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. . . . . . . . . . . . . . . . .8.4.1 The Mean-Value Theorem . . . . . . . . . . . . . . . . .8.4.2 A Sufficient Condition for Differentiability of a Functionof Several Variables . . . . . . . . . . . . . . . . . . . .8.4.3 Higher-Order Partial Derivatives . . . . . . . . . . . . .8.4.4 Taylor’s Formula . . .
. . . . . . . . . . . . . . . . . . .8.4.5 Extrema of Functions of Several Variables . . . . . . . .8.4.6 Some Geometric Images Connected with Functionsof Several Variables . . . . . . . . . . . . . . . . . . . .427427427428429431432432433436437438438441446448454454456457461462469xviiiContents8.4.7 Problems and Exercises . . .
. . . . . . . . . . . . . . .8.5 The Implicit Function Theorem . . . . . . . . . . . . . . . . . .8.5.1 Statement of the Problem and Preliminary Considerations8.5.2 An Elementary Version of the Implicit Function Theorem8.5.3 Transition to the Case of a Relation F (x 1 , . . . , x m , y) = 08.5.4 The Implicit Function Theorem . .
. . . . . . . . . . . .8.5.5 Problems and Exercises . . . . . . . . . . . . . . . . . .8.6 Some Corollaries of the Implicit Function Theorem . . . . . . .8.6.1 The Inverse Function Theorem . . . . . . . . . . . . . .8.6.2 Local Reduction of a Smooth Mapping to Canonical Form8.6.3 Functional Dependence . . . . . . . . . . . .
. . . . . .8.6.4 Local Resolution of a Diffeomorphism intoa Composition of Elementary Ones . . . . . . . . . . . .8.6.5 Morse’s Lemma . . . . . . . . . . . . . . . . . . . . . .8.6.6 Problems and Exercises . . . . . . . . . . . . . . . . . .8.7 Surfaces in Rn and the Theory of Extrema with Constraint . . .
.8.7.1 k-Dimensional Surfaces in Rn . . . . . . . . . . . . . . .8.7.2 The Tangent Space . . . . . . . . . . . . . . . . . . . . .8.7.3 Extrema with Constraint . . . . . . . . . . . . . . . . . .8.7.4 Problems and Exercises . . . . . . . . . . . . . . . . . .Some Problems from the Midterm Examinations . . . . . . .1Introduction to Analysis (Numbers, Functions, Limits)2One-Variable Differential Calculus . . .
. . . . . . .3Integration and Introduction to Several Variables . . .4Differential Calculus of Several Variables . . . . . . .....................Examination Topics . . . . . . . . . . . . . . . . . . . . . . . . . . .1First Semester . . . . . . . . . .
. . . . . . . . . . . . . . .1.1Introduction to Analysis and One-Variable DifferentialCalculus . . . . . . . . . . . . . . . . . . . . . . . .2Second Semester . . . . . . . . . . . . . . . . . . . . . . . .2.1Integration. Multivariable Differential Calculus . . . ......474480480482486489494498498503508509512515517517521526539.....545545546549550. ..
.555555. .. .. .555557557Appendix A Mathematical Analysis (Introductory Lecture) . . . . .A.1 Two Words About Mathematics . . . . . . . . . . . . . . . . .A.2 Number, Function, Law . . . . . . . . . . . . . . . . . . . . .A.3 Mathematical Model of a Phenomenon (Differential Equations,or We Learn How to Write) . . . . . . . . .
. . . . . . . . . .A.4 Velocity, Derivative, Differentiation . . . . . . . . . . . . . . .A.5 Higher Derivatives, What for? . . . . . . . . . . . . . . . . . .A.5.1 Again Toward Numbers . . . . . . . . . . . . . . . . .A.5.2 And What to Do Next? . . . . . . . . . . . . . . . . ....559559560.....561563565566567ContentsxixAppendix B Numerical Methods for Solving Equations(An Introduction) . .
. . . . . . . . . . . . . . . . . . .B.1 Roots of Equations and Fixed Points of Mappings .B.2 Contraction Mappings and Iterative Process . . . .B.3 The Method of Tangents (Newton’s Method) . . . .........569569569570Appendix C The Legendre Transform (First Discussion) . . . . . . .C.1 Initial Definition of the Legendre Transform and the GeneralYoung Inequality . . . . . . . . . . .
. . . . . . . . . . . . . .C.2 Specification of the Definition in the Case of Convex FunctionsC.3 Involutivity of the Legendre Transform of a Function . . . . . .C.4 Concluding Remarks and Comments . . . . . . . . . . . . . ..573....573574574575Appendix D The Euler–MacLaurin Formula . . . . .D.1 Bernoulli Numbers . . . . . . . . . . . . . . .
.D.2 Bernoulli Polynomials . . . . . . . . . . . . . .D.3 Some Known Operators and Series of OperatorsD.4 Euler–MacLaurin Series and Formula . . . . . .D.5 The General Euler–MacLaurin Formula . . . . .D.6 Applications . . . . . . . . . . . . . . . . . . .D.7 Again to the Actual Euler–MacLaurin Formula .........577577577578578579579580. .. .583583....................................................................................Appendix E Riemann–Stieltjes Integral, Delta Function,and the Concept of Generalized Functions . .
. . . . . . . . . .E.1 The Riemann–Stieltjes Integral . . . . . . . . . . . . . . . .E.2 Case in Which the Riemann–Stieltjes Integral Reducesto the Riemann Integral . . . . . . . . . . . . . . . . . . . .E.3 Heaviside Function and an Example of a Riemann–StieltjesIntegral Computation . . . . . . . . . . . .
. . . . . . . . .E.4 Generalized Functions . . . . . . . . . . . . . . . . . . . . .E.4.1 Dirac’s Delta Function. A Heuristic Description . . .E.5 The Correspondence Between Functions and Functionals . .E.6 Functionals as Generalized Functions . . . . . . . . .
. . . .E.7 Differentiation of Generalized Functions . . . . . . . . . . .E.8 Derivatives of the Heaviside Function and the Delta Function. .585..............586586586587588589589Appendix F The Implicit Function Theorem (An AlternativePresentation) . . . . . . . . . . . . . . . . . . . . . . . . . . .F.1 Formulation of the Problem . . . . .
. . . . . . . . . . . .F.2 Some Reminders of Numerical Methods to Solve EquationsF.2.1 The Principle of the Fixed Point . . . . . . . . . . .F.3 The Implicit Function Theorem . . . . . . . . . . . . . . .F.3.1 Statement of the Theorem . . . . . . . . . . . . .
.F.3.2 Proof of the Existence of an Implicit Function . . .F.3.3 Continuity of an Implicit Function . . . . . . . . .F.3.4 Differentiability of an Implicit Function . . . . . .F.3.5 Continuous Differentiability of an Implicit FunctionF.3.6 Higher Derivatives of an Implicit Function .
. . . .......................591591591593595595596597597598599...........xxReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Classic Works . . . . . . . . . . . . . . . . . . . . . . .1.1Primary Sources . . . . . . . . . . . . . . . . . .1.2Major Comprehensive Expository Works . . . . .1.3Classical Courses of Analysis from the First Halfof the Twentieth Century . . . . .
. . . . . . . .2Textbooks . . . . . . . . . . . . . . . . . . . . . . . . .3Classroom Materials . . . . . . . . . . . . . . . . . . . .4Further Reading . . . . . . . . . . . . . . . . . . . . . .Contents................601601601601................602602602603Subject Index . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .605Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .615Chapter 1Some General Mathematical Conceptsand Notation1.1 Logical Symbolism1.1.1 Connectives and BracketsThe language of this book, like the majority of mathematical texts, consists of ordinary language and a number of special symbols from the theories being discussed.Along with the special symbols, which will be introduced as needed, we use thecommon symbols of mathematical logic ¬, ∧, ∨, ⇒, and ⇔ to denote respectivelynegation (not) and the logical connectives and, or, implies, and is equivalent to.1For example, take three statements of independent interest:L.
If the notation is adapted to the discoveries. . . , the work of thought is marvelously shortened. (G. Leibniz)2P. Mathematics is the art of calling different things by the same name. (H. Poincaré).3G. The great book of nature is written in the language of mathematics. (Galileo).4Then, according to the notation given above, Table 1.1 relates L, P , G.1 The symbol & is often used in logic in place of ∧.
Logicians more often write the implicationsymbol ⇒ as → and the relation of logical equivalence as ←→ or ↔. However, we shall adhereto the symbolism indicated in the text so as not to overburden the symbol →, which has beentraditionally used in mathematics to denote passage to the limit.2 G.W.
Leibniz (1646–1716) – outstanding German scholar, philosopher, and mathematician towhom belongs the honor, along with Newton, of having discovered the foundations of the infinitesimal calculus.3 H. Poincaré (1854–1912) – French mathematician whose brilliant mind transformed many areasof mathematics and achieved fundamental applications of it in mathematical physics.4 Galileo Galilei (1564–1642) – Italian scholar and outstanding scientific experimenter. His workslie at the foundation of the subsequent physical concepts of space and time. He is the father ofmodern physical science.© Springer-Verlag Berlin Heidelberg 2015V.A. Zorich, Mathematical Analysis I, Universitext,DOI 10.1007/978-3-662-48792-1_1121 Some General Mathematical Concepts and NotationTable 1.1NotationMeaningL⇒PL implies PL⇔PL is equivalent to P((L ⇒ P ) ∧ (¬P )) ⇒ (¬L)If P follows from L and P is false, then L is false¬((L ⇔ G) ∨ (P ⇔ G))G is not equivalent either to L or to PWe see that it is not always reasonable to use only formal notation, avoidingcolloquial language.We remark further that parentheses are used in the writing of complex statementscomposed of simpler ones, fulfilling the same syntactical function as in algebraicexpressions.
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