Calculus, Vol I, 2ed (An Indian Adaptation)
ISBN: 9789354642630
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Description
Calculus, Second Edition, strikes a perfect balance between theory and technique. Although it differs from the majority of contemporary texts, treating integration before differentiation is the most accurate historical representation of the relationship between the integral and the derivative. This introduces the mean-value theorems and their applications earlier in the text, incorporates a treatment of linear algebra, and contains many new and easier exercises. As in the first edition, an interesting historical introduction precedes each important new concept.
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)