Foreword
Volume 1
1. Infinite Sequences
§ 1. General Remarks
§ 2. Theorems and Properties
Exercise 1
2. Number Series
§ 1. General Remarks
§ 2. Theorems and Properties
2.1 Necessary Condition for Convergence
2.2 Comparison Test
2.3 The Geometric Series
2.4 The Harmonic Series
2.5 The Harmonic Series of Order α
2.6 Two Groups of Series
§ 3. Series of Non-Negative Terms
3.1 Criterion for Convergence
3.2 Criterion for Divergence
3.3 D'Alembert's Ratio Test
3.4 Cauchy's Test
§ 4. Alternating Series
4.1 Leibniz's Test
4.2 Absolute Convergence
§ 5. Other Series
Exercise 2
3. Functions and their Derivatives
§ 1. Definition of a Function
§ 2. Limit of a Function
§ 3. Continuity
3.1 Properties of Continuous Functions
3.2 Examples of Continuous Functions
Exercise 3
§ 4. First Order Derivatives of Functions of One Variable
4.1 Definition
4.2 Definition
4.3 Geometrical Interpretation
4.4 Differentiability of Continuous Functions
4.5 Properties of First Derivatives
4.6 Function of a Function
4.7 Inverse Functions
4.8 A List of First Derivatives
Exercise 4
§ 5. Derivatives of Higher Order
5.1 Definition
5.2 Derivatives of Order n
Exercise 5
§ 6. Derivatives of a Function Given by Parametric Equations
6.1 First Derivatives
6.2 Second Derivatives
Exercise 6
4. Partial Differentiation
§ 1. Continuity of a Function of Two Variables
§ 2. First Order Derivatives
§ 3. Second and Higher Order Derivatives
3.1 Introduction
3.2 Theorem
3.3 Definition
3.4 Function of a Function
3.5 Functions of Several Variables
Exercise 7
§ 4. Derivatives of Implicit Functions
4.1 Definition
4.2 The Existence of Implicit Functions
Exercise 8
5. Algebra
§ 1. Complex Numbers
1.1 Definitions
1.2 Trigonometric Interpretation
1.3 De Moivre's Theorem
Exercise 9
§ 2. The Solution of Algebraic Equations
2.1 General Properties of Algebraic Equations
Exercise 10
2.2 Cubic Equations
Exercise 11
2.3 Quartic Equations
Exercise 12
6. Curve Tracing
§1. Maxima and Minima
1.1 Increasing and Decreasing Functions
1.2 Definition of Maxima and Minima
1.3 Stationary Points
1.4 Turning Points
1.5 Determination of the Nature of Turning Points
§ 2. Concavity
2.1 Definition
2.2 Points of Inflexion
§ 3. Asymptotes
3.1 Definition
3.2 Asymptotes Parallel to the Coordinate Axes
3.3 Oblique Asymptotes
§4. Curve Tracing
4.1 Procedure
4.2 General Notes
§ 5. Implicit Functions
5.1 Definition
5.2 Conditions for a Multiple Point
5.3 Tangents at Multiple Points
5.4 Two Rules for Finding Asymptotes
5.5 Technique
§ 6. Functions of the type yq = xp
6.1 Symmetry
6.2 p/q > 1
6.3 p/q < 1
Exercise 13
7. Power Series
§ 1. General Remarks
1.1 Definition
1.2 Radius of Convergence
1.3 Theorems
Exercise 14
§ 2. Taylor's Theorem
2.1 Definition
2.2 The Taylor Series of a Function
2.3 The Maclaurin Series of a Function
2.4 The Integration of Any Series
Exercise 15
8. Limiting Values of Indeterminate Forms
§ 1. The Indeterminate Symbols 0/0, ∞/∞
1.1 The rule of 1'Hôpital
1.2 Generalization of Rule
§ 2. The Indeterminate Symbols 0.∞, ∞ - ∞, 1∞, ∞°, 0°
2.1 Theorem (0.∞)
2.2 Theorem (∞ - ∞)
2.3 Theorem (1∞, ∞°, 0°)
Exercise 16
9. Approximation to Roots of Equations
§ 1. A Graphical Method
§ 2. Routh's Method
§ 3. The Methods of Chords
§ 4. The Method of Tangents (Newton's Method)
§ 5. A Combined Method
Exercise 17
Answers
Index
Volume 2
10. Indefinite Integrals
§ 1. Introduction
§ 2. Some Standard Forms
§ 3. Some General Properties of Indefinite Integrals
3.1. The Addition of Integrals
3.2. Multiplication by a Constant
3.3. Integration by Parts
3.4. Substitution
3.5. Two Important Results
Exercise 18
§ 4. The Integration of Rational Functions
4.1 Introduction
4.2 Integrals of the Type ∫ mx+n/ax2+bx+c dx
4.3 Practical Methods
Exercise 19
§ 5. The Integration of Irrational Functions
5.1 Integrals Containing Roots of Linear Expressions
Exercise 20
5.2 Functions of the Type 1/√ax2+bx+c
5.3 Functions of the Type Ax+B/√ax2+bx+c
5.4 Two Pairs of Related Functions
5.5 The Method of Undetermined Multipliers
5.6 Functions of the Type 1/(ax2+b) √(px2+q)
Exercise 21
§ 6. The Integration of Trigonometric Functions
6.1 Some Standard Integrals
6.2 Integrals of Linear Functions
6.3 Integrals of Squared Functions
6.4 Integrals of Products
6.5 Reduction Formulae
6.6 The Transformation of General Trigonometric Functions Into Rational Functions
Exercise 22
§ 7. The Integration of Inverse Circular Functions
Exercise 23
§ 8. The Integration of Logarithmic and Exponential Function
Exercise 24
11. Definite Integrals
§ 1. General Remarks
1.1 Definition
1.2 Geometric Interpretation of a Definite Integral
§ 2. Properties of Definite Integrals
2.1 The Law of Addition
2.2 Multiplication by a Constant
2.3 Sum of Integrals
2.4 A Mean Value Theorem
2.5 A Function Defined as a Definite Integral
2.6 The Relationship Between Definite and Indefinite Integrals
2.7 Integration by Parts
2.8 Integration by Substitution
2.9 Dummy Symbols
Exercise 25
§ 3. Applications of Definite Integrals
3.1 The Calculation of Areas Given Parametric or Polar Equations of Curves
Exercise 26
3.2 The Calculation of Lengths of Arc
Exercise 27
3.3 Volumes of Revolution and Surface Area
Exercise 28
3.4 Moment of Inertia and Centre of Gravity
Exercise 29
§4. Irregular Integrals
4.1 Integrals of Functions Not Defined at a Finite Number of Points
4.2 Integrals of Unbounded Functions
4.3 Integrals of Infinite Interval
4.4 Geometric Interpretation
Exercise 30
12. Maxima and Minima of a Function of Two Variables
§ 1. Definition
§2. Theorem
Exercise 31
13. Tangents and Normals. Curvature
§ 1. Tangents and Normals
1.1 Equation of Tangent and Normal
Exercise 32
§ 2. Curvature and Radius of Curvature
2.1 Definition of Curvature
2.2 Definition of Radius of Curvature
2.3 Formulae for Curvature and Radius of Curvature
2.4 Sign Convention
Exercise 33
§ 3. Evolute and Involute
3.1 Coordinates of the Centre of Curvature
3.2 Definition
3.3 Two Properties of the Evolute
Exercise 34
14. Fourier Series
§ 1. General Ideas
1 1 The Euler—Fourier Formulae
1.2 The Fourier Series
1.3 Dirichlet's Conditions
1.4 Definition
1.5 Odd and Even Functions
1.6 Functions of Period 2l
1.7 Functions Defined in the Interval [a, b]
Exercise 35
Answers
Index