Advanced calculus

목차 (Table of Contents)

• CONTENTS
• 1／FUNDAMENTALS OF ELEMENTARY CALCULUS
• 1.0 Introduction = 1
• 1.1 Functions = 2
• 1.1.1 Derivatives = 12
• 1.1.2 Maxima and Minima = 20
• 1.2 The Law of the Mean (The Mean-Value Theorem for Derivatives) = 26
• 1.3 Differentials = 32
• 1.4 The Inverse of Differentiation = 35
• 1.5 Definite Integrals = 38
• 1.5.1 The Mean-Value Theorem for Integrals = 45
• 1.5.2 Variable Limits of Integration = 46
• 1.5.3 The Integral of a Derivative = 49
• 1.6 Limits = 53
• 1.6.1 Limits of Functions of a Continuous Variable = 54
• 1.6.2 Limits of Sequences = 58
• 1.6.3 The Limit Defining a Definite Integral = 67
• 1.6.4 The Theorem on Limits of Sums, Products, and Quotients = 67
• 2／THE REAL NUMBER SYSTEM
• 2.0 Numbers = 72
• 2.1 The Field of Real Numbers = 72
• 2.2 Inequalities. Absolute Value = 74
• 2.3 The Principle of Mathematical Induction = 75
• 2.4 The Axiom of Continuity = 77
• 2.5 Rational and Irrational Numbers = 78
• 2.6 The Axis of Reals = 79
• 2.7 Least Upper Bounds = 80
• 2.8 Nested Intervals = 82
• 3／CONTINUOUS FUNCTIONS
• 3.0 Continuity = 85
• 3.1 Bounded Functions = 86
• 3.2 The Attainment of Extreme Values = 88
• 3.3 The Intermediate-Value Theorem = 90
• 4／EXTENSIONS OF THE LAW OF THE MEAN
• 4.0 Introduction = 95
• 4.1 Cauchy's Generalized Law of the Mean = 95
• 4.2 Taylor's Formula with Integral Remainder = 97
• 4.3 Other Forms of the Remainder = 99
• 4.4 An Extension of the Mean-Value Theorem for Integrals = 105
• 4.5 L'Hospital's Rule = 106
• 5／FUNCTIONS OF SEVERAL VARIABLES
• 5.0 Functions and Their Regions of Definition = 116
• 5.1 Point Sets = 117
• 5.2 Limits = 122
• 5.3 Continuity = 125
• 5.4 Modes of Representing a Function = 127
• 6／THE ELEMENTS OF PARTIAL DIFFERENTIATION
• 6.0 Partial Derivatives = 130
• 6.1 Implicit Functions = 132
• 6.2 Geometrical Significance of Partial Derivatives = 135
• 6.3 Maxima and Minima = 138
• 6.4 Differentials = 144
• 6.5 Composite Functions and the Chain Rule = 154
• 6.5.1 An Application in Fluid Mechanics = 162
• 6.5.2 Second Derivatives by the Chain Rule = 164
• 6.5.3 Homogeneous Functions. Euler's Theorem = 168
• 6.6 Derivatives of Implicit Functions = 172
• 6.7 Extremal Problems with Constraints = 177
• 6.8 Lagrange's Method = 182
• 6.9 Quadratic Forms = 189
• 7／GENERAL THEOREMS OF PARTIAL DIFFERENTIATION
• 7.0 Preliminary Remarks = 196
• 7.1 Sufficient Conditions for Differentiability = 197
• 7.2 Changing the Order of Differentiation = 199
• 7.3 Differentials of Composite Functions = 202
• 7.4 The Law of the Mean = 204
• 7.5 Taylor's Formula and Series = 207
• 7.6 Sufficient Conditions for a Relative Extreme = 211
• 8／IMPLICIT-FUNCTION THEOREMS
• 8.0 The Nature of the Problem of Implicit Functions = 222
• 8.1 The Fundamental Theorem = 224
• 8.2 Generalization of the Fundamental Theorem = 227
• 8.3 Simultaneous Equations = 230
• 9／THE INVERSE FUNCTION THEOREM WITH APPLICATIONS
• 9.0 Introduction = 237
• 9.1 The Inverse Function Theorem in Two Dimensions = 241
• 9.2 Mappings = 247
• 9.3 Successive Mappings = 252
• 9.4 Transformations of Co-ordinates = 255
• 9.5 Curvilinear Co-ordinates = 258
• 9.6 Identical Vanishing of the Jacobian. Functional Dependence = 263
• 10／VECTORS AND VECTOR FIELDS
• 10.0 Purpose of the Chapter = 268
• 10.1 Vectors in Euclidean Space = 268
• 10.1.1 Orthogonal Unit Vectors in $$R^3$$ = 273
• 10.1.2 The Vector Space $$R^n$$ = 274
• 10.2 Cross Products in $$R^3$$ = 280
• 10.3 Rigid Motions of the Axes = 283
• 10.4 Invariants = 286
• 10.5 Scalar Point Functions = 291
• 10.5.1 Vector Point Functions = 293
• 10.6 The Gradient of a Scalar Field = 295
• 10.7 The Divergence of a Vector Field = 300
• 10.8 The Curl of a Vector Field = 305
• 11／LINEAR TRANSFORMATIONS
• 11.0 Introduction = 309
• 11.1 Linear Transformations = 312
• 11.2 The Vector Space L($$R^n$$, $$R^m$$) = 313
• 11.3 Matrices and Linear Transformations = 313
• 11.4 Some Special Cases = 316
• 11.5 Norms = 318
• 11.6 Metrics = 319
• 11.7 Open Sets and Continuity = 320
• 11.8 A Norm on L($$R^n$$, $$R^m$$) = 324
• 11.9 L($$R^n$$) = 327
• 11.10 The Set of Invertible Operators = 330
• 12／DIFFERENTIAL CALCULUS OF FUNCTIONS FROM $$R^n$$ TO $$R^m$$
• 12.0 Introduction = 335
• 12.1 The Differential and the Derivative = 336
• 12.2 The Component Functions and Differentiability = 340
• 12.2.1 Directional Derivatives and the Method of Steepest Descent = 343
• 12.3 Newton's Method = 347
• 12.4 A Form of the Law of the Mean for Vector Functions = 350
• 12.4.1 The Hessian and Extreme Values = 352
• 12.5 Continuously Differentiable Functions = 354
• 12.6 The Fundamental Inversion Theorem = 355
• 12.7 The Implicit Function Theorem = 361
• 12.8 Differentiation of Scalar Products of Vector Valued Functions of a Vector Variable = 366
• 13／DOUBLE AND TRIPLE INTEGRALS
• 13.0 Preliminary Remarks = 376
• 13.1 Motivations = 376
• 13.2 Definition of a Double Integral = 379
• 13.2.1 Some Properties of the Double Integral = 381
• 13.2.2 Inequalities. The Mean-Value Theorem = 382
• 13.2.3 A Fundamental Theorem = 383
• 13.3 Iterated Integrals. Centroids = 384
• 13.4 Use of Polar Co-ordinates = 390
• 13.5 Applications of Double Integrals = 395
• 13.5.1 Potentials and Force Fields = 401
• 13.6 Triple Integrals = 404
• 13.7 Applications of Triple Integrals = 409
• 13.8 Cylindrical Co-ordinates = 412
• 13.9 Spherical Co-ordinates = 413
• 14／CURVES AND SURFACES
• 14.0 Introduction = 417
• 14.1 Representations of Curves = 417
• 14.2 Arc Length = 418
• 14.3 The Tangent Vector = 421
• 14.3.1 Principal normal. Curvature = 423
• 14.3.2 Binomial. Torsion = 425
• 14.4 Surfaces = 428
• 14.5 Curves on a Surface = 433
• 14.6 Surface Area = 437
• 15／LINE AND SURFACE INTEGRALS
• 15.0 Introduction = 445
• 15.1 Point Functions on Curves and Surfaces = 445
• 15.1.2 Line Integrals = 446
• 15.1.3 Vector Functions and Line Integrals. Work = 451
• 15.2 Partial Derivatives at the Boundary of a Region = 455
• 15.3 Green's Theorem in the Plane = 457
• 15.3.1 Comments on the Proof of Green's Theorem = 463
• 15.3.2 Transformations of Double Integrals = 465
• 15.4 Exact Differentials = 469
• 15.4.1 Line Integrals Independent of the Path = 474
• 15.5 Further Discussion of Surface Area = 478
• 15.5.1 Surface Integrals = 480
• 15.6 The Divergence Theorem = 484
• 15.6.1 Green's Identities = 492
• 15.6.2 Transformation of Triple Integrals = 494
• 15.7 Stokes's Theorem = 499
• 15.8 Exact Differentials in Three Variables = 505
• 16／POINT-SET THEORY
• 16.0 Preliminary Remarks = 512
• 16.1 Finite and Infinite Sets = 512
• 16.2 Point Sets on a Line = 514
• 16.3 The Bolzano-Weierstrass Theorem = 517
• 16.3.1 Convergent Sequences on a Line = 518
• 16.4 Point Sets in Higher Dimensions = 520
• 16.4.1 Convergent Sequences in Higher Dimensions = 521
• 16.5 Cauchy's Convergence Condition = 522
• 16.6 The Heine-Borel Theorem = 523
• 17／FUNDAMENTAL THEOREMS ON CONTINUOUS FUNCTIONS
• 17.0 Purpose of the Chapter = 527
• 17.1 Continuity and Sequential Limits = 527
• 17.2 The Boundedness Theorem = 529
• 17.3 The Extreme-Value Theorem = 529
• 17.4 Uniform Continuity = 529
• 17.5 Continuity of Sums, Products, and Quotients = 532
• 17.6 Persistence of Sign = 532
• 17.7 The Intermediate-Value Theorem = 533
• 18／THE THEORY OF INTEGRATION
• 18.0 The Nature of the Chapter = 535
• 18.1 The Definition of Integrability = 535
• 18.1.1 The Integrability of Continuous Functions = 539
• 18.1.2 Integrable Functions with Discontinuities = 540
• 18.2 The Integral as a Limit of Sums = 542
• 18.2.1 Duhamel's Principle = 545
• 18.3 Further Discussion of .Integrals = 548
• 18.4 The Integral as a Function of the Upper Limit = 548
• 18.4.1 The Integral of a Derivative = 550
• 18.5 Integrals Depending on a Parameter = 551
• 18.6 Riemann Double Integrals = 554
• 18.6.1 Double Integrals and Iterated Integrals = 557
• 18.7 Triple Integrals = 559
• 18.8 Improper Integrals = 559
• 18.9 Stieltjes Integrals = 560
• 19／INFINITE SERIES
• 19.0 Definitions and Notation = 566
• 19.1 Taylor's Series = 569
• 19.1.1 A Series for the Inverse Tangent = 572
• 19.2 Series of Nonnegative Terms = 573
• 19.2.1 The Integral Test = 577
• 19.2.2 Ratio Tests = 579
• 19.3 Absolute and Conditional Convergence = 581
• 19.3.1 Rearrangement of Terms = 585
• 19.3.2 Alternating Series = 587
• 19.4 Tests for Absolute Convergence = 590
• 19.5 The Binomial Series = 597
• 19.6 Multiplication of Series = 600
• 19.7 Dirichlet's Test = 604
• 20／UNIFORM CONVERGENCE
• 20.0 Functions Defined by Convergent Sequences = 610
• 20.1 The Concept of Uniform Convergence = 613
• 20.2 A Comparison Test for Uniform Convergence = 618
• 20.3 Continuity of the Limit Function = 620
• 20.4 Integration of Sequences and Series = 621
• 20.5 Differentiation of Sequences and Series = 624
• 21／POWER SERIES
• 21.0 General Remarks = 627
• 21.1 The Interval of Convergence = 627
• 21.2 Differentiation of Power Series = 632
• 21.3 Division of Power Series = 639
• 21.4 Abel's Theorem = 643
• 21.5 Inferior and Superior Limits = 647
• 21.6 Real Analytic Functions = 650
• 22／IMPROPER INTEGRALS
• 22.0 Preliminary Remarks = 654
• 22.1 Positive Integrands. Integrals of the First Kind = 656
• 22.1.1 Integrals of the Second Kind = 661
• 22.1.2 Integrals of Mixed Type = 664
• 22.2 The Gamma Function = 666
• 22.3 Absolute Convergence = 670
• 22.4 Improper Multiple Integrals. Finite Regions = 673
• 22.4.1 Improper Multiple Integrals. Infinite Regions = 678
• 22.5 Functions Defined by Improper Integrals = 682
• 22.5.1 Laplace Transforms = 690
• 22.6 Repeated Improper Integrals = 693
• 22.7 The Beta Function = 695
• 22.8 Stirling's Formula = 699
• ANSWERS TO SELECTED EXERCISES = 709
• INDEX = 727