Differential and Integral Calculus

CONTENTS
Preface 11
Chapter I. NUMBER. VARIABLE. FUNCTION
1. Real Numbers. Real Numbers as Points on a Number Scale .... 13
2. The Absolute Value of a Real Number 14
3. Variables and Constants 16
4. The Range of a Variable 16
5. Ordered Variables. Increasing and Decreasing Variables. Bounded Variables 18
6. Function 19
7. Ways of Representing Functions 20
8. Basic Elementary Functions. Elementary Functions 22
9. Algebraic Functions " 26
10. Polar Coordinate System 28
Exercises on Chapter 7 30
Chapter II. LIMIT. CONTINUITY OF A FUNCTION
1. The Limit of a Variable. An Infinitely Large Variable 32
2. The Limit of a Function 35
3. A Function that Approaches Infinity. Bounded Functions .:.... 38
4. Infinitesimals and Their Basic Properties 42
5. Basic Theorems on Limits 45
6. The Limit of the Function as 50
7. The Number e 51
8. Natural Logarithms 56
9. Continuity of Functions 57
10. Certain Properties of Continuous Functions 61
11. Comparing Infinitesimals 63
Exercises on Chapter 77 66
Chapter III. DERIVATIVE AND DIFFERENTIAL
1. Velocity of Motion 69
2. Definition of Derivative 71
3. Geometric Meaning of the Derivative 73
4. Differentiability of Functions 74
5. Finding the Derivatives of Elementary Functions. The Derivative of
the Function y = xn% Where n Is Positive and Integral ....... 76
6. Derivatives of the Functions y=s\nx\ y = cosx . 78
7. Derivatives of: a Constant, the Product of a Constant by a Function, a Sum, a Product, and a Quotient 80
8. The Derivative of a Logarithmic Function 84
9. The Derivative of a Composite Function . 85
10. Derivatives of the Functions y = ianx, y = coixt y=\n\x\ 83
11. An Implicit Function and Its Differentiation 89
12. Derivatives of a Power Function for an Arbitrary Real Exponent, of an Exponential Function, and a Composite Exponential Function . . 91
13. An Inverse Function and Its Differentiation 94
14. Inverse Trigonometric Functions and Their Differentiation 98
15. Table of Basic Differentiation Formulas 102
16. Parametric Representation of a Function 103
17. The Equations of Certain Curves in Parametric Form 105
18. The Derivative of a Function Represented Parametrically 108
19. Hyperbolic Functions 110
20. The Differential 113
21. The Geometric Significance of the Differential 117
22. Derivatives of Different Orders 118
23. Differentials of Various Orders 121
24. Different-Order Derivatives of Implicit Functions and of Functions Represented Parametrically 122
25. The Mechanical Significance of the Second Derivative 124
26. The Equations of a Tangent and of a Normal. The Lengths of the Subtangent and the Subnormal 126
27. The Geometric Significance of the Derivative of the Radius Vector with Respect to the Polar Angle 129
Exercises on Chapter III 130
Chapter IV. SOME THEOREMS ON DIFFERENTIABLE FUNCTIONS
1. A Theorem on the Roots of a Derivative (Rolle's Theorem) .... 140
2. A Theorem on Finite Increments (Lagrange's Theorem) 142
3. A Theorem on the Ratio of the Increments of Two Functions (Cauchy's Theorem) - 143
4. The Limit of a Ratio of Two Infinitesimals ( Evaluation of Indeterminate Forms of the Type -rr- J 144
5. The Limit of a Ratio of Two Infinitely Large Quantities (Evaluation of Indeterminate Forms of the Type 147
6. Taylor's Formula 152
7. Expansion of the Functions e*, sin*, and cos x in a Taylor Series . 156
Exercises on Chapter IV 159
Chapter V. INVESTIGATING THE BEHAVIOUR OF FUNCTIONS
1. Statement of the Problem 162
2. Increase and Decrease of a Function 163
3. Maxima and Minima of Functions 164
4. Testing a Differentiable Function for Maximum and Minimum with a First Derivative 171
5. Testing a Function for Maximum and Minimum with a Second Derivative 174
6. Maxima and Minima of a Function on an Interval 178
7. Applying the Theory of Maxima and Minima of Functions to the Solution of Problems 179
8. Testing a Function for Maximum and Minimum by Means of Taylor's Formula 181
9. Convexity and Concavity of a Curve. Points of Inflection 183
10. Asymptotes ' 189
11. General Plan for Investigating Functions and Constructing Graphs 194
12. Investigating Curves Represented Parametrically 199
Exercises on Chapter V . 203
Chapter VI. THE CURVATURE OF A CURVE
1. The Length of an Arc and Its Derivative 208
2. Curvature 210
3. Calculation of Curvature 212
4. Calculation of the Curvature of a Line Represented Parametrically . . 215
5. Calculation of the Curvature of a Line Given by an Equation of Polar Coordinates 215
6. The Radius and Circle of Curvature. Centre of Curvature. Evolute and Involute 217
7. The Properties of an Evolute 221
8. Approximating the Real Roots of an Equation 225
Exercises on Chapter VI -. . . . 229
Chapter VII. COMPLEX NUMBERS. POLYNOMIALS
1. Complex Numbers. Basic Definitions 233
2. Basic Operations on Complex Numbers : 234
3. Powers and Roots of Complex Numbers 237
4. Exponential Function with Complex Exponent and Its Properties . . 240
5. Euler's Formula. The Exponential Form of a Complex Number . . . 243
6. Factoring a Polynomial . . . . 244
7. The Multiple Roots of a Polynomial 247.
8. Factorisation of a Polynomial in the Case of Complex Roots .... 248
9. Interpolation. Lagrange's Interpolation Formula 250
10. On the Best Approximation of Functions by Polynomials. Chebyshev's Theory . . . . . 252
Exercises on Chapter VII 253
Chapter VIII. FUNCTIONS OF SEVERAL VARIABLES
1. Definition of a Function of Several Variables 255
2. Geometric Representation of a Function of Two Variables ..... 25§
3. Partial and Total Increment of a Function 259
4. Continuity of a Function of Several Variables 260
5. Partial Derivatives of a Function of Several Variables 263
6. The Geometric Interpretation of the Partial Derivatives of a Function of Two Variables 264
7. Total Increment and Total Differentials 265
8. Approximation by Total Differentials 268
9. Error Approximation by Differentials 270
10. The Derivative of a Composite Function. The Total Derivative . . . 273
11. The Derivative of a Function Defined Implicitly 276
12. Partial Derivatives of Different Orders 279
13. Level Surfaces .283
14. Directional Derivatives . 284
15. Gradient 286
16. Taylor's Formula for a Function of Two Variables 290
17. Maximum and-Minimum of a Function of Several Variables .... 292
18. Maximum and Minimum of a Function of Several Variables Related by Given Equations (Conditional Maxima and Minima) 300
19. Singular Points of a Curve 305
Exercises on Chapter VIII 310
Chapter IX. APPLICATIONS OF DIFFERENTIAL CALCULUS TO SOLID GEOMETRY
1. The Equations of a Curve in Space 314
2. The Limit and Derivative of the Vector Function of a Scalar Argument. The Equation of a Tangent to a Curve. The Equation of a Normal Plane 317
3. Rules for Differentiating Vectors (Vector Functions) 322
4. The First and Second Derivatives of a Vector with Respect to the Arc Length. The Curvature of a Curve. The Principal Normal .... 324
5. Osculating Plane. Binormal. Torsion 331
6. A Tangent Plane and Normal to a Surface 336
Exercises on Chapter IX 340
Chapter X, INDEFINITE INTEGRALS
1. Antiderivative and the Indefinite Integral 342
2. Table of Integrals 344
3. Some Properties of an Indefinite Integral 346
4. Integration by Substitution (Change of Variable) 348
5. Integrals of Functions Containing a Quadratic Trinomial 351
6. Integration by Parts 354
7. Rational Fractions. Partial Rational Fractions and Their Integration 357
8. Decomposition of a Rational Fraction into Partial Fractions .... 361
9. Integration of Rational Fractions 365
10. Ostrogradsky's Method 368
11. Integrals of Irrational Functions 371
12. Integrals of the Form [R(x> V'ax2+bx + c)dx . 372
13. Integration of Binomial Differentials 375
14. Integration of Certain Classes of Trigonometric Functions 378
15. Integration of Certain Irrational Functions by Means of Trigonometric Substitutions 383
16. Functions Whose Integrals Cannot Be Expressed in Terms of Elementary Functions 385
Exercises on Chapter X 386
Chapter XI. THE DEFINITE INTEGRAL
1. Statement of the Problem. The Lower and Upper Integral Sums . . . 396
2. The Definite Integral 398
3. Basic Properties o? the Definite Integral . 404
4. Evaluating a Definite Integral. Newton-Leibniz Formula 407
5. Changing the Variable in the Definite Integral 412
6. Integration by Parts 413
7. Improper Integrals 416
8. Approximating Definite Integrals 424
9. Chebyshev's Formula 430
10. Integrals Dependent on a Parameter 435
Exercises on Chapter XI 438
Chapter XII. GEOMETRIC AND MECHANICAL APPLICATIONS OF THE DEFINITE INTEGRAL
1. Computing Areas in Rectangular Coordinates 442
2. The Area of a Curvilinear Sector in Polar Coordinates ' . 445
3. The Arc Length of a Curve . 447
4. Computing the Volume of a Solid from the Areas of Parallel Sections
(Volumes by Slicing) 453
5. The Volume of a Solid of Revolution 455
6. The Surface of a Solid of Revolution 455
7. Computing Work by the Definite Integral 457
8. Coordinates of the Centre of Gravity 459
Exercises on Chapter XII 462
Chapter XIII. DIFFERENTIAL EQUATIONS
1. Statement of the Problem. The Equation of Motion of a Body with
Resistance of the Medium Proportional to the Velocity. The
Equation of a Catenary 469
2. Definitions \ . . , 472
3. First-Order Differential Equations (General Notions) . 473
4. Equations with Separated and Separable Variables. The Problem of
the Disintegration of Radium 478
5. Homogeneous First-Order Equations 482
6. Equations Reducible to Homogeneous Equations 484
7. First-Order Linear Equations 487
8. Bernoulli's Equation 490
9. Exact Differential Equations 492
10. Integrating Factor 495
11. The Envelope of a Family of Curves 497
12. Singular Solutions of a First-Order Differential Equation 504
13. Clairaut's Equation 505
14. Lagrange's Equation 507
15. Orthogonal and Isogonal Trajectories 509
16. Higher-Order Differential Equations (Fundamentals) 514
17. An Equation of the Form y = f(x) 516
18. Some Types of Second-Order Differential Equations Reducible to
First-Order Equations . . 518
19. Graphical Method of Integrating Second-Order Differential Equations 527
20. Homogeneous Linear Equations. Definitions and General Properties 528
21. Second-Order Homogeneous Linear Equations with Constant Coefficients 535
22. Homogeneous Linear Equations of the Aith Order with Constant
Coefficients 539
23. Nonhomogeneous Second-Order Linear Equations 541
24. Nonhomogeneous Second-Order Linear Equations with Constant Coefficients 545
25. Higher-Order Nonhomogeneous Linear Equations 551
26. The Differential -Equation of Mechanical Vibrations 555
27. Free Oscillations' 557
28. Forced Oscillations 559
29. Systems of Ordinary Differential Equations 563
30. Systems of Linear Differential Equations with Constant Coefficients 569
31. On Lyapunov's Theory of Stability 576
32. Euler's Method of Approximate Solution of First-Order Differential Equations . 581
33. A Difference Method for Approximate Solution of Differential Equations Based on Taylor's Formula. Adams Method 584
34. An Approximate Method for Integrating Systems of First-Order Differential Equations 591
Exercises on Chapter XIII 595
Chapter XIV. MULTIPLE INTEGRALS
1. Double Integrals . 608
2. Calculating Double Integrals 610
3. Calculating Double Integrals (Continued) 617
4. Calculating Areas and Volumes by Means of Double Integrals .... 623
5. The Double Integral in Polar Coordinates 626
6. Changing Variables in a Double Integral (General Case) 633
7. Computing the Area of a Surface 638
8. The Density of Distribution of Matter and the Double Integral . . . 642
9. The Moment of Inertia of the Area of a Plane Figure 643
10. The Coordinates of the Centre of Gravity of the Area of a Plane Figure 648
11. Triple Integrals 650
12. Evaluating a Triple Integral 651
13. Change of Variables in a Triple Integral 656
14. The Moment of Inertia and the Coordinates of the Centre of Gravity of a Solid . 660
15. Computing Integrals Dependent on a Parameter . 662
Exercises on Chapter XIV . . . 663
Chapter XV. LINE INTEGRALS AND SURFACE INTEGRALS
1. Line Integrals 670
2. Evaluating a Line Integral 673
3. Green's Formula 679
4. Conditions for a Line Integral Being Independent of the Path of Integration 681
5. Surface Integrals . 687
6. Evaluating Surface Integrals 689
7. Stokes' Formula 692
8. Ostrogradsky's Formula 697
9. The Hamiltonian Operator and Certain Applications of It 700
Exercises on Chapter XV 703
Chapter XVI. SERIES
1. Series. Sum of a Series 710
2. Necessary Condition for Convergence of a Series 713
3. Comparing Series with Positive Terms 716
4. D'Alembert's Test 718
5. Cauchy's Test 721
6. The Integral Test for Convergence of a Series .723
7. Alternating Series. Leibniz' Theorem 727
8. PIus-and-Minus Series. Absolute and Contitional Convergence .... 729
9. Functional Series 733
10. Majorised Series 734
11. The Continuity of the Sum of a Series 736
12. Integration and Differentiation of Series 739
13. Power Series. Interval of Convergence 742
14. Differentiation of Power Series 747
15. Series in Powers of 748
16. Taylor's Series and Maclaurin's Series .' 750
17. Examples of Expansion of Functions in Series 751
18. Euler's Formula 753
19. The Binomial Series 754
20. Expansion of the Function In (\+x) in a Power Series. Computing Logarithms 756
21. Integration by Use of Series (Calculating Definite Integrals) .... 758
22. Integrating Differential Equations by Means of Series ....... 760
23. Bessel's Equation 763
Exercises on Chapter XVI 768
Chapter XVII. FOURIER SERIES
1. Definition. Statement of the Problem 776
2. Expansions of Functions in Fourier Series 780
3. A Remark on the Expansion of a Periodic Function in a Fourier Series 785
4. Fourier Series for Even and Odd Functions 787
5. The Fourier Series for a Function with Period 2/ 789
6. On the Expansion of a Nonperiodic Function in a Fourier Series . . 791
7. Approximation by a Trigonometric Polynomial of a Function Represented in the Mean 792
8. The Dirichlet Integral 798
9. The Convergence of a Fourier Series at a Given Point . . . . . . .801
10. Certain Sufficient Conditions for the Convergence of a Fourier Series 802
11. Practical Harmonic Analysis 805
12. Fourier Integral 810
13. The Fourier Integral in Complex Form 810
Exercises on Chapter XVII 812
Chapter XVIII. EQUATIONS OF MATHEMATICAL PHYSICS
1. Basic Types of Equations of Mathematical Physics . 815
2. Derivation of the Equation of Oscillations of a String. Formulation
of the Boundary-Value Problem. Derivation of Equations of Electric
Oscillations in Wires : 816
3. Solution of the Equation of Oscillations of a String by the Method
of Separation of Variables (The Fourier Method) 820
4. The Equation for Propagation of Heat in a Rod. Formulation of the
Boundary-Value Problem 823
5. Heat Propagation in Space 825
6. Solution of the First Boundary-Value Problem for the Heat-
Conductivity Equation by the Method of Finite Differences 829
7. Propagation of Heat in an Unbounded Rod 831
8. Problems That Reduce to Investigating Solutions of the Laplace
Equation. Stating Boundary-Value Problems 836
9. The Laplace Equation in Cylindrical Coordinates. Solution of the Dirichlet Problem for a Ring with Constant Values of the Desired Function on the Inner and Outer Circumferences 841
10. The Solution of Dirichlet's Problem for a Circle 843
11. Solution of the Dirichlet Problem by the Method of Finite Differences 847
Exercises on Chapter XVIII 850
Chapter XIX. OPERATIONAL CALCULUS AND CERTAIN OF ITS APPLICATIONS
1. The Initial Function and Its Transform 854
2. Transforms of the Functions.oQ(t), sin/, cost 855
3. The Transform of a Function with Changed Scale of the Independent
Variable. Transforms of the Functions sin at, cos at 856
4. The Linearity Property of a Transform 857
5. The Shift Theorem r 858
6. Transforms of the Functionse~at, sinh at, cosh at, e-aisinatt e~a/ cos at 858
7. Differentiation of Transforms 860
8. The Transforms of Derivatives 861
9. Table of Transforms ~. 862
10. An Auxiliary Equation for a Given Differential Equation 864
11. Decomposition Theorem . 867
12. Examples of Solutions of Differential Equations and Systems of
Differential Equations by the Operational /Method 869
13. The Convolution Theorem 871
14. The Differential Equations of Mechanical Oscillations. The Differential Equations of Electric-Circuit Theory 873
15. Solution of the Differential Oscillation Equation 874
16. Investigating Free Oscillations 875
17. Investigating Mechanical and Electrical Oscillations in the Case of a Periodic External Force 876
18. Solving the Oscillation Equation in the Case of Resonance . . . . 878
19. The Delay Theorem 879
Exercises on Chapter XIX 880
Subject Index