Multivariable Calculus

1 About the book 
1.1 The audience
1.1.1 Engineering
1.1.2 Physics
1.1.3 Mathematics
1.1.4 Statistics
1.1.5 Sciences
1.2 How to read the book
1.2.1 Definitions, Theorems, Formulas, Equations
1.2.2 Figures
1.2.3 Examples and Exercises
1.2.4 Remarks
1.2.5 Notation
1.2.6 Software-based calculations and plotting
1.3 Logic is important!
1.4 Prerequisites

2 Functions of several variables: introductory concepts
2.1 Functions
2.2 Functions of 1 variable
2.3 Functions of 2 variables
2.4 Functions of 3 variables
2.5 Functions of n variables
2.6 Limits and continuity
2.7 Partial derivatives
2.8 Gradient
2.9 Directional derivatives
2.10 Tangent (hyper)plane and linear approximation
2.11 The chain rule
2.12 Examples
2.12.1 Lines, planes, hyperplanes (linear, degree 1)
2.12.2 Quadric surfaces (degree 2)
2.12.3 Real numbers versus complex numbers
2.12.4 Cubic surfaces and the donuts
2.12.5 Nonpolynomial examples

3 Optimization of functions of several variables
3.1 Local theory
3.2 Global theory: closed and bounded domains
3.3 Constraints and Lagrange multipliers
3.4 Linear systems versus nonlinear system

4 Integration
4.1 Definite integrals of functions of 1 variable
4.2 Definite integrals of functions of 2 variables
4.3 Double integrals and probability—Monte Carlo methods
4.4 Double integrals in polar coordinates
4.5 Triple integrals
4.6 Cylindrical coordinates
4.7 Spherical coordinates
4.8 Riemann integrals in sciences
4.8.1 Center of mass
4.8.2 Three statistics formulae
4.8.3 Electromagnetic energy
4.9 Riemann integrals: a few general results
4.9.1 Integrals in more than 3 variables
4.9.2 Change of variables in multiple integrals

5 Parametric curves
5.1 Basic facts
5.2 Arc length
5.3 Line integrals of scalar valued functions
5.4 Curvature

6 Vector fields
6.1 Introduction
6.2 Vector fields and ordinary differential equations
6.3 A few words about fields in physics
6.4 The three basic differential operators on vector fields and functions
6.5 Rotations, compression, expansions
6.6 Line integrals of vector fields
6.7 Two physics laws expressed with line integrals

7 Conservative vector fields 
7.1 Fundamental theorem of conservative vector fields
7.2 Conservative vector fields in 2D
7.3 Conservative vector fields in 3D

8 Surfaces 
8.1 Cross product
8.2 Parametrizations, orientations 
8.3 Surface integrals
8.4 Flux of a vector field
8.5 A few words about curvatures of surfaces

9 Green’s, Stokes’, and Gauss’ Theorems
9.1 Green’s Theorem
9.2 Areas and Green’s Theorem
9.3 Stokes’ Theorem
9.4 Gauss’ Theorem
9.5 Beyond 3D

10 Conclusions
10.1 From 1 variable to several variables
10.2 Differential versus integral calculus
10.3 Differential calculus
10.4 Coordinates
10.5 Integral calculus
10.6 Infinite dimensional spaces

11 Exercises
11.1 Chapter 1
11.2 Chapter 2 
11.3 Chapter 3
11.4 Chapter 4
11.5 Chapter 5
11.6 Chapter 6
11.7 Chapter 7
11.8 Chapter 8
11.9 Chapter 9
11.10 Chapter 10

Bibliography