Contents
Preface
Chapter 1 Euclidean Spaces
The Euclidean Space Rn as a Vector Space
Convergence of Sequences in Rn
Open Sets and Closed Sets
Interior, Exterior, Boundary and Closure
Limit Points and Isolated Points
Chapter 2 Limits of Multivariable Functions and Continuity
Multivariable Functions
Polynomials and Rational Functions
Component Functions of a Mapping
Invertible Mappings
Linear Transformations
Quadratic Forms
Limits of Functions
Continuity
Uniform Continuity
Contraction Mapping Theorem
Chapter 3 Continuous Functions on Connected Sets and Compact Sets
Path-Connectedness and Intermediate Value Theorem
Connectedness and Intermediate Value Property
Sequential Compactness and Compactness
Applications of Compactness
The Extreme Value Theorem
Distance Between Sets
Uniform Continuity
Linear Transformations and Quadratic Forms
Lebesgue Number Lemma
Chapter 4 Differentiating Functions of Several Variables
Partial Derivatives
Differentiability and First Order Approximation
Differentiability
First Order Approximations
Tangent Planes
Directional Derivatives
The Chain Rule and the Mean Value Theorem
Second Order Approximations
Local Extrema
Chapter 5 The Inverse and Implicit Function Theorems
The Inverse Function Theorem
The Proof of the Inverse Function Theorem
The Implicit Function Theorem
Extrema Problems and the Method of Lagrange Multipliers
Chapter 6 Multiple Integrals
Riemann Integrals
Properties of Riemann Integrals
Jordan Measurable Sets and Riemann Integrable Functions
Iterated Integrals and Fubini's Theorem
Change of Variables Theorem
Translations and Linear Transformations
Polar Coordinates
Spherical Coordinates
Other Examples
Proof of the Change of Variables Theorem
Some Important Integrals and Their Applications
Chapter 7 Fourier Series and Fourier Transforms
Orthogonal Systems of Functions and Fourier Series
The Pointwise Convergence of a Fourier Series
The L2 Convergence of a Fourier Series
The Uniform Convergence of a Trigonometric Series
Fourier Transforms
Appendix A Sylvester's Criterion
Appendix B Volumes of Parallelepipeds
Appendix C Riemann Integrability
References