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Chapter I. Calculus for Functions of One Variable |
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3 | (72) |
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0. Prerequisites Properties of the real numbers, limits and convergence of sequences of real numbers, exponential function and logarithm |
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3 | (10) |
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1. Limits and Continuity of Functions Definitions of continuity, uniform continuity, properties of continuous functions, intermediate value theorem, Holder and Lipschitz continuity |
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13 | (8) |
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2. Differentiability Definitions of differentiability, differentiation rules, differentiable functions are continuous, higher order derivatives |
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21 | (10) |
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3. Characteristic Properties of Differentiable Functions. Differential Equations Characterization of local extrema by the vanishing of the derivative, mean value theorems, the differential equation f(1) = XXXf, uniqueness of solutions of differential equations, characterization of local maxima and minima via second derivatives, Taylor expansion |
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31 | (10) |
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4. The Banach Fixed Point Theorem. The Concept of Banach Space Banach fixed point theorem, definition of norm, metric, Cauchy sequence, completeness |
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41 | (4) |
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5. Uniform Convergence. Interchangeability of Limiting Processes. Examples of Banach Spaces. The Theorem of Arzela-Ascoli Convergence of sequences of functions, power series, convergence theorems, uniformly convergent sequences, norms on function spaces, theorem of Arzela-Ascoli on the uniform convergence of sequences of uniformly bounded and equicontinuous functions |
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45 | (14) |
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6. Integrals and Ordinary Differential Equations Primitives, Riemann integral, integration rules, integration by parts, chain rule, mean value theorem, integral and area, ODEs, theorem of Picard-Lindelof on the local existence and uniqueness of solutions of ODEs with a Lipschitz condition |
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59 | (16) |
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Chapter II. Topological Concepts |
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75 | (24) |
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7. Metric Spaces: Continuity, Topological Notions, Compact Sets Definition of a metric space, open, closed, convex, connected, compact sets, sequential compactness, continuous mappings between metric spaces, bounded linear operators, equivalence of norms in R(d), definition of a topological space |
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75 | (24) |
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Chapter III. Calculus in Euclidean and Banach Spaces |
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99 | (52) |
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8. Differentiation in Banach Spaces Definition of differentiability of mappings between Banach spaces, differentiation rules, higher derivatives, Taylor expansion |
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99 | (12) |
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9. Differential Calculus in R(d) |
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111 | (16) |
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A. Scalar valued functions Gradient, partial derivatives, Hessian, local extrema, Laplace operator, partial differential equations |
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B. Vector valued functions Jacobi matrix, vector fields, divergence, rotation |
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10. The Implicit Function Theorem. Applications Implicit and inverse function theorems, extrema with constraints, Lagrange multipliers |
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127 | (12) |
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11. Curves in R(d). Systems of ODEs Regular and singular curves, length, rectifiability, arcs, Jordan arc theorem, higher order ODE as systems of ODEs |
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139 | (12) |
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Chapter IV. The Lebesgue Integral |
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151 | (80) |
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12. Preparations. Semicontinuous Functions Theorem of Dini, upper and lower semicontinuous functions, the characteristic function of a set |
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151 | (8) |
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13. The Lebesgue Integral for Semicontinuous Functions. The Volume of Compact Sets The integral of continuous and semicontinuous functions, theorem of Fubini, volume, integrals of rotationally symmetric functions and other examples |
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159 | (18) |
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14. Lebesgue Integrable Functions and Sets Upper and lower integral, Lebesgue integral, approximation of Lebesgue integrals, integrability of sets |
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177 | (12) |
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15. Null Functions and Null Sets. The Theorem of Fubini Null functions, null sets, Cantor set, equivalence classes of integrable functions, the space L(1), Fubini's theorem for integrable functions |
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189 | (10) |
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16. The Convergence Theorems of Lebesgue Integration Theory Monotone convergence theorem of B. Levi, Fatou's lemma, dominated convergence theorem of H. Lebesgue, parameter dependent integrals, differentiation under the integral sign |
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199 | (10) |
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17. Measurable Functions and Sets. Jensen's Inequality. The Theorem of Egorov Measurable functions and their properties, measurable sets, measurable functions as limits of simple functions, the composition of a measurable function with a continuous function is measurable, Jensen's inequality for convex functions, theorem of Egorov on almost uniform convergence of measurable functions |
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209 | (10) |
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18. The Transformation Formula Transformation of multiple integrals under diffeomorphisms, integrals in polar coordinates |
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219 | (12) |
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Chapter V. L(p) and Sobolev Spaces |
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231 | (40) |
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19. The L(p)-Spaces L(p)-functions, Holder's inequality, Minkowski's inequality, completeness of L(p)-spaces, approximation of L(p)-functions by smooth functions through mollification, test functions |
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231 | (18) |
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20. Integration by Parts. Weak Derivatives. Sobolev Spaces Weak derivatives defined by an integration by parts formula, Sobolev functions have weak derivatives in L(p)-spaces, calculus for Sobolev functions, Sobolev embedding theorem on the continuity of Sobolev functions whose weak derivatives are integrable to a sufficiently high power, Poincare inequality, compactness theorem of Rellich-Kondrachov on the L(p)-convergence of sequences with bounded Sobolev norm |
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249 | (22) |
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Chapter VI. Introduction to the Calculus of Variations and Elliptic Partial Differential Equations |
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271 | (76) |
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21. Hilbert Spaces. Weak Convergence Definition and properties of Hilbert spaces, Riesz representation theorem, weak convergence, weak compactness of bounded sequences, Banach-Saks lemma on the convergence of convex combinations of bounded sequences |
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271 | (10) |
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22. Variational Principles and Partial Differential Equations Dirichlet's principle, weakly harmonic functions, Dirichlet problem, Euler-Lagrange equations, variational problems, weak lower semicontinuity of variational integrals with convex integrands, examples from physics and continuum mechanics, Hamilton's principle, equilibrium states, stability, the Laplace operator in polar coordinates |
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281 | (32) |
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23. Regularity of Weak Solutions Smoothness of weakly harmonic functions and of weak solutions of general elliptic PDEs, boundary regularity, classical solutions |
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313 | (16) |
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24. The Maximum Principle Weak and strong maximum principle for solutions of elliptic PDEs, boundary point lemma of E. Hopf, gradient estimates, theorem of Liouville |
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329 | (12) |
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25. The Eigenvalue Problem for the Laplace Operator Eigenfunctions of the Laplace operator form a complete orthonormal basis of L(2) as an application of the Rellich compactness theorem |
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341 | (6) |
| Index of Notation |
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347 | (2) |
| Index |
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349 | |
