Mathematical Analysis. Volume II

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