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

Part 1. Linear Analysis

1. Linear Spaces

1.1 Introduction

1.2 Linear space axioms

1.3 Elementary consequences of the axioms

1.4 Exercises

1.5 Subspaces

1.6 Dependent and independent sets

1.7 Bases and dimension

1.8 Components

1.9 Exercises

1.10 Inner products, Euclidean spaces: L norms

1.11 Orthogonality in a Euclidean space

1.12 Exercises

1.13 Construction of orthogonal sets: The Gram-Schmidt process

1.14 Orthogonal complements: Projections

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

1.16 Exercises

2. Linear Transformations and Matrices

2.1 Linear transformations

2.2 Null space and range

2.3 Nullity and rank

2.4 Exercises

2.5 Algebraic operations on linear transformations

2.6 Inverses

2.7 One-to-one linear transformations

2.8 Exercises

2.9 Linear transformations with prescribed values

2.10 Matrix representations of linear transformations

2.11 Construction of a matrix representation in diagonal form

2.12 Exercises

2.13 Linear spaces of matrices

2.14 Isomorphism between linear transformations and matrices

2.15 Multiplication of matrices

2.16 Exercises

2.17 Systems of linear equations

2.18 Computation techniques

2.19 Inverses of square matrices

2.20 Transpose of a matrix

2.21 Hadamard matrices

2.22 Exercises

2.23 Miscellaneous exercises

3. Determinants

3.1 Introduction

3.2 Motivation for the choice of axioms

3.3 A set of axioms

3.4 Computation of determinants

3.5 The uniqueness theorem

3.6 Exercises

3.7 The product formula

3.8 The determinant of the inverse of a nonsingular matrix

3.9 Determinants and independence of vectors

3.10 The determinant of a block-diagonal matrix

3.11 Exercises

3.12 Expansion formulas for determinants: Minors and cofactors

3.13 Existence of the determinant function

3.14 The determinant of a transpose

3.15 The cofactor matrix

3.16 Cramer’s rule

3.17 Exercises

4. Eigenvalues and Eigenvectors

4.1 Linear transformations with diagonal matrix representations

4.2 Eigenvectors and eigenvalues

4.3 Linear independence of eigenvectors corresponding to distinct eigenvalues

4.4 Exercises

4.5 The finite-dimensional case: Characteristic polynomials

4.6 Eigenvalues and eigenvectors in the finite-dimensional case

4.7 Trace of a matrix

4.8 Exercises

4.9 Similar matrices

4.10 Exercises

5. Eigenvalues of Operators Acting on Euclidean Spaces

5.1 Eigenvalues and inner products

5.2 Hermitian and skew-Hermitian transformations

5.3 Eigenvalues and eigenvectors of Hermitian and skew-Hermitian operators

5.4 Orthogonality of eigenvectors corresponding to distinct eigenvalues

5.5 Exercises

5.6 Existence of an orthonormal set of eigenvectors

5.7 Matrix representations for Hermitian and skew-Hermitian operators

5.8 The adjoint of a matrix

5.9 Diagonalization of a matrix

5.10 Unitary and orthogonal matrices

5.11 Exercises

5.12 Quadratic forms

5.13 Reduction of a real quadratic form to a diagonal form

5.14 Applications to analytic geometry

5.15 Exercises

5.16 Eigenvalues of a symmetric transformation

5.17 Extremal properties of eigenvalues of a symmetric transformation

5.18 The finite-dimensional case

5.19 Unitary transformations

5.20 Exercises

6. Linear Differential Equations

6.1 Historical introduction

6.2 Review of results concerning linear equations of first and second orders

6.3 Exercises

6.4 Linear differential equations of order n

6.5 The existence-uniqueness theorem

6.6 The dimension of the solution space of a homogeneous linear equation

6.7 The algebra of constant-coefficient operators

6.8 Basis of solutions for linear equations with constant coefficients

6.9 Exercises

6.10 The relation between the homogeneous and nonhomogeneous equations

6.11 The method of variation of parameters

6.12 Nonsingularity of the Wronskian matrix

6.13 Reduction of nonhomogeneous equations to a system of first-order linear equations

6.14 The annihilator method for determining a particular solution of the nonhomogeneous equation

6.15 Exercises

6.16 Miscellaneous exercises on linear differential equations

6.17 Linear equations of second order with analytic coefficients

6.18 The Legendre equation

6.19 The Legendre polynomials

6.20 Rodrigues’ formula for the Legendre polynomials

6.21 Exercises

6.22 The method of Frobenius

6.23 The Bessel equation

6.24 Exercises

7. Systems of Differential Equations

7.1 Introduction

7.2 Calculus of matrix functions

7.3 Infinite series of matrices. Norms of matrices

7.4 Exercises

7.5 The exponential matrix

7.6 The differential equation satisfied by et A

7.7 Uniqueness theorem for the matrix differential equation F(t) = AF(t)

7.8 The law of exponents for exponential matrices

7.9 Existence and uniqueness theorems for homogeneous linear systems with constant coefficients

7.10 The Cayley-Hamilton theorem

7.11 Exercises

7.12 Nonhomogeneous linear systems with constant coefficients

7.13 Exercises

7.14 The general linear system Y’(t) = P(t)Y(t) + Q(t)

7.15 A power-series method for solving homogeneous linear systems

7.16 Exercises

7.17 Proof of the existence theorem by the method of successive approximations

7.18 The method of successive approximations applied to first-order nonlinear systems

7.19 Proof of an existence-uniqueness theorem for first-order nonlinear systems

7.20 Exercises

7.21 Successive approximations and fixed points of operators

7.22 Normed linear spaces

7.23 Contraction operators

7.24 Fixed-point theorem for contraction operators

7.25 Applications of the fixed-point theorem

Part 2. Nonlinear Analysis

8. Differential Calculus of Scalar and Vector Fields

8.1 Functions from Rn to Rm

8.2 Open balls and open sets

8.3 Exercises

8.4 Limits and continuity

8.5 Exercises

8.6 The derivative of a scalar field with respect to a vector

8.7 Directional derivatives and partial derivatives

8.8 Partial derivatives of higher order

8.9 Exercises

8.10 Directional derivatives and continuity

8.11 The total derivative

8.12 The gradient of a scalar field

8.13 A sufficient condition for differentiability

8.14 Exercises

8.15 A chain rule for derivatives of scalar fields

8.16 Applications to geometry

8.17 Exercises

8.18 Derivatives of vector fields

8.19 Differentiability implies continuity

8.20 The chain rule for derivatives of vector fields

8.21 Matrix form of the chain rule

8.22 Exercises

8.23 Sufficient conditions for the equality of mixed partial derivatives

8.24 Miscellaneous exercises

9. Applications of Differential Calculus

9.1 Partial differential equations

9.2 A first-order partial differential equation with constant coefficients

9.3 Exercises

9.4 The one-dimensional wave equation

9.5 Exercises

9.6 Derivatives of functions defined implicitly

9.7 Exercises

9.8 Maxima, minima, and saddle points

9.9 Second-order Taylor formula for scalar fields

9.10 Eigenvalues of the Hessian matrix

9.11 Second-derivative test for extrema of functions of two variables

9.12 Exercises

9.13 Extrema with constraints. Lagrange’s multipliers

9.14 Exercises

9.15 The extreme-value theorem for continuous scalar fields

9.16 The small-span theorem for continuous scalar fields (uniform continuity)

10. Line Integrals

10.1 Introduction

10.2 Paths and line integrals

10.3 Other notations for line integrals

10.4 Basic properties of line integrals

10.5 Exercises

10.6 The concept of work as a line intregal

10.7 Line integrals with respect to arc length

10.8 Further applications of line integrals

10.9 Exercises

10.10 Open connected sets

10.11 The second fundamental theorem of calculus for line integrals

10.12 Applications to mechanics

10.13 Exercises

10.14 The first fundamental theorem of calculus for line integrals

10.15 Necessary and sufficient conditions for a vector field to be a gradient

10.16 Special methods for constructing potential functions

10.17 Exercises

10.18 Applications to exact differential equations of first order

10.19 Exercises

10.20 Potential functions on convex sets

11. Multiple Integrals

11.1 Introduction

11.2 Partitions of rectangles. Step functions

11.3 The double integral of a step function

11.4 The double integral of a function defined and bounded on a rectangle

11.5 Upper and lower double integrals

11.6 The double integral by repeated one-dimensional integration

11.7 Geometric interpretation of the double integral as a volume

11.8 Exercises

11.9 Integrability of continuous functions

11.10 Integrability of bounded functions with discontinuities

11.11 Double integrals extended over more general regions

11.12 Applications to area and volume

11.13 Exercises

11.14 Further applications of double integrals

11.15 Pappus’ theorems

11.16 Exercises

11.17 Green’s theorem in the plane

11.18 A necessary and sufficient condition for a two-dimensional vector field to be a gradient

11.19 Exercises

11.20 Green’s theorem for multiply connected regions

11.21 The winding number

11.22 Exercises

11.23 Change of variables in a double integral

11.24 Special cases of the transformation formula

11.25 Exercises

11.26 Proof of the transformation formula in the general case and a special case

11.27 Extensions to higher dimensions

11.28 Change of variables in an n-fold integral

11.29 Exercises

12. Surface Integrals

12.1 Parametric representation of a surface

12.2 The fundamental vector product

12.3 The fundamental vector product as a normal to the surface

12.4 Exercises

12.5 Area of a parametric surface

12.6 Exercises

12.7 Surface integrals

12.8 Change of parametric representation

12.9 Other notations for surface integrals

12.10 Exercises

12.11 Stokes’ theorem

12.12 The curl and divergence of a vector field

12.13 Exercises

12.14 Further properties of the curl and divergence

12.15 Exercises

12.16 Reconstruction of a vector field from its curl

12.17 Exercises

12.18 Extensions of Stokes’ theorem

12.19 The divergence theorem (Gauss’ theorem)

12.20 Applications of the divergence theorem

12.21 Exercises

Part 3. Special Topics

13. Set Functions and Elementary Probability

13.1 Historical introduction

13.2 Finitely additive set functions

13.3 Finitely additive measures

13.4 Exercises

13.5 The definition of probability for finite sample spaces

13.6 Special terminology peculiar to probability theory

13.7 Exercises

13.8 Exercises

13.9 Some basic principles of combinatorial analysis

13.10 Exercises

13.11 Conditional probability: Bayes’ theorem

13.12 Independence

13.13 Exercises

13.14 Compound experiments

13.15 Bernoulli trials

13.16 The most probable number of successes in n Bernoulli trials

13.17 Exercises

13.18 Countable and uncountable sets

13.19 Exercises

13.20 The definition of probability for countably infinite sample spaces

13.21 Exercises

13.22 Miscellaneous exercises

14. Calculus of Probabilities

14.1 The definition of probability for uncountable sample spaces

14.2 Countability of the set of points with positive probability

14.3 Random variables

14.4 Exercises

14.5 Distribution functions

14.6 Discontinuities of distribution functions

14.7 Discrete distributions. Probability mass functions

14.8 Exercises

14.9 Continuous distributions. Density functions

14.10 Uniform distribution over an interval

14.11 Cauchy’s distribution

14.12 Exercises

14.13 Exponential distributions

14.14 Normal distributions

14.15 Remarks on more general distributions

14.16 Exercises

14.17 Distributions of functions of random variables

14.18 Exercises

14.19 Distributions of two-dimensional random variables

14.20 Two-dimensional discrete distributions

14.21 Two-dimensional continuous distributions. Density functions

14.22 Exercises

14.23 Distributions of functions of two random variables

14.24 Exercises

14.25 Expectation and variance

14.26 Expectation of a function of a random variable

14.27 Exercises

14.28 Chebyshev’s inequality

14.29 Laws of large numbers

14.30 The central limit theorem

14.31 Exercises

15. Introduction to Numerical Analysis

15.1 Historical introduction

15.2 Approximations by polynomials

15.3 Polynomial approximation and normed linear spaces

15.4 Fundamental problems in polynomial approximation

15.5 Exercises

15.6 Interpolating polynomials

15.7 Equally spaced interpolation points

15.8 Error analysis in polynomial interpolation

15.9 Exercises

15.10 Newton’s interpolation formula

15.11 Equally spaced interpolation points. The forward difference operator

15.12 Factorial polynomials

15.13 Exercises

15.14 A minimum problem relative to the max norm

15.15 Chebyshev polynomials

15.16 A minimal property of Chebyshev polynomials

15.17 Application to the error formula for interpolation

15.18 Exercises

15.19 Approximate integration

15.20 Exercises

Suggested References

Index

Answers to Problems (online)