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. \newline Table
Table of Contents (for multivariable part): \ 8 Euclidean Spaces \ 8
8 Euclidean Spaces
8.1: Algebraic Structure \ 8
8.2: Planes and Linear Transformations \ 8
8.3: Topology of $\mathbb{R}^n$ \ 8
8.4: Interior, closure, and boundary \ \vspace*{1cm} 9 Convergence in $\mathbb{R}^n$ \ 9
9 Convergence in $\mathbb{R}^n$
9.1: Limits of sequences \ 9
9.2: Limits of functions \ 9
9.3: Continuous functions \ 9
9.4: Compact sets \ 9
9.5: Applications \ \vspace*{1cm} 10 Metric Spaces \ 10
10 Metric Spaces
10.1: Introduction \ 10
10.2: Limits of functions \ 10
10.3: Interior, closure, boundary \ 10
10.4: Compact sets \ 10
10.5: Connected sets \ 10
10.6: Continuous functions \ \vspace*{1cm} 11 Differentiability in $\mathbb{R}^n$ \ 11
11 Differentiability in $\mathbb{R}^n$
11.1: Partial derivatives and partial integrals \ 11
11.2: Definition of differentiability \ 11
11.3: Derivatives, differentials, and tangent planes \ 11
11.4: Chain rule \ 11
11.5: Mean Value Theorem and Taylor's Formula \ 11
11.6: Inverse Function Theorem \ 11
11.7: Optimization (Lagrange Multipliers) \ \vspace*{1cm} 12 Integration on $\mathbb{R}^n$ \ 12
12 Integration on $\mathbb{R}^n$
12.1: Jordan regions \ 12
12.2: Riemann integration on Jordan regions \ 12
12.3: Iterated integrals \ 12
12.4: Change of variables \ 12
12.5: Partitions of unity \ 12
12.6: Gamma function and volume \ \vspace*{1cm} 13 Fundamental Theorem of Vector Calculus \ 13
13 Fundamental Theorem of Vector Calculus
13.1: Curves \ 13
13.2: Oriented curves \ 13
13.3: Surfaces \ 13
13.4: Oriented surfaces \ 13
13.5: Theorems of Green and Gauss \ 13
13.6: Stokes's Theorem \ \vspace*{1cm} 14 Fourier Series \ 14
14 Fourier Series
14.1: Introduction \ 14
14.2: Summability of Fourier series \ 14
14.3: Growth of Fourier coefficients \ 14
14.4: Convergence of Fourier series \ 14
14.5: Uniqueness \ \vspace*{1cm} 15 Differentiable Manifolds \ 15
15 Differentiable Manifolds
15.1: Differential forms on $\mathbb{R}^n$ \ 15
15.2: Differentiable manifolds \ 15
15.3: Stokes's Theorem on manifolds