About our author William Wade received his PhD in harmonic analysis from the University of California - Riverside. He has been a professor of the Department of Mathematics at the University of Tennessee for more than 40 years. During that time, he has received multiple awards including 2 Fulbright Scholarships, the Chancellor's Award for Research and Creative Achievements, the Dean's Award for Extraordinary Service, and the National Alumni Association Outstanding Teaching Award.Wade's research interests include problems of uniqueness, growth and dyadic harmonic analysis, on which he has published numerous papers, two books and given multiple presentations on three continents. His current publication, An Introduction to Analysis, is now in its 4th edition.In his spare time, Wade loves to travel and take photographs to document his trips. He is also musically inclined, and enjoys playing classical music, mainly baroque on the trumpet, recorder, and piano.

• PrefacePart I. ONE-DIMENSIONAL THEORY  1. The Real Number System1.1 Introduction1.2 Ordered field axioms1.3 Completeness Axiom1.4 Mathematical Induction1.5 Inverse functions and images1.6 Countable and uncountable sets  2. Sequences in R2.1 Limits of sequences2.2 Limit theorems2.3 Bolzano-Weierstrass Theorem2.4 Cauchy sequences*2.5 Limits supremum and infimum  3. Continuity on R3.1 Two-sided limits3.2 One-sided limits and limits at infinity3.3 Continuity3.4 Uniform continuity  4. Differentiability on R4.1 The derivative4.2 Differentiability theorems4.3 The Mean Value Theorem4.4 Taylor's Theorem and l'Hôpital's Rule4.5 Inverse function theorems  5 Integrability on R5.1 The Riemann integral5.2 Riemann sums5.3 The Fundamental Theorem of Calculus5.4 Improper Riemann integration*5.5 Functions of bounded variation*5.6 Convex functions  6. Infinite Series of Real Numbers6.1 Introduction6.2 Series with nonnegative terms6.3 Absolute convergence6.4 Alternating series*6.5 Estimation of series*6.6 Additional tests  7. Infinite Series of Functions7.1 Uniform convergence of sequences7.2 Uniform convergence of series7.3 Power series7.4 Analytic functions*7.5 Applications  Part II. MULTIDIMENSIONAL THEORY  8. Euclidean Spaces8.1 Algebraic structure8.2 Planes and linear transformations8.3 Topology of Rn8.4 Interior, closure, boundary  9. Convergence in Rn9.1 Limits of sequences9.2 Heine-Borel Theorem9.3 Limits of functions9.4 Continuous functions*9.5 Compact sets*9.6 Applications  10. Metric Spaces10.1 Introduction10.2 Limits of functions10.3 Interior, closure, boundary10.4 Compact sets10.5 Connected sets10.6 Continuous functions10.7 Stone-Weierstrass Theorem  11. Differentiability on Rn11.1 Partial derivatives and partial integrals11.2 The definition of differentiability11.3 Derivatives, differentials, and tangent planes11.4 The Chain Rule11.5 The Mean Value Theorem and Taylor's Formula11.6 The Inverse Function Theorem*11.7 Optimization  12. Integration on Rn12.1 Jordan regions12.2 Riemann integration on Jordan regions12.3 Iterated integrals12.4 Change of variables*12.5 Partitions of unity*12.6 The gamma function and volume  13. Fundamental Theorems of Vector Calculus13.1 Curves13.2 Oriented curves13.3 Surfaces13.4 Oriented surfaces13.5 Theorems of Green and Gauss13.6 Stokes's Theorem  *14. Fourier Series*14.1 Introduction*14.2 Summability of Fourier series*14.3 Growth of Fourier coefficients*14.4 Convergence of Fourier series*14.5 Uniqueness  AppendicesA. Algebraic lawsB. TrigonometryC. Matrices and determinantsD. Quadric surfacesE. Vector calculus and physicsF. Equivalence relations  ReferencesAnswers and Hints to ExercisesSubject IndexSymbol Index  *Enrichment section

About our author William Wade received his PhD in harmonic analysis from the University of California - Riverside. He has been a professor of the Department of Mathematics at the University of Tennessee for more than 40 years. During that time, he has received multiple awards including 2 Fulbright Scholarships, the Chancellor's Award for Research and Creative Achievements, the Dean's Award for Extraordinary Service, and the National Alumni Association Outstanding Teaching Award.Wade's research interests include problems of uniqueness, growth and dyadic harmonic analysis, on which he has published numerous papers, two books and given multiple presentations on three continents. His current publication, An Introduction to Analysis, is now in its 4th edition.In his spare time, Wade loves to travel and take photographs to document his trips. He is also musically inclined, and enjoys playing classical music, mainly baroque on the trumpet, recorder, and piano.

• PrefacePart I. ONE-DIMENSIONAL THEORY  1. The Real Number System1.1 Introduction1.2 Ordered field axioms1.3 Completeness Axiom1.4 Mathematical Induction1.5 Inverse functions and images1.6 Countable and uncountable sets  2. Sequences in R2.1 Limits of sequences2.2 Limit theorems2.3 Bolzano-Weierstrass Theorem2.4 Cauchy sequences*2.5 Limits supremum and infimum  3. Continuity on R3.1 Two-sided limits3.2 One-sided limits and limits at infinity3.3 Continuity3.4 Uniform continuity  4. Differentiability on R4.1 The derivative4.2 Differentiability theorems4.3 The Mean Value Theorem4.4 Taylor's Theorem and l'Hôpital's Rule4.5 Inverse function theorems  5 Integrability on R5.1 The Riemann integral5.2 Riemann sums5.3 The Fundamental Theorem of Calculus5.4 Improper Riemann integration*5.5 Functions of bounded variation*5.6 Convex functions  6. Infinite Series of Real Numbers6.1 Introduction6.2 Series with nonnegative terms6.3 Absolute convergence6.4 Alternating series*6.5 Estimation of series*6.6 Additional tests  7. Infinite Series of Functions7.1 Uniform convergence of sequences7.2 Uniform convergence of series7.3 Power series7.4 Analytic functions*7.5 Applications  Part II. MULTIDIMENSIONAL THEORY  8. Euclidean Spaces8.1 Algebraic structure8.2 Planes and linear transformations8.3 Topology of Rn8.4 Interior, closure, boundary  9. Convergence in Rn9.1 Limits of sequences9.2 Heine-Borel Theorem9.3 Limits of functions9.4 Continuous functions*9.5 Compact sets*9.6 Applications  10. Metric Spaces10.1 Introduction10.2 Limits of functions10.3 Interior, closure, boundary10.4 Compact sets10.5 Connected sets10.6 Continuous functions10.7 Stone-Weierstrass Theorem  11. Differentiability on Rn11.1 Partial derivatives and partial integrals11.2 The definition of differentiability11.3 Derivatives, differentials, and tangent planes11.4 The Chain Rule11.5 The Mean Value Theorem and Taylor's Formula11.6 The Inverse Function Theorem*11.7 Optimization  12. Integration on Rn12.1 Jordan regions12.2 Riemann integration on Jordan regions12.3 Iterated integrals12.4 Change of variables*12.5 Partitions of unity*12.6 The gamma function and volume  13. Fundamental Theorems of Vector Calculus13.1 Curves13.2 Oriented curves13.3 Surfaces13.4 Oriented surfaces13.5 Theorems of Green and Gauss13.6 Stokes's Theorem  *14. Fourier Series*14.1 Introduction*14.2 Summability of Fourier series*14.3 Growth of Fourier coefficients*14.4 Convergence of Fourier series*14.5 Uniqueness  AppendicesA. Algebraic lawsB. TrigonometryC. Matrices and determinantsD. Quadric surfacesE. Vector calculus and physicsF. Equivalence relations  ReferencesAnswers and Hints to ExercisesSubject IndexSymbol Index  *Enrichment section

About our author William Wade received his PhD in harmonic analysis from the University of California - Riverside. He has been a professor of the Department of Mathematics at the University of Tennessee for more than 40 years. During that time, he has received multiple awards including 2 Fulbright Scholarships, the Chancellor's Award for Research and Creative Achievements, the Dean's Award for Extraordinary Service, and the National Alumni Association Outstanding Teaching Award.Wade's research interests include problems of uniqueness, growth and dyadic harmonic analysis, on which he has published numerous papers, two books and given multiple presentations on three continents. His current publication, An Introduction to Analysis, is now in its 4th edition.In his spare time, Wade loves to travel and take photographs to document his trips. He is also musically inclined, and enjoys playing classical music, mainly baroque on the trumpet, recorder, and piano.

• PrefacePart I. ONE-DIMENSIONAL THEORY  1. The Real Number System1.1 Introduction1.2 Ordered field axioms1.3 Completeness Axiom1.4 Mathematical Induction1.5 Inverse functions and images1.6 Countable and uncountable sets  2. Sequences in R2.1 Limits of sequences2.2 Limit theorems2.3 Bolzano-Weierstrass Theorem2.4 Cauchy sequences*2.5 Limits supremum and infimum  3. Continuity on R3.1 Two-sided limits3.2 One-sided limits and limits at infinity3.3 Continuity3.4 Uniform continuity  4. Differentiability on R4.1 The derivative4.2 Differentiability theorems4.3 The Mean Value Theorem4.4 Taylor's Theorem and l'Hôpital's Rule4.5 Inverse function theorems  5 Integrability on R5.1 The Riemann integral5.2 Riemann sums5.3 The Fundamental Theorem of Calculus5.4 Improper Riemann integration*5.5 Functions of bounded variation*5.6 Convex functions  6. Infinite Series of Real Numbers6.1 Introduction6.2 Series with nonnegative terms6.3 Absolute convergence6.4 Alternating series*6.5 Estimation of series*6.6 Additional tests  7. Infinite Series of Functions7.1 Uniform convergence of sequences7.2 Uniform convergence of series7.3 Power series7.4 Analytic functions*7.5 Applications  Part II. MULTIDIMENSIONAL THEORY  8. Euclidean Spaces8.1 Algebraic structure8.2 Planes and linear transformations8.3 Topology of Rn8.4 Interior, closure, boundary  9. Convergence in Rn9.1 Limits of sequences9.2 Heine-Borel Theorem9.3 Limits of functions9.4 Continuous functions*9.5 Compact sets*9.6 Applications  10. Metric Spaces10.1 Introduction10.2 Limits of functions10.3 Interior, closure, boundary10.4 Compact sets10.5 Connected sets10.6 Continuous functions10.7 Stone-Weierstrass Theorem  11. Differentiability on Rn11.1 Partial derivatives and partial integrals11.2 The definition of differentiability11.3 Derivatives, differentials, and tangent planes11.4 The Chain Rule11.5 The Mean Value Theorem and Taylor's Formula11.6 The Inverse Function Theorem*11.7 Optimization  12. Integration on Rn12.1 Jordan regions12.2 Riemann integration on Jordan regions12.3 Iterated integrals12.4 Change of variables*12.5 Partitions of unity*12.6 The gamma function and volume  13. Fundamental Theorems of Vector Calculus13.1 Curves13.2 Oriented curves13.3 Surfaces13.4 Oriented surfaces13.5 Theorems of Green and Gauss13.6 Stokes's Theorem  *14. Fourier Series*14.1 Introduction*14.2 Summability of Fourier series*14.3 Growth of Fourier coefficients*14.4 Convergence of Fourier series*14.5 Uniqueness  AppendicesA. Algebraic lawsB. TrigonometryC. Matrices and determinantsD. Quadric surfacesE. Vector calculus and physicsF. Equivalence relations  ReferencesAnswers and Hints to ExercisesSubject IndexSymbol Index  *Enrichment section