Mathematical analysis I

Description

1 Basic notions 1 -- 1.1 Sets 1 -- 1.2 Elements of mathematical logic 5 -- 1.2.1 Connectives 5 -- 1.2.2 Predicates 6 -- 1.2.3 Quantifiers 7 -- 1.3 Sets of numbers 8 -- 1.3.1 The ordering of real numbers 12 -- 1.3.2 Completeness of R 17 -- 1.4 Factorials and binomial coefficients 18 -- 1.5 Cartesian product 21 -- 1.6 Relations in the plane 23 -- 2 Functions 31 -- 2.1 Definitions and first examples 31 -- 2.2 Range and pre-image 36 -- 2.3 Surjective and injective functions inverse function 38 -- 2.4 Monotone functions 41 -- 2.5 Composition of functions 43 -- 2.5.1 Translations, rescalings, reflections 45 -- 2.6 Elementary functions and properties 47 -- 2.6.1 Powers 48 -- 2.6.2 Polynomial and rational functions 50 -- 2.6.3 Exponential and logarithmic functions 50 -- 2.6.4 Trigonometric functions and inverses 51 -- 3 Limits and continuity I 65 -- 3.1 Neighbourhoods 65 -- 3.2 Limit of a sequence 66 -- 3.3 Limits of functions continuity 72 -- 3.3.1 Limits at infinity 72 -- 3.3.2 Continuity. Limits at real points 74 -- 3.3.3 One-sided limits points of discontinuity 82 -- 3.3.4 Limits of monotone functions 85 -- 4 Limits and continuity II 89 -- 4.1 Theorems on limits 89 -- 4.1.1 Uniqueness and sign of the limit 89 -- 4.1.2 Comparison theorems 91 -- 4.1.3 Algebra of limits. Indeterminate forms of algebraic type 96 -- 4.1.4 Substitution theorem 102 -- 4.2 More fundamental limits. Indeterminate forms of exponential type 105 -- 4.3 Global features of continuous maps 108 -- 5 Local comparison of functions. Numerical sequences and series 123 -- 5.1 Landau symbols 123 -- 5.2 Infinitesimal and infinite functions 130 -- 5.3 Asymptotes 135 -- 5.4 Further properties of sequences 137 -- 5.5 Numerical series 141 -- 5.5.1 Positive-term series 146 -- 5.5.2 Alternating series 149 -- 6 Differential calculus 167 -- 6.1 The derivative 167 -- 6.2 Derivatives of the elementary functions. Rules of differentiation 170 -- 6.3 Where differentiability fails 175 -- 6.4 Extrema and critical points 178 -- 6.5 Theorems of Rolle and of the Mean Value 181 -- 6.6 First and second finite increment formulas 183 -- 6.7 Monotone maps 185 -- 6.8 Higher-order derivatives 187 -- 6.9 Convexity and inflection points 189 -- 6.9.1 Extension of the notion of convexity 192 -- 6.10 Qualitative study of a function 193 -- 6.10.1 Hyperbolic functions 195 -- 6.11 The Theorem of de l'Hopital 197 -- 6.11.1 Applications of de l'Hopital's theorem 199 -- 7 Taylor expansions and applications 223 -- 7.1 Taylor formulas 223 -- 7.2 Expanding the elementary functions 227 -- 7.3 Operations on Taylor expansions 234 -- 7.4 Local behaviour of a map via its Taylor expansion 242 -- 8 Geometry in the plane and in space 257 -- 8.1 Polar, cylindrical, and spherical coordinates 257 -- 8.2 Vectors in the plane and in space 260 -- 8.2.1 Position vectors 260 -- 8.2.2 Norm and scalar product 263 -- 8.2.3 General vectors 268 -- 8.3 Complex numbers 269 -- 8.3.1 Algebraic operations 270 -- 8.3.2 Cartesian coordinates 271 -- 8.3.3 Trigonometric and exponential form 273 -- 8.3.4 Powers and nth roots 275 -- 8.3.5 Algebraic equations 277 -- 8.4 Curves in the plane and in space 279 -- 8.5 Functions of several variables 284 -- 8.5.1 Continuity 284 -- 8.5.2 Partial derivatives and gradient 286 -- 9 Integral calculus I 299 -- 9.1 Primitive functions and indefinite integrals 300 -- 9.2 Rules of indefinite integration 304 -- 9.2.1 Integrating rational maps 310 -- 9.3 Definite integrals 317 -- 9.4 The Cauchy integral 318 -- 9.5 The Riemann integral 320 -- 9.6 Properties of definite integrals 326 -- 9.7 Integral mean value 328 -- 9.8 The Fundamental Theorem of integral calculus 331 -- 9.9 Rules of definite integration 335 -- 9.9.1 Application: computation of areas 337 -- 10 Integral calculus II 355 -- 10.1 Improper integrals 355 -- 10.1.1 Unbounded domains of integration 355 -- 10.1.2 Unbounded integrands 363 -- 10.2 More improper integrals 367 -- 10.3 Integrals along curves 368 -- 10.3.1 Length of a curve and arc length 373 -- 10.4 Integral vector calculus 376 -- 11 Ordinary differential equations 387 -- 11.1 General definitions 387 -- 11.2 First order differential equations 388 -- 11.2.1 Equations with separable variables 392 -- 11.2.2 Linear equations 394 -- 11.2.3 Homogeneous equations 397 -- 11.2.4 Second order equations reducible to first order 398 -- 11.3 Initial value problems for equations of the first order 399 -- 11.3.1 Lipschitz functions 399 -- 11.3.2 A criterion for solving initial value problems 402 -- 11.4 Linear second order equations with constant coefficients 404.