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Author Browder, Andrew. author Mathematical Analysis [electronic resource] : An Introduction / by Andrew Browder New York, NY : Springer New York : Imprint: Springer, 1996 http://dx.doi.org/10.1007/978-1-4612-0715-3 XIV, 335 p. online resource

SUMMARY

This is a textbook suitable for a year-long course in analysis at the adยญ vanced undergraduate or possibly beginning-graduate level. It is intended for students with a strong background in calculus and linear algebra, and a strong motivation to learn mathematics for its own sake. At this stage of their education, such students are generally given a course in abstract algebra, and a course in analysis, which give the fundamentals of these two areas, as mathematicians today conceive them. Mathematics is now a subject splintered into many specialties and subยญ specialties, but most of it can be placed roughly into three categories: alยญ gebra, geometry, and analysis. In fact, almost all mathematics done today is a mixture of algebra, geometry and analysis, and some of the most inยญ teresting results are obtained by the application of analysis to algebra, say, or geometry to analysis, in a fresh and surprising way. What then do these categories signify? Algebra is the mathematics that arises from the ancient experiences of addition and multiplication of whole numbers; it deals with the finite and discrete. Geometry is the mathematics that grows out of spatial experience; it is concerned with shape and form, and with measurยญ ing, where algebra deals with counting

CONTENT

1 Real Numbers -- 1.1 Sets, Relations, Functions -- 1.2 Numbers -- 1.3 Infinite Sets -- 1.4 Incommensurability -- 1.5 Ordered Fields -- 1.6 Functions on R -- 1.7 Intervals in R -- 1.8 Algebraic and Transcendental Numbers -- 1.9 Existence of R -- 1.10 Exercises -- 1.11 Notes -- 2 Sequences and Series -- 2.1 Sequences -- 2.2 Continued Fractions -- 2.3 Infinite Series -- 2.4 Rearrangements of Series -- 2.5 Unordered Series -- 2.6 Exercises -- 2.7 Notes -- 3 Continuous Functions on Intervals -- 3.1 Limits and Continuity -- 3.2 Two Fundamental Theorems -- 3.3 Uniform Continuity -- 3.4 Sequences of Functions -- 3.5 The Exponential function -- 3.6 Trigonometric Functions -- 3.7 Exercises -- 3.8 Notes -- 4 Differentiation -- 4.1 Derivatives -- 4.2 Derivatives of Some Elementary Functions -- 4.3 Convex Functions -- 4.4 The Differential Calculus -- 4.5 Lโ{128}{153}Hospitalโ{128}{153}s Rule -- 4.6 Higher Order Derivatives -- 4.7 Analytic Functions -- 4.8 Exercises -- 4.9 Notes -- 5 The Riemann Integral -- 5.1 Riemann Sums -- 5.2 Existence Results -- 5.3 Properties of the Integral -- 5.4 Fundamental Theorems of Calculus -- 5.5 Integrating Sequences and Series -- 5.6 Improper Integrals -- 5.7 Exercises -- 5.8 Notes -- 6 Topology -- 6.1 Topological Spaces -- 6.2 Continuous Mappings -- 6.3 Metric Spaces -- 6.4 Constructing Topological Spaces -- 6.5 Sequences -- 6.6 Compactness -- 6.7 Connectedness -- 6.8 Exercises -- 6.9 Notes -- 7 Function Spaces -- 7.1 The Weierstrass Polynomial Approximation Theorem . . . -- 7.2 Lengths of Paths -- 7.3 Fourier Series -- 7.4 Weylโ{128}{153}s Theorem -- 7.5 Exercises -- 7.6 Notes -- 8 Differentiable Maps -- 8.1 Linear Algebra -- 8.2 Differentials -- 8.3 The Mean Value Theorem -- 8.4 Partial Derivatives -- 8.5 Inverse and Implicit Functions -- 8.6 Exercises -- 8.7 Notes -- 9 Measures -- 9.1 Additive Set Functions -- 9.2 Countable Additivity -- 9.3 Outer Measures -- 9.4 Constructing Measures -- 9.5 Metric Outer Measures -- 9.6 Measurable Sets -- 9.7 Exercises -- 9.8 Notes -- 10 Integration -- 10.1 Measurable Functions -- 10.2 Integration -- 10.3 Lebesgue and Riemann Integrals -- 10.4 Inequalities for Integrals -- 10.5 Uniqueness Theorems -- 10.6 Linear Transformations -- 10.7 Smooth Transformations -- 10.8 Multiple and Repeated Integrals -- 10.9 Exercises -- 10.10 Notes -- 11 Manifolds -- 11.1 Definitions -- 11.2 Constructing Manifolds -- 11.3 Tangent Spaces -- 11.4 Orientation -- 11.5 Exercises -- 11.6 Notes -- 12 Multilinear Algebra -- 12.1 Vectors and Tensors -- 12.2 Alternating Tensors -- 12.3 The Exterior Product -- 12.4 Change of Coordinates -- 12.5 Exercises -- 12.6 Notes -- 13 Differential Forms -- 13.1 Tensor Fields -- 13.2 The Calculus of Forms -- 13.3 Forms and Vector Fields -- 13.4 Induced Mappings -- 13.5 Closed and Exact Forms -- 13.6 Tensor Fields on Manifolds -- 13.7 Integration of Forms in Rn -- 13.8 Exercises -- 13.9 Notes -- 14 Integration on Manifolds -- 14.1 Partitions of Unity -- 14.2 Integrating k-Forms -- 14.3 The Brouwer Fixed Point Theorem -- 14.4 Integrating Functions on a Manifold -- 14.5 Vector Analysis -- 14.6 Harmonic Functions -- 14.7 Exercises -- 14.8 Notes -- References

Mathematics Mathematical analysis Analysis (Mathematics) Functions of real variables Manifolds (Mathematics) Complex manifolds Mathematics Analysis Real Functions Manifolds and Cell Complexes (incl. Diff.Topology)

Location

Office of Academic Resources, Chulalongkorn University, Phayathai Rd. Pathumwan Bangkok 10330 Thailand