This question already has an answer here:
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.
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.
Table of Contents (for multivariable part):
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.