# (Theoretical) Multivariable Calculus Textbooks

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A book fitting your description quite well is

Multidimensional Real Analysis by Duistermaat and Kolk, a 2-volume set: Differentiation and Integration.

It has rigorous, slick proofs, is highly theoretical, but with lots of (advanced) examples and many, many exercises. Much attention is given to the Inverse and Implicit Function theorem, and submanifolds of $\mathbb{R}^n$. The book is used in a second-year course at Utrecht University. I have to admit that it was quite hard to read for me when I took the course. But it is great as a reference, and years later I still consult it now and then.

Another nice book is Loomis & Sternberg – Advanced Calculus (freely available from Sternberg’s website.)

This is a lazy answer from a guy, who in his first and second year felt the need for a good exact rigorous and intuitive book in calculus, both one and several variables.

I haven’t read any of the following books, but I have browsed through them.

• Mathematical Analysis I, Zorich, amazon, 578 pages
• Mathematical Analysis II, Zorich, amazon, 688 pages

From my impressions, Zorich 1&2 is, as is said on the street, “the shit”. I was impressed to no end by his table of contents:

The second half of the book “An Introduction to Analysis” by William Wade supplies what you ask. (The first half is single variable.) There is even a section on elementary Fourier analysis.

8 Euclidean Spaces

8.1: Algebraic Structure

8.2: Planes and Linear Transformations

8.3: Topology of $\mathbb{R}^n$

8.4: Interior, closure, and boundary

9 Convergence in $\mathbb{R}^n$

9.1: Limits of sequences

9.2: Limits of functions

9.3: Continuous functions

9.4: Compact sets

9.5: Applications

10 Metric Spaces

10.1: Introduction

10.2: Limits of functions

10.3: Interior, closure, boundary

10.4: Compact sets

10.5: Connected sets

10.6: Continuous functions

11 Differentiability in $\mathbb{R}^n$

11.1: Partial derivatives and partial integrals

11.2: Definition of differentiability

11.3: Derivatives, differentials, and tangent planes

11.4: Chain rule

11.5: Mean Value Theorem and Taylor’s Formula

11.6: Inverse Function Theorem

11.7: Optimization (Lagrange Multipliers)

12 Integration on $\mathbb{R}^n$

12.1: Jordan regions

12.2: Riemann integration on Jordan regions

12.3: Iterated integrals

12.4: Change of variables

12.5: Partitions of unity

12.6: Gamma function and volume

13 Fundamental Theorem of Vector Calculus

13.1: Curves

13.2: Oriented curves

13.3: Surfaces

13.4: Oriented surfaces

13.5: Theorems of Green and Gauss

13.6: Stokes’s Theorem

14 Fourier Series

14.1: Introduction

14.2: Summability of Fourier series

14.3: Growth of Fourier coefficients

14.4: Convergence of Fourier series

14.5: Uniqueness

15 Differentiable Manifolds

15.1: Differential forms on $\mathbb{R}^n$

15.2: Differentiable manifolds

15.3: Stokes’s Theorem on manifolds

There is a recent book Functions of Several Real Variables. It has lot of good examples and exercises and is certainly theoretical.

I’ve been reading Mathematical Analysis by Tom Apostol to review much of this material. He seems to discuss and explain things more than Rudin. He even includes a few pictures, which are understandably primitive given the age of the book but I think they help. He also seems to have a lot of exercises.