Elementary Analysis: The Theory of CalculusSpringer Science & Business Media, 17/04/2013 - 264 من الصفحات Designed for students having no previous experience with rigorous proofs, this text on analysis can be used immediately following standard calculus courses. It is highly recommended for anyone planning to study advanced analysis, e.g., complex variables, differential equations, Fourier analysis, numerical analysis, several variable calculus, and statistics. It is also recommended for future secondary school teachers. A limited number of concepts involving the real line and functions on the real line are studied. Many abstract ideas, such as metric spaces and ordered systems, are avoided. The least upper bound property is taken as an axiom and the order properties of the real line are exploited throughout. A thorough treatment of sequences of numbers is used as a basis for studying standard calculus topics. Optional sections invite students to study such topics as metric spaces and Riemann-Stieltjes integrals. |
المحتوى
3 The Set ℝ of Real Numbers | |
4 The Completeness Axiom | |
5 The Symbols + and 6 A Development of | |
ChapterII Sequences 7 Limits of Sequences | |
8 A Discussion about Proofs | |
9 Limit Theorems for Sequences | |
19 Uniform Continuity | |
20 Limits of Functions | |
Continuity | |
Connectedness | |
Sequences and Series of Functions | |
23 Power Series | |
24 Uniform Convergence 25 More on Uniform Convergence | |
26 Differentiation and Integration of Power Series | |
10 Monotone Sequences and Cauchy Sequences | |
11 Subsequences | |
12 lim sups and lim infs 13 Some Topological Concepts in Metric Spaces | |
14 Series | |
15 Alternating Series and Integral Tests | |
16 Decimal Expansions of Real Numbers | |
Continuity 17 Continuous Functions | |
18 Properties of Continuous Functions | |
27 Weierstrasss Approximation Theorem | |
ChapterV Differentiation 28 Basic Properties of the Derivative | |
29 The Mean Value Theorem | |
30 LHospitals Rule | |
31 Taylors Theorem | |
Integration 32 The Riemann Integral 33 Properties of the Riemann Integral | |
طبعات أخرى - عرض جميع المقتطفات
Elementary Analysis: The Theory of Calculus <span dir=ltr>Kenneth A. Ross</span> معاينة محدودة - 1980 |
Elementary Analysis: The Theory of Calculus <span dir=ltr>Kenneth A. Ross</span> لا تتوفر معاينة - 2010 |
عبارات ومصطلحات مألوفة
algebraic apply Theorem assertion bounded function calculate Cauchy sequence completeness axiom Consider continuous function continuous on ℝ converges uniformly Corollary decimal expansion Dedekind cuts Definition Discussion diverges dom(f dom(g Example f and g f is continuous f is integrable f uniformly finite Fintegrable follows function defined function f Hence Hint holds implies infinite L’Hospital’s rule Lemma Let f Let f(x lim inf lim s n lim sup mathematical induction Mean Value Theorem metric space natural numbers nonnegative notation Note obtain open interval open sets partial sums partition pointwise polynomials power series proof of Theorem properties radius of convergence Ratio Test rational numbers real numbers Repeat Exercise Riemann integral ROOF Root Test satisfies series converges Show that f show that lim strictly increasing subsequence subsequential limits subset of ℝ Theorems 9.3 triangle inequality uniformly continuous upper bound