# Multivariable Calculus

**Author(s):**
Fabrizio
Donzelli

**Edition:
**
1

**Copyright:
**
2022

**Pages:
**
236

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## $31.50

** Multivariable Calculus** is an introduction to the basic concepts and tools of multivariable calculus, which are usually covered in the second and third parts of the undergraduate calculus course sequence. The book started as series of lecture notes for the MAT2322 University of Ottawa course (Calculus for Engineers). Since the content became large and complete, I decided to convert the lecture notes into a book that can be used as a textbook and a reference for all students of North-American universities.

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

**Fabrizio Donzelli**

Fabrizio Donzelli earned his BS in Physics from the Universita' degli studi di Milano (Italy) and his Ph. D. in Mathematics from the University of Miami (Florida). He started his career as a researcher by working in the field of affine algebraic geometric, and then expanded his research to the field of numerical analysis, focusing on stochastic and deterministic domain decomposition solution methods of partial differential equations. The author has almost two decades of teaching experience of undergraduate and graduate courses in university of Canada and United States. His interest and passion for education does not limit to university: he works as well in the tutoring business, providing assistance to students from grade 4 to grade 12.

** Multivariable Calculus** is an introduction to the basic concepts and tools of multivariable calculus, which are usually covered in the second and third parts of the undergraduate calculus course sequence. The book started as series of lecture notes for the MAT2322 University of Ottawa course (Calculus for Engineers). Since the content became large and complete, I decided to convert the lecture notes into a book that can be used as a textbook and a reference for all students of North-American universities.

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

**Fabrizio Donzelli**

Fabrizio Donzelli earned his BS in Physics from the Universita' degli studi di Milano (Italy) and his Ph. D. in Mathematics from the University of Miami (Florida). He started his career as a researcher by working in the field of affine algebraic geometric, and then expanded his research to the field of numerical analysis, focusing on stochastic and deterministic domain decomposition solution methods of partial differential equations. The author has almost two decades of teaching experience of undergraduate and graduate courses in university of Canada and United States. His interest and passion for education does not limit to university: he works as well in the tutoring business, providing assistance to students from grade 4 to grade 12.