The Way of AnalysisThe Way of Analysis gives a thorough account of real analysis in one or several variables, from the construction of the real number system to an introduction of the Lebesgue integral. The text provides proofs of all main results, as well as motivations, examples, applications, exercises, and formal chapter summaries. Additionally, there are three chapters on application of analysis, ordinary differential equations, Fourier series, and curves and surfaces to show how the techniques of analysis are used in concrete settings. |
Contents
Preliminaries | 1 |
Differential Calculus | 5 |
Sequences and Series of Functions | 7 |
Transcendental Functions | 8 |
Implicit Functions Curves and Surfaces | 13 |
Construction of the Real Number System | 25 |
Topology of the Real Line | 73 |
Definitions | 97 |
plex Exponentials | 323 |
Euclidean Space and Metric Spaces | 355 |
Differential Calculus in Euclidean Space | 445 |
Ordinary Differential Equations | 459 |
Fourier Series | 517 |
The Lebesgue Integral | 625 |
Multiple Integrals | 691 |
Index | 722 |
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Common terms and phrases
affine function approximation assume axiom of Archimedes Cą function Cauchy criterion Cauchy sequence closed set compact complete complex numbers compute construction contains continuous function convergence countable curve Dedekind cut defined definition denote derivative difference quotient differentiable disjoint domain endpoints equation equivalence class error 1/n example exercise f and g f is continuous f(xo fact finite number fn(x formula Fourier series function f graph infinite decimal expansions intersection Lebesgue lemma Let f limit limit-point limn limsup limx Lipschitz condition m₁ mean value theorem measure metric space natural number non-negative obtain open interval open set partition polynomial positive power series proof properties Prove rational numbers real number system Riemann integrable satisfies sequence of rationals sequence x1 shown in Figure solution subintervals subset Suppose triangle inequality uniformly unique upper bound vector zero レー