The Theory of Calculus To order this title, and for more information, click here Second Edition
By John Fridy, Kent State University, Ohio, U.S.A.
Description Introductory Analysis, Second Edition, is intended for the standard course on calculus limit theories that is taken after a problem
solving first course in calculus (most often by junior/senior mathematics majors). Topics studied include sequences, function limits,
derivatives, integrals, series, metric spaces, and calculus in n-dimensional Euclidean space
Audience
Researchers, professionals, the general public, and librarians who want to expand or enhance their knowledge of calculus limit theories.
Contents Introduction: Mathematical Statements and Proofs * Types of Mathematical Statements * The Structure of Proofs Ordering of the Real Numbers
* The Order Axiom * Least Upper Bounds * The Density of the Rational Numbers * Sequence Limits * Convergent Sequences * Algebraic Combinations
of Sequences * Infinite Limits * Subsequences and Limit Points * Monotonic Sequences * Completeness of the Real Numbers * The Bolzano--Weierstrass
Theorem * Cauchy Sequences * The Nested Intervals Theorem * The Heine--Borel Covering Theorem * Continuous Functions * Continuity * The
Sequential Criterion for Continuity * Combinations of Continuous Functions * One-Sided Continuity * Function Limits * The Sequential
Criterion for Function Limits * Variations of Function Limits * Consequences of Continuity * The Range of a Continuous Function * The
Intermediate Value Property * Uniform Continuity * The Sequential Criterion for Uniform * Continuity * The Derivative * Difference Quotients
* The Chain Rule * The Law of the Mean * Cauchy Law of the Mean * Taylor's Formula with Remainder * L'Hopital's Rule * The Riemann Integral
* Riemann Sums and Integrable Functions * Basic Properties * The Darboux Criterion for Integrability * Integrability of Continuous Functions
* Products of Integrable Functions * The Fundamental Theorem of Calculus * Improper Integrals * Types of Improper Integrals * Integrals
over Unbounded Domains * Integrals of Unbounded Functions * The Gamma Function * The Laplace Transform * Infinite Series * Convergent
and Divergent Series * Comparison Tests * The Cauchy Condensation Test * Elementary Tests * Delicate Tests * Absolute and Conditional
Convergence * Regrouping and Rearranging Series * Multiplication of Series * The Riemann--Stieltjes Integral * Functions of Bounded Variation
* The Total Variation Function * Riemann--Stieltjes Sums and Integrals * Integration by Parts * Integrability of Continuous Functions
Function Sequences * Pointwise Convergence * Uniform Convergence * Sequences of Continuous Functions * Sequences of Integrable Functions
* Sequences of Differentiable Functions * The Weierstrass Approximation Theorem * Function Series Power Series * Convergence of Power
Series * Integration and Differentiation of Power Series * Taylor Series * The Remainder Term * Taylor Series of Some Elementary Functions
* Metric Spaces and Euclidean Spaces * Metric Spaces * Euclidean n-Space * Metric Space Topology * Connectedness * Point Sequences *
Completeness of En * Dense Subsets of En * Continuous Transformations * Transformations and Functions * Criteria for Continuity * The
Range of a Continuous Transformation * Continuity in En * Linear Transformations * Differential Calculus in Euclidean Spaces * Partrial
Derivatives and Directional * Derivatives * Differentials and the Approximation Property * The Chain Rule * The Law of the Mean * Mixed
Partial Derivatives * The Implicit Function Theorem * Area and Integration in E2 * Integration on a Bounded Set * Inner and Outer Area
* Properties of the Double Integral * Line Integrals * Independence of Path and Exact Differentials * Green's Theorem * Analogs of Green's
Theorem * Appendix A Mathematical Induction * Appendix B Countable and Uncountable Sets * Appendix C Infinite Products * Appendix D List
of Mathematicians * Appendix E Glossary of Symbols * Index *
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