A really complicated calculus book

I've been studying math as a hobby, just for fun for years, and I had my goal to understand nearly every good undergraduate textbook and I think, I finally reached it. So now I need an another goal. I've just found a very nice book /S. Ramanan – Global Calculus/ from the "Graduate Studies in Mathematics" series and it looks nearly awesome:

• Sheaves and presheaves
• Differential manifolds
• Lie groups
• Differential operators
• Tensor fields
• Sheaf cohomology
• Linear connections
• Complex manifolds
• Ricci curvature tensor
• Elliptic operators

But it's only 316 pages and it seemed not very fundamental and detailed for me (but yes, it's still great). So here's my question: what huge complicated calculus textbooks like this one do you know that I should aim to understand? The Big Creepy Books, you know :) I'm very interested in algebraic and differential geometry, general and algebraic topology, Lie groups and algebras, pseudo- and differential operators. I don't know very much about all of this yet but I'm trying so hard to do, it's so exciting! ;)

I've already covered:

• Algebra: Chapter 0 (Graduate Studies in Mathematics) by Paolo Aluffi
• A Course in Algebra (Graduate Studies in Mathematics, Vol. 56) by E. B. Vinberg
• Linear Algebra and Geometry (Algebra, Logic and Applications) by P. K. Suetin, Alexandra I. Kostrikin and Yu I Manin
• Topology from the Differentiable Viewpoint by John Willard Milnor
• Topology and Geometry for Physicists by Charles Nash and Siddhartha Sen
• Mathematical Analysis I and II by V. A. Zorich and R. Cooke
• Complex Analysis by Serge Lang
• Ordinary Differential Equations by Vladimir I. Arnold and R. Cooke
• Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics) by Sigurdur Helgason

So I'm looking for something like Ramanan's book, but maybe more detailed and fundamental.

Solution 1:

As a cure for your desire I recommend a few pages per day of Madsen and Tornehave's From Calculus to Cohomology.
The book starts at the level of advanced calculus and introduces an amazing set of concepts and results: de Rham cohomology, degree, Poincaré-Hopf theorem, characteristic classes, Thom isomorphism, Gauss-Bonnet theorem,...

The style is austere but very honest: none of the odious "it is easy to see" or "left as an exercise for the reader" here!
On the contrary, you will find some very explicit computations rarely done elsewhere : for an example look at pages 74-75, where the authors very explicitly analyze the tangent bundles and some differential forms on numerical spaces $\mathbb R^n$, spheres $S^{n-1}$ and projective spaces $\mathbb R\mathbb P^{n-1}$.

All in all, a remarkable book that should appeal to you by its emphasis on algebraic topology done with calculus tools .

Edit:September 25, 2016
An even more complicated book is Nickerson, Spencer and Steenrod's Advanced Calculus, of which you can find a review by Allendoerfer here.
The book is an offspring of lecture notes given in Princeton around 1958 for an honours course on advanced calculus.
The book starts very tamely with an introduction to vector spaces in which students are requested to prove (on page 5) that for a vector $A$, one has $A+A=2A$.
On page 232 however (the book has 540 pages) the authors introduce the notion of graded tensor algebra, on page 258 Hodge's star operator, then come potential theory, Laplace-Beltrami operators, harmonic forms and cohomology, Grothendieck-Dolbeault's version of the Poincaré lemma and Kähler metrics.
I suspect that a teacher who tried to give such a course at the undergraduate level today would run a serious risk of being tarred-and-feathered.

Solution 2:

The obvious "scary book" for linear partial differential operators (and pseudodifferential operators, and Fourier integral operators, and distribution theory and...) is

Hörmander, Lars, Analysis of linear partial differential operators, 1-4, Springer Verlag.

Solution 3:

Another ambitious book is Raghavan Narasimhan's Analysis on real and Complex Manifolds
It consists of three chapters.
Chapter 2 contains more or less standard material on real and complex manifolds, but the other two chapters are quite unusual:

Chapter 1 contains some hard analysis on $\mathbb R^n$.
You will find there some classical results like Sard's theorem but also tough results like Borel's theorem, according to which there exists a smooth (generally non-analytic) function with any prescribed set of coefficients as its Taylor development at the origin.
In other words, the morphism of $\mathbb R$-algebras $ C^\infty(\mathbb R^n) \to \mathbb R[[x_1,...,X_n]]$ from smooth functions to formal power series given by Taylor's formula is surjective .
The chapter also contains theorems of Whitney approximating smooth functions by analytic ones: these results are very rarely presented in books.

Chapter 3 is devoted to linear elliptic differential operators.
As an application of that theory, Narasimhan proves Behnke-Stein's theorem (it is the last result of the book) according to which every non-compact connected Riemann surface is Stein.

This is a difficult but great book by a great mathematician.

Solution 4:

The five volume set by Spivak should keep you busy for a while.

Solution 5:

I love Morita's "Geometry of Differential Forms". Don't remember whether it covers the linear algebra prerequisites (alternating tensor product) thoroughly or not and while those are just linear algebra, they aren't covered in most books and having them down really solidly makes a dramatic difference in how difficult or easy differential forms is. Regardless, I remember it as a beautiful book.