Go to Introduction to Methods of Applied Mathematics (2004, 2321 pages, 9.23 MB pdf file, you can also download it in other formats). Additional, there are some Mathematica notebooks that solve some of the exercises of the text.
In the author’s own words, the text is „neither complete nor polished“ and is “in some stage of development”. However, the completed parts of this 2321 page work look very nice and cover a lot of topics. You can download the text for free, so you should have a look and check if the topics that interest you most are complete enough. As a general observation, it is in the advanced parts of the text (Partial Differential Equations, Calculus of Variations and Nonlinear Differential Equations) where most of the material is still missing. The other parts (Algebra, Calculus, Functions of a Complex Variable, Ordinary Differential Equations ) are more complete, please see the "Status of the Text" (at the bottom of the page) for details.
For an “Applied” text (the alternative title of the text is “Advanced Mathematical Methods for Scientists and Engineers”) I would have wished for less proofs and a bit more intuitive explanations. On the positive side, the text contains lots of examples and exercises. The exercises do not only have detailed solutions, but also good hints (though some of them are missing). Even if you choose to learn from another text you should come back to this one and try to do the exercises.
Other than many “Advanced Math for Scientists and Engineers” texts, this one does not assume prior knowledge of calculus. There are chapters on calculus and even algebra. However, if you are new to calculus, I would recommend that you rather look at some of the easier material that I have compiled on the Introductions to Calculus page. But don’t miss chapter 3.6.1 about Taylor’s theorem: This is a very good explanation to this important topic, with some pictures that demonstrate the approximation of functions very clearly.
If you are looking for some relief after working through the text and the exercises, take a look at chapter 0.3, Warnings and Disclaimers, and Appendix V, Glossary – the explanation of “one can prove that…” and other phrases is quite interesting...
If you find mistakes in the text and want to send them to the author (Mauch gives his Email Adress in chapter 0.3 and welcomes constructive criticism), please note that the text was last updated in 2004. I would speculate that this is due to lack of time, not lack of interest: In his bio, Mauch says that after his Ph.D. in 2003 and a one-year postdoc he became a staff scientist at Caltech's CACR and in 2009 joined a computational biology startup. Maybe he will continue to work on the text one day. I think it’s a really great text that would deserve to be completed and polished.
You might also be interested in some other Mathematics for Physics and Engineering resources.
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