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E-grāmata: Topology, Calculus and Approximation

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Topology, Calculus and ApproximationPalielināt
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Presenting basic results of topology, calculus of several variables, and approximation theory which are rarely treated in a single volume, this textbook includes several beautiful, but almost forgotten, classical theorems of Descartes, Erdos, Fejér, Stieltjes, and Turán. 

The exposition style of Topology, Calculus and Approximation follows the Hungarian mathematical tradition of Paul Erdos and others. In the first part, the classical results of Alexandroff, Cantor, Hausdorff, Helly, Peano, Radon, Tietze and Urysohn illustrate the theories of metric, topological and normed spaces. Following this, the general framework of normed spaces and Carathéodory's definition of the derivative are shown to simplify the statement and proof of various theorems in calculus and ordinary differential equations. The third and final part is devoted to interpolation, orthogonal polynomials, numerical integration, asymptotic expansions and the numerical solution of algebraic and differential equations.

Students of both pure and applied mathematics, as well as physics and engineering should find this textbook useful. Only basic results of one-variable calculus and linear algebra are used, and simple yet pertinent examples and exercises illustrate the usefulness of most theorems. Many of these examples are new or difficult to locate in the literature, and so the original sources of most notions and results are given to help readers understand the development of the field.
Part I Topology
1 Metric Spaces
3(34)
1.1 Definitions and Examples
3(5)
1.2 Convergence, Limits and Continuity
8(6)
1.3 Completeness: A Fixed Point Theorem
14(9)
1.4 Compactness
23(7)
1.5 Exercises
30(7)
2 Topological Spaces
37(28)
2.1 Definitions and Examples
37(6)
2.2 Neighborhoods: Continuous Functions
43(3)
2.3 Connectedness
46(4)
2.4 * Compactness
50(5)
2.5 * Convergence of Nets
55(6)
2.6 Exercises
61(4)
3 Normed Spaces
65(32)
3.1 Definitions and Examples
65(8)
3.2 Metric and Topological Properties
73(5)
3.3 Finite-Dimensional Normed Spaces
78(4)
3.4 Continuous Linear Maps
82(3)
3.5 Continuous Linear Functionals
85(3)
3.6 Complex Normed Spaces
88(1)
3.7 Exercises
89(8)
Part II Differential Calculus
4 The Derivative
97(20)
4.1 Definitions and Elementary Properties
97(8)
4.2 Mean Value Theorems
105(6)
4.3 The Functions Rm → Rn
111(4)
4.4 Exercises
115(2)
5 Higher-order derivatives
117(24)
5.1 Continuous multilinear maps
117(3)
5.2 Higher-order derivatives
120(6)
5.3 Taylor's formula
126(4)
5.4 Local extrema
130(2)
5.5 Convex functions
132(5)
5.6 The functions Rm → Rn
137(2)
5.7 Exercises
139(2)
6 Ordinary Differential Equations
141(24)
6.1 Integrals of Vector-Valued Functions
141(2)
6.2 Definitions and Examples
143(5)
6.3 The Cauchy--Lipschitz Theorem
148(5)
6.4 Additional Results and Linear Equations
153(4)
6.5 Explicit Solutions
157(3)
6.6 Exercises
160(5)
7 Implicit Functions and Their Applications
165(28)
7.1 Implicit Functions
165(6)
7.2 Lagrange Multipliers
171(3)
7.3 The Spectral Theorem
174(2)
7.4 * The Inverse Function Theorem
176(6)
7.5 * The Implicit Function Theorem
182(2)
7.6 * Lagrange Multipliers: General Case
184(1)
7.7 * Differential Equations: Dependence on the Initial Data
185(3)
7.8 Exercises
188(5)
Part III Approximation Methods
8 Interpolation
193(26)
8.1 Lagrange Interpolation
194(2)
8.2 Minimization of Errors: Chebyshev Polynomials
196(4)
8.3 Divided Differences: Newton's Interpolating Formula
200(3)
8.4 Hermite Interpolation
203(4)
8.5 Theorems of Weierstrass and Fejer
207(5)
8.6 Spline Functions
212(2)
8.7 Exercises
214(5)
9 Orthogonal Polynomials
219(12)
9.1 Gram-Schmidt Orthogonalization
219(1)
9.2 Orthogonal Polynomials
220(3)
9.3 Roots of Orthogonal Polynomials
223(4)
9.4 Exercises
227(4)
10 Numerical Integration
231(36)
10.1 Lagrange Formulas
232(2)
10.2 Newton-Cotes Rules
234(1)
10.3 Gauss Rules
235(3)
10.4 Theorems of Stieltjes and Erdos--Turan
238(3)
10.5 Euler's and Stirling's Formulas
241(5)
10.6 Bernoulli Polynomials
246(5)
10.7 Euler's General Formula
251(4)
10.8 Asymptotic Expansions: Stirling's Series
255(5)
10.9 The Trapezoidal Rule: Romberg's Method
260(3)
10.10 Exercises
263(4)
11 Finding Roots
267(16)
11.1 * Descartes's Rule of Signs
267(2)
11.2 * Sturm Sequences
269(2)
11.3 * Roots of Polynomials
271(1)
11.4 * The Method of Householder and Bauer
272(3)
11.5 * Givens' Method
275(2)
11.6 Newton's Method
277(3)
11.7 Exercises
280(3)
12 Numerical Solution of Differential Equations
283(16)
12.1 Approximation of Solutions
283(6)
12.2 The Runge--Kutta Method
289(1)
12.3 The Dirichlet Problem
290(3)
12.4 The Monte Carlo Method
293(1)
12.5 The Heat Equation
294(1)
12.6 Exercises
295(4)
Hints and Solutions to Some Exercises
299(38)
Comments and Historical References
337(12)
Metric Spaces
337(2)
Topological Spaces
339(1)
Normed Spaces
340(1)
The Derivative
341(1)
Higher-Order Derivatives
342(1)
Ordinary Differential Equations
343(1)
Implicit Functions and Their Applications
344(2)
Interpolation
346(1)
Orthogonal Polynomials
346(1)
Numerical Integration
347(1)
Finding Roots
347(1)
Numerical Solution of Differential Equations
348(1)
Bibliography 349(20)
Teaching Suggestions 369(2)
Subject Index 371(6)
Name Index 377
Vilmos Komornik has studied in Budapest, Hungary, and has taught in Hungary and France for nearly 40 years. His main research fields are control theory of partial differential equations and combinatorial number theory. He has made a number of contributions to the theory of J.L. Lions on exact controllability and stabilization and has co-authored several papers on expansions in noninteger bases with P. Erds. He is an external member of the Hungarian Academy of Sciences.
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