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| Background | |
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| The Language of Mathematics | |
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| Sets and Functions | |
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| Calculus | |
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| Linear Algebra | |
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| The Role of Proofs | |
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| Appendix: Equivalence Relations | |
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| Abstract Analysis | |
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| The Real Numbers | |
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| An Overview of the Real Numbers | |
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| Infinite Decimals | |
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| Limits | |
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| Basic Properties of Limits | |
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| Upper and Lower Bounds | |
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| Subsequences | |
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| Cauchy Sequences | |
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| Appendix: Cardinality | |
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| Series | |
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| Convergent Series | |
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| Convergence Tests for Series | |
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| The Numbere | |
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| Absolute and Conditional Convergence | |
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| The Topology of R^n | |
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| n-dimensional Space | |
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| Convergence and Completeness inR n | |
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| Closed and Open Subsets ofR n | |
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| Compact Sets and the Heine-Borel Theorem | |
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| Functions | |
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| Limits and Continuity | |
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| Discontinuous Functions | |
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| Properties of Continuous Functions | |
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| Compactness and Extreme Values | |
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| Uniform Continuity | |
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| The Intermediate Value Theorem | |
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| Monotone Functions | |
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| Normed Vector Spaces | |
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| Differentiable Functions | |
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| The Mean Value Theorem | |
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| Riemann Integration | |
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| The Fundamental Theorem of Calculus | |
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| Wallis's Product and Stirling's Formula | |
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| Measure Zero and Lebesgue's Theorem | |
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| Differentiation and Integration | |
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| Definition and Examples | |
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| Topology in Normed Spaces | |
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| Inner Product Spaces | |
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| Orthonormal Sets | |
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| Orthogonal Expansions in Inner Product Spaces | |
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| Finite-Dimensional Normed Spaces | |
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| TheL P Norms | |
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| Limits of Functions | |
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| Limits of Functions | |
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| Uniform Convergence and Continuity | |
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| Uniform Convergence and Integration | |
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| Series of Functions | |
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| Power Series | |
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| Compactness and Subsets ofC(K) | |
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| Metric Spaces | |
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| Definitions and Examples | |
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| Compact Metric Spaces | |
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| Complete Metric Spaces | |
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| Connectedness | |
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| Metric Completion | |
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| TheL P Spaces and Abstract Integration | |
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| Applications | |
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| Approximation by Polynomials | |
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| Taylor Series | |
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| How Not to Approximate a Function | |
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| Bernstein's Proof of the Weierstrass Theorem | |
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| Accuracy of Approximation | |
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| Existence of Best Approximations | |
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| Characterizing Best Approximations | |
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| Expansions Using Chebychev Polynomials | |
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| Splines | |
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| Uniform Approximation by Splines | |
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| Appendix: The Stone-Weierstrass Theorem | |
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| Discrete Dynamical Systems | |
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| Fixed Points and the Contraction Principle | |
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| Newton's Method | |
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| Orbits of a Dynamical System | |
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| Periodic Points | |
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| Chaotic Systems | |
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| Topological Conjugacy | |
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| Iterated Function Systems and Fractals | |
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| Differential Equations | |
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| Integral Equations and Contractions | |
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| Calculus of Vector-Valued Functions | |
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| Differential Equations and Fixed Points | |
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| Solutions of Differential Equations | |
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| Local Solutions | |
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| Linear Differential Equations | |
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| Perturbation and Stability of Des | |
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| Existence without Uniqueness | |
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| Fourier Series and Physics | |
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| The Steady-State Heat Equation | |
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| Formal Solution | |
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| Orthogonality Relations | |
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| Convergence in the Open Disk | |
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| The Poisson Formula | |
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| Poisson's Theorem | |
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| The Maximum Principle | |
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| The Vibrating String (Formal Solution) | |
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| The Vibrating String (Rigourous Solution) | |
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| Appendix: The Complex Exponential | |
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| Fourier Series and Approximation | |
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| Least Squares Approximations | |
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| The Isoperimetric Problem | |
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| The Riemann-Lebesgue Lemma | |
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| Pointwise Convergence of Fourier Series | |
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| Gibbs's Phenomenon | |
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| Cesaro Summation of Fourier Series | |