Elementary Classical Analysis, 2nd Edition

Introduction: Sets And Functions -- 1. The Real Line And Euclidean Space -- 1.1. Ordered Fields And The Number System -- 1.2. Completeness And The Real Number System -- 1.3. Least Upper Bounds -- 1.4. Cauchy Sequences -- 1.5. Cluster Points; Lim Inf And Lim Sup -- 1.6. Euclidean Space -- 1.7. Norms, Inner Products, And Metrics -- 1.8. The Complex Numbers -- 2. The Topology Of Euclidean Space -- 2.1. Open Sets -- 2.2. Interior Of A Set -- 2.3. Closed Sets -- 2.4. Accumulation Points -- 2.5. Closure Of A Set -- 2.6. Boundary Of A Set -- 2.7. Sequences -- 2.8. Completeness -- 2.9. Series Of Real Numbers And Vectors -- 3. Compact And Connected Sets -- 3.1. Compactness -- 3.2. The Heine-borel Theorem -- 3.3. Nested Set Property -- 3.4. Path-connected Sets -- 3.5. Connected Sets -- 4. Continuous Mappings -- 4.1. Continuity -- 4.2. Images Of Compact And Connected Sets -- 4.3. Operations On Continuous Mappings -- 4.4. The Boundedness Of Continuous Functions On Compact Sets --^ 4.5. The Intermediate Value Theorem -- 4.6. Uniform Continuity -- 4.7. Differentiation Of Functions Of One Variable -- 4.8. Integration Of Functions Of One Variable -- 5. Uniform Convergence -- 5.1. Pointwise And Uniform Convergence -- 5.2. The Weierstrass M Test -- 5.3. Integration And Differentiation Of Series -- 5.4. The Elementary Functions -- 5.5. The Space Of Continuous Functions -- 5.6. The Arzela-ascoli Theorem -- 5.7. The Contraction Mapping Principle And Its Applications -- 5.8. The Stone-weierstrass Theorem -- 5.9. The Dirichlet And Abel Tests -- 5.10. Power Series And Cesaro And Abel Summability -- 6. Differentiable Mappings -- 6.1. Definition Of The Derivative -- 6.2. Matrix Representation -- 6.3. Continuity Of Differentiable Mappings; Differentiable Paths -- 6.4. Conditions For Differentiability -- 6.5. The Chain Rule -- 6.6. Product Rule And Gradients -- 6.7. The Mean Value Theorem -- 6.8. Taylor's Theorem And Higher Derivatives -- 6.9. Maxima And Minima --^ 7. The Inverse And Implicit Function Theorems And Related Topics -- 7.1. Inverse Function Theorem -- 7.2. Implicit Function Theorem -- 7.3. The Domain-straightening Theorem -- 7.4. Further Consequences Of The Implicit Function Theorem -- 7.5. An Existence Theorem For Ordinary Differential Equations -- 7.6. The Morse Lemma -- 7.7. Constrained Extrema And Lagrange Multipliers -- 8. Integration -- 8.1. Integrable Functions -- 8.2. Volume And Sets Of Measure Zero -- 8.3. Lebesgue's Theorem -- 8.4. Properties Of The Integral -- 8.5. Improper Integrals -- 8.6. Some Convergence Theorems -- 8.7. Introduction To Distributions -- 9. Fubini's Theorem And The Change Of Variables Formula -- 9.2. Fubini's Theorem -- 9.3. Change Of Variables Theorem -- 9.4. Polar Coordinates -- 9.5. Spherical And Cylindrical Coordinates -- 9.6. A Note On The Lebesgue Integral -- 9.7. Interchange Of Limiting Operations -- 10. Fourier Analysis -- 10.1. Inner Product Spaces -- 10.2. Orthogonal Families Of Functions --^ 10.3. Completeness And Convergence Theorems -- 10.4. Functions Of Bounded Variation And Fejer Theory (optional) -- 10.5. Computation Of Fourier Series -- 10.6. Further Convergence Theorems -- 10.7. Applications -- 10.8. Fourier Integrals -- 10.9. Quantum Mechanical Formalism. Jerrold E. Marsden, Michael J. Hoffman. Includes Bibliographical References (p. [677]-682) And Index. 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of the math without vector calculus or complex analysis.