数理解析学1:1変数関数<br>Mathematical Analysis, Functions of One Variable (2003. XII, 353 p. 24 cm)

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数理解析学1:1変数関数
Mathematical Analysis, Functions of One Variable (2003. XII, 353 p. 24 cm)

  • ウェブストア価格 ¥17,222(本体¥15,657)
  • BIRKHÄUSER(2003発売)
  • 外貨定価 EUR 106.95
  • Kinokuniya Point Card 誕生10周年記念キャンペーン
  • ポイント 312pt
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  • ウェブストア価格 ¥15,969(本体¥14,518)
  • BIRKHÄUSER(2003発売)
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  • Kinokuniya Point Card 誕生10周年記念キャンペーン
  • ポイント 290pt
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  • 製本 Hardcover:ハードカバー版/ページ数 350 p.
  • 商品コード 9780817643126

基本説明

Interesting and valuable historical account of ideas and methods in analysis with beautiful illustrations.

Table of Contents

Preface                                            v
Numbers, Functions and their Graphs 1 (64)
Real Numbers: a Description 1 (8)
Algebraic operations and order 2 (1)
Continuity property 2 (2)
Absolute value 4 (1)
Intervals 4 (1)
Distance on the line 5 (1)
Square root and second order equations 6 (1)
Rational numbers 7 (1)
n-th roots 8 (1)
The Cartesian Plane 9 (21)
References in the plane 9 (1)
References on the straight line 9 (1)
Cartesian references in the plane 10 (2)
Translations 12 (1)
Change of frame 12 (1)
Symmetries 13 (1)
Rotations 14 (1)
Relations, graphs and elementary 14 (1)
geometry
Cartesian plane 15 (2)
The inner product and orthogonal vectors 17 (1)
The space 18 (1)
Straight lines and conics 19 (1)
Circles 19 (1)
The parametric equation of a straight 20 (1)
line
The implicit equation of a straight line 21 (1)
Parabolas 22 (1)
Polynomials of degree 2 23 (1)
Parabolas in polar coordinates 23 (1)
Ellipses 24 (1)
Eccentricity 25 (1)
Ellipse in polar coordinates 25 (1)
Hyperbolas 26 (1)
Asymptotes 27 (1)
Hyperbolas in polar coordinates 28 (1)
Rectangular hyperbolas 29 (1)
Elementary Functions 30 (20)
The graph of a function 30 (1)
Surjectivity and injectivity 31 (1)
Real valued functions of one real 32 (2)
variable
Odd and even functions 34 (1)
Trigonometric functions 35 (1)
Sine and cosine 35 (5)
Solving a triangle 40 (1)
Tangent 40 (1)
Cotangent 41 (1)
Operating with functions 41 (1)
Composition 41 (1)
Inverse function 42 (1)
Inverse of composition 42 (1)
Graph of the inverse function 43 (1)
Inverse trigonometric functions 43 (1)
The function are sine 43 (2)
The function are cosine 45 (1)
The function arc tangent 45 (1)
The function arc cotangent 46 (1)
Exponentials and logarithms 46 (1)
Exponential functions 46 (1)
Logarithms 47 (1)
Logarithmic coordinates 48 (2)
Remarks on Common Language and the 50 (8)
Language of Mathematics
Or, and, the, a, an 50 (1)
Constants 51 (1)
Variables 51 (1)
Quantifiers 52 (1)
Propositions and predicates 52 (1)
Negation and quantifiers 53 (1)
Propositional calculus 53 (1)
Implication and proof 54 (1)
Necessary and sufficient conditions 55 (1)
Proof by contradiction 55 (1)
Sets: some terminology 56 (1)
How to define sets 56 (1)
Complement, union, and intersection 57 (1)
Cartesian products 58 (1)
Exercises 58 (7)
Limits and Continuity 65 (28)
Limits 65 (13)
Definitions 66 (1)
Finite limits 66 (3)
Infinite limits and limits at infinity 69 (1)
Basic properties of limits 70 (1)
Uniqueness, locality, and junction 70 (2)
Information from the limit to the 72 (1)
function
The calculus of limits 73 (3)
Limits of monotone functions 76 (2)
Continuous Functions 78 (2)
Continuous functions on an interval 80 (7)
The intermediate value theorem 80 (2)
Continuity of the inverse function 82 (2)
Continuity of elementary functions 84 (3)
Weierstrass's Theorem 87 (3)
Summing Up 90 (1)
Exercises 91 (2)
The Fundamental Ideas of the Differential 93 (52)
and Integral Calculus
Differential Calculus 93 (18)
The derivative 93 (2)
Geometric meaning of the derivative 95 (1)
The tangent to the graph 96 (2)
Nondifferentiable functions 98 (1)
Rates of change and kinematics 99 (1)
Logarithmic derivative 100(1)
Elasticity 101(1)
The differential 101(2)
Fermat and Lagrange theorems 103(1)
Fermat's theorem 103(3)
Lagrange's theorem 106(2)
Constant, increasing and decreasing 108(3)
functions
Integral Calculus 111(12)
Riemann's integral 111(1)
The sum of the total effect of a process 111(1)
The method of exhaustion of Eudoxus and 112(2)
Archimedes revisited
Riemann's integral 114(3)
Properties of the integral 117(3)
Additivity of the integral with respect 120(1)
to the domain
The oriented integral 121(1)
Classes of Riemann integrable functions 121(1)
Monotonic functions 122(1)
Continuous functions 122(1)
The Fundamental Theorem of Calculus 123(8)
The mean value theorem of the integral 126(1)
calculus
Primitives 127(2)
Remarks on Riemann's theorem and the 129(2)
fundamental theorem of calculus
Calculus: Some Historical Remarks 131(8)
The two problems 132(4)
The problem of foundations 136(3)
Summing Up 139(1)
Exercises 140(5)
The Calculus of Derivatives and of Integrals 145(62)
Computation of Derivatives 145(14)
Differentiation of elementary functions 145(4)
Rules for differentiation 149(1)
Derivative of sums, products and 149(1)
quotients
Differentiation of compositions: the 149(5)
chain rule
Derivative of the inverse function 154(2)
Higher order derivatives 156(2)
Higher derivatives of the inverse 158(1)
function
Integrals and Primitives 159(11)
Definite integrals 159(1)
Integration by parts 159(1)
Integration by substitution 160(1)
Primitives 161(1)
A table of indefinite integrals 162(1)
Integration by parts 162(2)
Integration by substitution 164(1)
Integration of rational functions 165(3)
Integration by substitution and 168(2)
rationalization
A Definition of the Trigonometric, 170(5)
Logarithmic and Exponential Functions
Trigonometric functions 170(3)
Logarithms and exponentials 173(2)
Some Differential Equations 175(10)
Prescribing the derivative 176(1)
Uniform linear motion 177(1)
Exponential growth and decay 177(2)
Newtonian cooling 179(1)
Power laws 179(1)
Logistic equation 180(1)
Simple harmonic motion 181(3)
The equation y'' -- ω2y = 0 184(1)
Epidemic models 184(1)
Generalized Integrals 185(11)
Measurable and integrable functions 185(1)
Truncations 185(1)
Measurable functions 186(1)
Integrable and summable functions 187(1)
Properties of the generalized integral 188(3)
Summable functions 191(1)
Criteria for summability 191(3)
The improper integral 194(2)
Summing Up 196(4)
Exercises 200(7)
Further Developments in Calculus 207(54)
Taylor's Formula 207(6)
The Calculus of Limits 213(12)
Taylors' formula with Peano's remainder 213(1)
Landau's notation &fnot; = o(g), &fnot; 214(2)
= O(g), &fnot; ‾ g
The calculus of Taylor's expansions 216(2)
Asymptotic expansions 218(3)
de I'Hopital's theorems 221(3)
Computing limits 224(1)
Convex Functions 225(5)
Some Inequalities 230(13)
A few elementary inequalities 230(2)
Cauchy inequalities 232(1)
Young inequality 233(2)
Jensen's inequality 235(1)
Discrete Jensen inequality 236(1)
Some consequences of Jensen's inequality 237(1)
Entropy 238(1)
Mean values 239(4)
Graphing a Function 243(4)
Summing Up 247(5)
Exercises 252(9)
Toward Differential Equations and Minimum 261(80)
Principles
Linear Ordinary Differential Equations 261(16)
A few elementary facts and definitions 263(2)
Linear first order differential 265(1)
equations
The homogeneous case 265(1)
The nonhomogeneous case 266(2)
Second order linear ODEs with constant 268(1)
coefficients
The homogeneous case 268(2)
The nonhomogeneous case 270(1)
Linear vibrations 271(1)
Free oscillations 271(2)
Forced oscillations 273(3)
Electrical oscillations 276(1)
First Order ODEs 277(8)
Cauchy problem: existence and uniqueness 278(2)
Separation of variables 280(4)
Qualitative methods 284(1)
One-Dimensional Motions 285(16)
Trajectory and motion 286(1)
Conservation of energy 287(1)
The pendulum 288(2)
Uniform circular motion 290(2)
Central motions 292(4)
Trajectory of motion 296(1)
Elastic field 297(1)
Gravitational field and Kepler's laws 298(2)
Planetary law of motion 300(1)
Optimization Problems 301(33)
Minimum principles 303(1)
Triangles 303(1)
Heron's principle 304(1)
Fermat's principle 305(1)
Direct and indirect methods 306(3)
Focal properties of conics 309(3)
Parabola 312(1)
Ellipse 313(1)
Hyperbola 314(1)
Elementary isoperimetric problems 315(2)
The isoperimetric triangle 317(1)
Isoperimetric polygons 318(2)
The isoperimetric property of the circle 320(2)
Polyhedra in space 322(3)
Minimal connections 325(1)
Steiner's problem 325(4)
Steiner's networks 329(1)
Minimal connections on graphs 330(4)
Summing Up 334(3)
Exercises 337(4)
Matematicians and Other Scientists 341(2)
Bibliographical Notes 343(4)
Index 347