Calculus, Vol I, 2ed (An Indian Adaptation)

Introduction

Part 1. Historical Introduction

I 1.1 The two basic concepts of calculus

I 1.2 Historical background

I 1.3 The method of exhaustion for the area of a parabolic segment

I 1.4 Exercises

I 1.5 A critical analysis of Archimedes’ method

I 1.6 The approach to calculus to be used in this book

Part 2. Basic Concepts of Set Theory

I 2.1 Introduction

I 2.2 Notations for designating sets

I 2.3 Subsets

I 2.4 Set operations

I 2.5 Exercises

I 2.6 The inclusion-exclusion principle

Part 3. Axioms for the Real-Number System

I 3.1 Introduction

I 3.2 The field axioms

I 3.3 Exercises

I 3.4 The order axioms

I 3.5 Exercises

I 3.6 Integers and rational numbers

I 3.7 Geometric interpretation of real numbers as points on a line

I 3.8 Upper bound of a set, maximum element, least upper bound (supremum)

I 3.9 The least-upper-bound axiom (completeness axiom)

I 3.10 The Archimedean property of the real-number system

I 3.11 Fundamental properties of the supremum and infimum

I 3.12 Exercises

I 3.13 Existence of square roots of nonnegative real numbers

I 3.14 Roots of higher order. Rational powers

I 3.15 Representation of real numbers by decimals

Part 4. Mathematical Induction, Summation Notation, and Related Topics

I 4.1 An example of a proof by mathematical induction

I 4.2 The principle of mathematical induction

I 4.3 The well-ordering principle

I 4.4 Exercises

I 4.5 The summation notation

I 4.6 Exercises

I 4.7 Absolute values and the triangle inequality

I 4.8 Exercises

I 4.9 Miscellaneous Exercises

1. Integral Calculus

1.1 The basic ideas of cartesian geometry

1.2 Functions: Informal description and examples

1.3 Functions: Formal definition and examples

1.4 Exercises

1.5 The concept of area as a set function

1.6 Exercises

1.7 Intervals and ordinate sets

1.8 Partitions and step functions

1.9 Sum and product of step functions

1.10 Exercises

1.11 The integral for step functions and its properties

1.12 Other notations for integrals

1.13 Exercises

1.14 The integral of more general functions

1.15 Upper and lower integrals

1.16 The area of an ordinate set expressed as an integral

1.17 Remarks on the theory and technique of integration

1.18 Monotonic and piecewise monotonic functions: Definitions and examples

1.19 Integrability of bounded monotonic functions

1.20 Integral of a bounded monotonic function

1.21 Integral when p is a positive integer

1.22 The basic properties of the integral

1.23 Integration of polynomials

1.24 Exercises

2. Applications of Integration

2.1 Introduction

2.2 The area of a region between two graphs expressed as an integral

2.3 Exercises

2.4 The trigonometric functions

2.5 Integration formulas for the sine and cosine

2.6 A geometric description of the sine and cosine functions

2.7 Exercises

2.8 Polar coordinates

2.9 The integral for area in polar coordinates

2.10 Exercises

2.11 Application of integration to the calculation of volume

2.12 Exercises

2.13 Application of integration to the concept of work

2.14 Exercises

2.15 Average value of a function

2.16 Exercises

2.17 Indefinite integrals

2.18 Exercises

3. Continuous Functions

3.1 Continuity of a function: An informal description

3.2 The definition of the limit of a function

3.3 The definition of continuity of a function

3.4 The basic limit theorems

3.5 Exercises

3.6 Composite functions and continuity

3.7 Exercises

3.8 Bolzano’s theorem for continuous functions

3.9 The intermediate-value theorem for continuous functions

3.10 Exercises

3.11 Inverse of a function

3.12 Properties of functions preserved by inversion

3.13 Inverses of piecewise monotonic functions

3.14 Exercises

3.15 The extreme-value theorem for continuous functions

3.16 The small-span theorem for continuous functions (uniform continuity)

3.17 The integrability theorem for continuous functions

3.18 Mean-value theorems for integrals of continuous functions

3.19 Exercises

4. Differential Calculus

4.1 Historical introduction

4.2 A problem involving velocity

4.3 The derivative of a function

4.4 Examples of derivatives

4.5 The algebra of derivatives

4.6 Exercises

4.7 Geometric interpretation of the derivative as a slope

4.8 Other notations for derivatives

4.9 Exercises

4.10 The chain rule for differentiating composite functions

4.11 Applications of the chain rule

4.12 Exercises

4.13 Applications of differentiation to extreme values of functions

4.14 The mean-value theorem for derivatives

4.15 Exercises

4.16 Applications of the mean-value theorem

4.17 Second-derivative test for extrema

4.18 Curve sketching

4.19 Exercises

4.20 Examples of extremum problems

4.21 Exercises

4.22 Partial derivatives

4.23 Exercises

5. Relation Between Integration and Differentiation

5.1 The first fundamental theorem of calculus

5.2 The zero-derivative theorem

5.3 Primitive functions and the second fundamental theorem of calculus

5.4 Properties of a function deduced from properties of its derivative

5.5 Exercises

5.6 The Leibniz notation for primitives

5.7 Integration by substitution

5.8 Exercises

5.9 Integration by parts

5.10 Exercises

5.11 Miscellaneous exercises

6. The Logarithm, the Exponential, and the Inverse Trigonometric Functions

6.1 Introduction

6.2 Motivation for the definition of the natural logarithm as an integral

6.3 Logarithm function: Definition and basic properties

6.4 The graph of the natural logarithm

6.5 Consequences of the functional equation L(ab) = L(a) + L(b)

6.6 Logarithms referred to any positive base b ≠ 1

6.7 Differentiation and integration formulas involving logarithms

6.8 Logarithmic differentiation

6.9 Exercises

6.10 Polynomial approximations to the logarithm

6.11 Exercises

6.12 The exponential function

6.13 Exponentials expressed as powers of e

6.14 The definition of ex for arbitrary real x

6.15 The definition of ax for a > 0 and x real

6.16 Differentiation and integration formulas involving exponentials

6.17 Exercises

6.18 The hyperbolic functions

6.19 Exercises

6.20 Derivatives of inverse functions

6.21 Inverses of the trigonometric functions

6.22 Exercises

6.23 Integration by partial fractions

6.24 Integrals which can be transformed into integrals of rational functions

6.25 Exercises

6.26 Miscellaneous exercises

7. Polynomial Approximations to Functions

7.1 Introduction

7.2 The Taylor polynomials generated by a function

7.3 Calculus of Taylor polynomials

7.4 Exercises

7.5 Taylor’s formula with remainder

7.6 Maclaurin’s formula

7.7 Estimates for the error in Taylor’s formula

7.8 Other forms of the remainder in Taylor’s formula

7.9 Exercises

7.10 Further remarks on the error in Taylor’s formula. The o-notation

7.11 Applications to indeterminate forms

7.12 Exercises

7.13 L’Hôpital’s rule for the indeterminate form 0∕0

7.14 Exercises

7.15 The symbols +∞ and −∞. Extension of L’Hôpital’s rule

7.16 Infinite limits

7.17 The behavior of log x and ex for large x

7.18 Exercises

8. Differential Equations

8.1 Introduction

8.2 Terminology and notation

8.3 A first-order differential equation for the exponential function

8.4 First-order separable equations

8.5 Exercises

8.6 Homogeneous first-order equations

8.7 Exercises

8.8 Applications of first-order equations

8.9 First-order linear differential equations

8.10 Exercises

8.11 Applications of first-order linear differential equations

8.12 Exercises

8.13 Linear equations of second-order with constant coefficients

8.14 Existence of solutions of the equation y′ + by = 0

8.15 Reduction of the general equation to the special case y′ + by = 0

8.16 Uniqueness theorem for the equation y′ + by = 0

8.17 Complete solution of the equation y′ + by = 0

8.18 Complete solution of the equation y′ + ay′ + by = 0

8.19 Exercises

8.20 Nonhomogeneous linear equations of second-order with constant coefficients

8.21 Special methods for determining a particular solution of the nonhomogeneous equation y′ + ay′ + by = R

8.22 Exercises

8.23 Applications of linear second-order equations with constant coefficients

8.24 Exercises

8.25 Remarks concerning nonlinear differential equations

8.26 Integral curves and direction fields

8.27 Exercises

8.28 Miscellaneous exercises

9. Complex Numbers

9.1 Historical introduction

9.2 Definitions and field properties

9.3 The complex numbers as an extension of the real numbers

9.4 The imaginary unit i

9.5 Geometric interpretation of the complex number

9.6 Exercises

9.7 Complex exponentials

9.8 Complex-valued functions

9.9 Examples of differentiation and integration formulas

9.10 Exercises

10. Sequences, Infinite Series, Improper Integrals

10.1 Zeno’s paradox

10.2 Sequences

10.3 Monotonic sequences of real numbers

10.4 Exercises

10.5 Infinite series

10.6 The linearity property of convergent series

10.7 Telescoping series

10.8 The geometric series

10.9 Exercises

10.10 Decimal expansions

10.11 Tests for convergence

10.12 Comparison tests for series of nonnegative terms

10.13 The integral test

10.14 Exercises

10.15 The root test and the ratio test for series of nonnegative terms

10.16 Exercises

10.17 Alternating series

10.18 Conditional and absolute convergence

10.19 The convergence tests of dirichlet and abel

10.20 Exercises

10.21 Rearrangements of series

10.22 Improper integrals

10.23 Exercises

10.24 Miscellaneous review exercises

11. Sequences and Series of Functions

11.1 Pointwise convergence of sequences of functions

11.2 Uniform convergence of sequences of functions

11.3 Uniform convergence and continuity

11.4 Uniform convergence and integration

11.5 A sufficient condition for uniform convergence

11.6 Power series. Circle of convergence

11.7 Exercises

11.8 Properties of functions represented by real power series

11.9 The Taylor’s series generated by a function

11.10 A sufficient condition for Convergence of a Taylor’s Series

11.11 Power-series expansions for the exponential and trigonometric functions

11.12 Bernstein’s theorem

11.13 Exercises

11.14 Power series and differential equations

11.15 The binomial series

11.16 Exercises

12. Vector Algebra

12.1 Historical introduction

12.2 The vector space of n-tuples of real numbers

12.3 Geometric interpretation for n ≤ 3

12.4 Exercises

12.5 The dot product

12.6 Length or norm of a vector

12.7 Orthogonality of vectors

12.8 Exercises

12.9 Vector projections

12.10 The unit coordinate vectors

12.11 Exercises

12.12 The linear span of a finite set of vectors

12.13 Linear independence

12.14 Bases

12.15 Exercises

12.16 The vector space Vn(C) of n-tuples of complex numbers

12.17 Exercises

13. Applications of Vector Algebra to Analytic Geometry

13.1 Introduction

13.2 Lines in n-space

13.3 Some simple properties of straight lines

13.4 Lines and vector-valued functions

13.5 Exercises

13.6 Planes in Euclidean n-space

13.7 Planes and vector-valued functions

13.8 Exercises

13.9 The cross product

13.10 The cross product expressed as a determinant

13.11 Exercises

13.12 The scalar triple product
13.13 Cramer’s rule for solving a system of three linear equations
13.14 Exercises

13.15 Normal vectors to planes

13.16 Linear cartesian equations for planes

13.17 Exercises

13.18 The conic sections

13.19 Eccentricity of conic sections

13.20 Polar equations for conic sections

13.21 Conic sections symmetric about the origin

13.22 Exercises

13.23 Cartesian equations for the conic sections

13.24 Exercises

13.25 Miscellaneous Exercises

14. Calculus of Vector-Valued Functions

14.1 Vector-valued functions of a real variable

14.2 Algebraic operations

14.3 Limits, derivatives, and integrals

14.4 Exercises

14.5 Applications to curves

14.6 Applications to curvilinear motion

14.7 Exercises

14.8 The unit tangent, the principal normal, and the osculating plane of a curve

14.9 Exercises

14.10 The definition of arc length

14.11 Additivity of arc length

14.12 The arc-length function

14.13 Exercises

14.14 Curvature of a curve

14.15 Exercises

14.16 Evolute and involute

14.17 Velocity and acceleration in polar coordinates

14.18 Plane motion with radial acceleration

14.19 Cylindrical coordinates

14.20 Exercises

14.21 Applications to planetary motion

14.22 Miscellaneous exercises

15. Linear Spaces

15.1 Introduction

15.2 The definition of a linear space

15.3 Examples of linear spaces

15.4 Elementary consequences of the axioms

15.5 Exercises

15.6 Subspaces of a linear space

15.7 Dependent and independent sets in a linear space

15.8 Bases and dimension

15.9 Exercises

15.10 Inner products, Euclidean spaces. Norms

15.11 Orthogonality in a Euclidean space

15.12 Exercises

15.13 Construction of orthogonal sets: The Gram–Schmidt process

15.14 Orthogonal complements: Projections

15.15 Best approximation of elements in a Euclidean space by elements in a finite-dimensional subspace

15.16 Exercises

16. Linear Transformations and Matrices

16.1 Linear transformations

16.2 Null space and range

16.3 Nullity and rank

16.4 Exercises

16.5 Algebraic operations on linear transformations

16.6 Inverses

16.7 One-to-one linear transformations

16.8 Exercises

16.9 Linear transformations with prescribed values

16.10 Matrix representations of linear transformations

16.11 Construction of a matrix representation in diagonal form

16.12 Exercises

16.13 Linear spaces of matrices

16.14 Isomorphism between linear transformations and matrices

16.15 Multiplication of matrices

16.16 Exercises

16.17 Systems of linear equations

16.18 Computation techniques

16.19 Inverses of square matrices

16.20 Exercises

16.21 Miscellaneous exercises

Index

Answers to Problems (online)