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Table of Contents | |

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Preface | |

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Analysis of Functions of a Single Real Variable | |

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The Real Numbers | |

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Field Axioms | |

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Order Axioms | |

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Lowest Upper and Greatest Lower Bounds | |

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Natural Numbers, Integers, and Rational Numbers | |

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Recursion, Induction, Summations, and Products | |

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Sequences of Real Numbers | |

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Limits | |

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Limit Laws | |

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Cauchy Sequences | |

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Bounded Sequences | |

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Infinite Limits | |

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Continuous Functions | |

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Limits of Functions | |

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Limit Laws | |

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One-Sided Limits and Infinite Limits | |

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Continuity | |

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Properties of Continuous Functions | |

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Limits at Infinity | |

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Differentiable Functions | |

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Differentiability | |

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Differentiation Rules | |

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Rolle's Theorem and the Mean Value Theorem | |

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The Riemann Integral I | |

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Riemann Sums and the Integral | |

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Uniform Continuity and Integrability of Continuous Functions | |

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The Fundamental Theorem of Calculus | |

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The Darboux Integral | |

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Series of Real Numbers I | |

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Series as a Vehicle To Define Infinite Sums | |

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Absolute Convergence and Unconditional Convergence | |

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Some Set Theory | |

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The Algebra of Sets | |

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Countable Sets | |

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Uncountable Sets | |

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The Riemann Integral II | |

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Outer Lebesgue Measure | |

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Lebesgue's Criterion for Riemann Integrability | |

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More Integral Theorems | |

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Improper Riemann Integrals | |

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The Lebesgue Integral | |

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Lebesgue Measurable Sets | |

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Lebesgue Measurable Functions | |

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Lebesgue Integration | |

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Lebesgue Integrals versus Riemann Integrals | |

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Series of Real Numbers II | |

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Limits Superior and Inferior | |

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The Root Test and the Ratio Test | |

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Power Series | |

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Sequences of Functions | |

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Notions of Convergence | |

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Uniform Convergence | |

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Transcendental Functions | |

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The Exponential Function | |

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Sine and Cosine | |

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L'Hopital's Rule | |

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Numerical Methods | |

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Approximation with Taylor Polynomials | |

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Newton's Method | |

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Numerical Integration | |

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Analysis in Abstract Spaces | |

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Integration on Measure Spaces | |

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Measure Spaces | |

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Outer Measures | |

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Measurable Functions | |

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Integration of Measurable Functions | |

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Monotone and Dominated Convergence | |

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Convergence in Mean, in Measure, and Almost Everywhere | |

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Product [sigma]-Algebras | |

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Product Measures and Fubini's Theorem | |

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The Abstract Venues for Analysis | |

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Abstraction I: Vector Spaces | |

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Representation of Elements: Bases and Dimension | |

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Identification of Spaces: Isomorphism | |

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Abstraction II: Inner Product Spaces | |

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Nicer Representations: Orthonormal Sets | |

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Abstraction III: Normed Spaces | |

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Abstraction IV: Metric Spaces | |

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L[superscript p] Spaces | |

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Another Number Field: Complex Numbers | |

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The Topology of Metric Spaces | |

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Convergence of Sequences | |

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Completeness | |

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Continuous Functions | |

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Open and Closed Sets | |

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Compactness | |

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The Normed Topology of R[superscript d] | |

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Dense Subspaces | |

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Connectedness | |

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Locally Compact Spaces | |

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Differentiation in Normed Spaces | |

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Continuous Linear Functions | |

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Matrix Representation of Linear Functions | |

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Differentiability | |

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The Mean Value Theorem | |

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How Partial Derivatives Fit In | |

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Multilinear Functions (Tensors) | |

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Higher Derivatives | |

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The Implicit Function Theorem | |

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Measure, Topology, and Differentiation | |

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Lebesgue Measurable Sets in R[superscript d] | |

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C[infinity] and Approximation of Integrable Functions | |

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Tensor Algebra and Determinants | |

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Multidimensional Substitution | |

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Introduction to Differential Geometry | |

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Manifolds | |

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Tangent Spaces and Differentiable Functions | |

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Differential Forms, Integrals Over the Unit Cube | |

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k-Forms and Integrals Over k-Chains | |

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Integration on Manifolds | |

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Stokes' Theorem | |

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Hilbert Spaces | |

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Orthonormal Bases | |

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Fourier Series | |

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The Riesz Representation Theorem | |

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Applied Analysis | |

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Physics Background | |

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Harmonic Oscillators | |

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Heat and Diffusion | |

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Separation of Variables, Fourier Series, and Ordinary Differential Equations | |

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Maxwell's Equations | |

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The Navier Stokes Equation for the Conservation of Mass | |

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Ordinary Differential Equations | |

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Banach Space Valued Differential Equations | |

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An Existence and Uniqueness Theorem | |

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Linear Differential Equations | |

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The Finite Element Method | |

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Ritz-Galerkin Approximation | |

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Weakly Differentiable Functions | |

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Sobolev Spaces | |

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Elliptic Differential Operators | |

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Finite Elements | |

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Conclusion and Outlook | |

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Appendices | |

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Logic | |

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Statements | |

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Negations | |

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Set Theory | |

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The Zermelo-Fraenkel Axioms | |

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Relations and Functions | |

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Natural Numbers, Integers, and Rational Numbers | |

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The Natural Numbers | |

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The Integers | |

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The Rational Numbers | |

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Bibliography | |

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Index | |