Reddit mentions of Advanced Calculus (5th Edition)

I want to pay someone else to write the following book. And if someone else won't write it, maybe I will.

Applied Mathematics for Economists (max 700 pages)

aka Simon and Blume, but more mature, more compact, more focused.

Chapters:

1. Linear Algebra I (matricies, systems of equations, vector spaces, linear transformations, eigenthings; with applications to linear economic models)
   
2. Partial Differential Calculus (limits and continuity in R^(n); gradient, Jacobian, Hessian; maps from R^(n) to R^(m); implicit and inverse function theorems; unconstrained optimization, Lagrange multipliers, and equilibrium concepts; the focus is on systems of equations as seen throughout economics)
   
3. Multiple Integrals (standard topics with emphasis on applications to probability; the last section would cover Bayesian inference)
   
4. Linear Algebra II (least squares; singular value decomposition; LU, Cholesky, QR, Schur, and other funky matrix decompositions you occasionally use in econometrics; some extended attention is given to quadratic forms and symmetric matricies)
   
5. Theory of Optimization (Lagrangeans, Hamiltonians, Bellmans; the classical results regarding existence and uniqueness; the classical results regarding characterization, regularity, and sensitivity; Berge's theorem)
   
6. Complex Variables and Fourier Series (for the time-series folks)
   
   Prerequisite: Calculus I-II are assumed.
   
   Idea: Calculus III and linear algebra are developed in the first three chapters, with special emphasis on applications to economics. The fourth chapter covers advanced material in linear algebra and even touches on numerical issues. The fifth chapter develops optimization, both static and dynamic, in discrete and continuous time, with applications to economics throughout. The sixth chapter develops the complex variable theory that you need to do proper time-series analysis.
   
   Maybe sneak a chapter on analysis and topology in there. Maybe cut the optimization chapter into two or three pieces.
   
   The point of the book is to put all of the applied math that you'll actually use on a day-to-day basis in a single, relatively brief textbook. I think it can be done in less than 700 pages. Kaplan crammed all of advanced calculus for physics into 700 pages, so surely it can be done for economics.
   
   What isn't here: There is a market for books on advanced calculus. However, they are all geared towards engineering and physics students. They have a ton of material on differential equations and vector theory that are irrelevant for economists. So I cut all of those bits out and replace them with additional material on matrix analysis and optimization. Basically, I trade the theorems of Green, Gauss, and Stokes for the theorems of Lagrange, Hamilton, and Bellman. It's a reasonable tradeoff. I also throw away the (i,j,k) notation and spend more time on least squares, which would be brought back to its proper home in linear algebra.
   
   I have deliberately left out a detailed treatment of probability and statistics, as they could get their own book (of about 500 pages). I have left out a detailed treatment of numerical methods because they could get their own book as well.

It's hard to say, in my day, Advanced Calculus was a class that was taught out of a book like this which basically covered what might be termed "Vector Calculus" i.e. the study of functions f:R^n -->R^m . There would be little, if any, talk of the material in a book like Apostol. That would be reserved for a class called "Real Analysis".

For grad school in physics I'd say that the material in Kaplan (above) would be absolutely essential. I'm not sure how useful the Apostol book is going to be for a physicist, but I'm not a physicist... That said, Spivak, which was mentioned elsewhere, strikes me a being very useful for a physicist and it would be a lot easier to digest after a book like Apostol.

Kaplan's Advanced Calculus might be for you, then. Link. I've found this through more nefarious sources :) PM me if you can't find it then. Just be forewarned that this is a very proof-heavy book!