO Rudin is your man.

Halmos, 'Finite Dimensional Vector Spaces' Rudin, 'Real and Complex Analysis'

"Can you please suggest on-line resource to learn great beginning and advanced probability concepts?" Yup! The subject at the level you ask is a major topic in applied math but is not very popular in US universities. So, you will not get many good answers. In particular, the 'computer science' community, with 'machine learning' and 'artificial' this, Bayesian that, likely won't have good answers. Your "on-line" part is asking a bit much; I can give you references to books but not all on-line. There may be some PDF files on-line, from TeX, that have such material; try some Google searching with the keywords used here. The intuitive foundations of probability went back to gambling. 'Probability' is a field of 'applied' math and as such is well defined: About 100 years ago, E. Borel student H. Lebesgue invented 'measure theory' which essentially 'rewrote' classic calculus, especially the part about integration. For the simple cases, what Lebesgue did gets the same numerical values as the classic Riemann integral. The difference is that in theoretical work Lebesgue's integral is much more general and much better 'behaved'.…