Ask HN: Serious mathematics books that can replace a good teacher?
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!<p>For first calculus, read one of the well respected books.  In my experience and opinion, don&#x27;t need high school <i>advanced placement</i> 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&#x27;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&#x27;s old, but calculus hasn&#x27;t changed much since 1950 or so about when it gave up on teaching <i>infinitesimals</i>.<p>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.<p>For linear algebra, sure, Nering (a student of E. Artin at Princeton) and then Halmos (a finite dimensional introduction to Hilbert space and written when Halmos was an assistant to von Neumann, office not far from Einstein), etc.  For still more, R. Bellman has a good book -- he went all over active and put in everything including the kitchen sink.  Then for numerical work, old but a start, Forsythe and
Moler and&#x2F;or the documentation of LINPAK.<p>For multivariate calculus through the divergence theorem and as needed, e.g., for Maxwell&#x27;s equations, start with the most elementary treatments of <i>vector analysis</i> and f&#x27;get about the lack of proofs.  Actually notice, can see via a Google search, that the divergence theorem on a box is just a dirt simple application of the fundamental theorem of freshman calculus.  Then the difficulty is proving the thing on other shapes.  Then do multivariate calculus again via, say, Apostol, <i>Mathematical Analysis: A Modern Approach to Advanced Calculus</i>.  That he uses the Jordan curve theorem is a bit much but just go with the flow.  Fleming&#x27;s <i>Functions of Several Variables</i> does it with both measure theory and exterior algebra.  Or W. Rudin&#x27;s, as I recall, third edition, <i>Principles of Mathematical Analysis</i>.  For calculus done via Lebesgue&#x27;s measure theory, Royden&#x27;s <i>Real Analysis</i> and&#x2F;or the first half of Rudin&#x27;s <i>Real and Complex Analysis</i>.<p>My favorite author on ordinary differential equations is Coddington, and his elementary <i>An Introduction to Ordinary
Differential Equations</i> is nicely done and likely enough for nearly any applications might encounter.  Will want some of this if do A&#x2F;C circuit theory or deterministic optimal control.<p>For basic abstract algebra Herstein, <i>Topics in Algebra</i> should usually be enough and is well written.  With that will be well prepared for, say, algebraic coding theory which actually has some important applications.<p>Once have a background in measure theory, for probability, of course, Loeve or either of his students Breiman or Neveu.  Breiman is really nice to read. Neveu is a good candidate for  the most polished and elegantly done math book or writing of any kind in all of civilization.   Be sure to get to the good stuff on the Radon-Nikodym theorem, conditional expectation, martingales, and ergodic processes.<p>For relatively good results, better than hardly any students get just from courses, study the theorems and proofs until can do them yourself and can also explain them to any common man in the street and also explain where they fit in more generally.  Also be able to do the exercises.  E.g., Rudin&#x27;s <i>Principles</i> has the inverse and implicit function theorems as exercises!  Fleming applies them to Lagrange multipliers. These two theorems are the main prerequisites of differential geometry -- some parts of physics, now popular, touch on this math!<p>In my Ph.D. program I led the class in 4 of the five qualifying exams, and for the analysis and linear algebra exams, the books above were the secret.  After I did that, the department tried to have the students do better by offering a course from Rudin&#x27;s <i>Principles</i>; the course didn&#x27;t work; that book is too hard for a course and needs weeks of quiet time where a student can chew on the text slowly line by line. Rudin has some of the most precise presentations on Fourier theory.  So when physics uses Fourier theory or engineers use the fast Fourier transform, you will likely know Fourier theory quite a lot better than they do.  Right, the uncertainty principle in quantum mechanics is just a basic result in Fourier theory.  With Fourier theory and linearity, Schroedinger&#x27;s equation starts to make sense.  A good exercise in early measure theory is differentiation under the integral sign, Leibniz&#x27;s rule -- be sure to cover that with a good proof.<p>With the background from the books above will be well prepared for a wide range of more narrow topics in pure and applied math.<p>Nearly all these books are available used in nearly new condition for low prices.  In a move I lost some of my library and rebuilt with used copies.