- #1
- 22,183
- 3,324
- Author: Apostol
- Title: Calculus
- Amazon Link:
https://www.amazon.com/dp/0471000051/?tag=pfamazon01-20
https://www.amazon.com/dp/0471000078/?tag=pfamazon01-20 - Prerequisities: PreCalculus/Algebra & Trig ; Ideal with a basic knowledge of calculus however, it's fine without it.
Table of Contents of Volume I:
Code:
[LIST]
[*] Introduction
[LIST]
[*] Historical Introduction
[LIST]
[*] The two basic concepts of calculus
[*] Historical background
[*] The method of exhaustion for the area of a parabolic segment
[*] Exercises
[*] A critical analysis of Archimedes’ method
[*] The approach to calculus to be used in this book
[/LIST]
[*] Some Basic Concepts of the Theory of Sets
[LIST]
[*] Introduction to set theory
[*] Notations for designating sets
[*] Subsets
[*] Unions, intersections, complements
[*] Exercises
[/LIST]
[*] A Set of Axioms for the Real-Number System
[LIST]
[*] Introduction
[*] The field axioms
[*] Exercises
[*] The order axioms
[*] Exercises
[*] Integers and rational numbers
[*] Geometric interpretation of real numbers as points on a line
[*] Upper bound of a set, maximum element, least upper bound (supremum)
[*] The least-Upper-bound axiom (completeness axiom)
[*] The Archimedean property of the real-number system
[*] Fundamental properties of the supremum and infimum
[*] Exercises
[*] Existence of square roots of nonnegative real numbers
[*] Roots of higher order. Rational powers
[*] Representation of real numbers by decimals
[/LIST]
[*] Mathematical Induction, Summation Notation, and Related Topics
[LIST]
[*] An example of a proof by mathematical induction
[*] The principle of mathematical induction
[*] The well-ordering principle
[*] Exercises
[*] Proof of the well-ordering principle
[*] The summation notation
[*] Exercises
[*] Absolute values and the triangle inequality
[*] Exercises
[*] Miscellaneous exercises involving induction
[/LIST]
[/LIST]
[*] The Concepts of Integral Calculus
[LIST]
[*] The basic ideas of Cartesian geometry
[*] Functions: Informal description and examples
[*] Functions. Formal definition as a set of ordered pairs
[*] More examples of real functions
[*] Exercises
[*] The concept of area as a set function
[*] Exercises
[*] Intervals and ordinate sets
[*] Partitions and step functions
[*] Sum and product of step functions
[*] Exercises
[*] The definition of the integral for step functions
[*] Properties of the integral of a step function
[*] Other notations for integrals
[*] Exercises
[*] The integral of more general functions
[*] Upper and lower integrals
[*] The area of an ordinate set expressed as an integral
[*] Informal remarks on the theory and technique of integration
[*] Monotonic and piecewise monotonic functions. Definitions and examples
[*] Integrability of bounded monotonic functions
[*] Calculation of the integral of a bounded monotonic function
[*] Calculation of the integral [itex]\int_0^b x^pdx[/itex] when [itex]p[/itex] is a positive integer
[*] The basic properties of the integral
[*] Integration of polynomials
[*] Exercises
[*] Proofs of the basic properties of the integral
[/LIST]
[*] Some Applications of Integration
[LIST]
[*] Introduction
[*] The area of a region between two graphs expressed as an integral
[*] Worked examples
[*] Exercises
[*] The trigonometric functions
[*] Integration formulas for the sine and cosine
[*] A geometric description of the sine and cosine functions
[*] Exercises
[*] Polar coordinates
[*] The integral for area in polar coordinates
[*] Exercises
[*] Application of integration to the calculation of volume
[*] Exercises
[*] Application of integration to the concept of work
[*] Exercises
[*] Average value of a function
[*] Exercises
[*] The integral as a function of the Upper limit. Indefinite integrals
[*] Exercises
[/LIST]
[*] Continuous Functions
[LIST]
[*] Informal description of continuity
[*] The definition of the limit of a function
[*] The definition of continuity of a function
[*] The basic limit theorems. More examples of continuous functions
[*] Proofs of the basic limit theorems
[*] Exercises
[*] Composite functions and continuity
[*] Exercises
[*] Bolzano’s theorem for continuous functions
[*] The intermediate-value theorem for continuous
[*] Exercises
[*] The process of inversion
[*] Properties of functions preserved by inversion
[*] Inverses of piecewise monotonic functions
[*] Exercises
[*] The extreme-value theorem for continuous functions
[*] The small-span theorem for continuous functions (uniform continuity)
[*] The integrability theorem for continuous functions
[*] Mean-value theorems for integrals of continuous functions
[*] Exercises
[/LIST]
[*] Differential Calculus
[LIST]
[*] Historical introduction
[*] A problem involving velocity
[*] The derivative of a function
[*] Examples of derivatives
[*] The algebra of derivatives
[*] Exercises
[*] Geometric interpretation of the derivative as a slope
[*] Other notations for derivatives
[*] Exercises
[*] The chain rule for differentiating composite functions
[*] Applications of the chain rule. Related rates and implicit differentiation
[*] Exercises
[*] Applications of differentiation to extreme values of functions
[*] The mean-value theorem for derivatives
[*] Exercises
[*] Applications of the mean-value theorem to geometric properties of functions
[*] Second-derivative test for extrema
[*] Curve sketching
[*] Exercises
[*] Exercises
[*] Partial derivatives
[*] Exercises
[/LIST]
[*] The Relation Between Integration and Differentiation
[LIST]
[*] The derivative of an indefinite integral. The first fundamental theorem of calculus
[*] The zero-derivative theorem
[*] Primitive functions and the second fundamental theorem of calculus
[*] Properties of a function deduced from properties of its derivative
[*] Exercises
[*] The Leibniz notation for primitives
[*] Integration by substitution
[*] Exercises
[*] Integration by parts
[*] Exercises
[*] Miscellaneous review exercises
[/LIST]
[*] The Logarithm, the Exponential, and the Inverse Trigonometric Functions
[LIST]
[*] Introduction
[*] Motivation for the definition of the natural logarithm as an integral
[*] The definition of the logarithm. Basic properties
[*] The graph of the natural logarithm
[*] Consequences of the functional equation [itex]L(ab)=L(a)+L(b)[/itex]
[*] Logarithms referred to any positive base [itex]b\neq 1[/itex]
[*] Differentiation and integration formulas involving logarithms
[*] Logarithmic differentiation
[*] Exercises
[*] Polynomial approximations to the logarithm
[*] Exercises
[*] The exponential function
[*] Exponentials expressed as powers of [itex]e[/itex]
[*] The definition of [itex]e^x[/itex] for arbitrary real [itex]x[/itex]
[*] The definition of [itex]a^x[/itex] for [itex]a>0[/itex] and [itex]x[/itex] real
[*] Differentiation and integration formulas involving exponentials
[*] Exercises
[*] The hyperbolic functions
[*] Exercises
[*] Derivatives of inverse functions
[*] Inverses of the trigonometric functions
[*] Exercises
[*] Integration by partial fractions
[*] Integrals which can be transformed into integrals of rational functions
[*] Exercises
[*] Miscellaneous review exercises
[/LIST]
[*] Polynomial Approximations to Functions
[LIST]
[*] Introduction
[*] The Taylor polynomials generated by a function
[*] Calculus of Taylor polynomials
[*] Exercises
[*] Taylor's formula with remainder
[*] Estimates for the error in Taylor’s formula
[*] Other forms of the remainder in Taylor’s formula
[*] Exercises
[*] Further remarks on the error in Taylor’s formula. The o-notation
[*] Applications to indeterminate forms
[*] Exercises
[*] L’Hôpital’s rule for the indeterminate form [itex]0/0[/itex]
[*] Exercises
[*] The symbols [itex]+\infty[/itex] and [itex]-\infty[/itex]. Extension of L’Hôpital’s rule
[*] Infinite limits
[*] The behavior of [itex]\log x[/itex] and [itex]e^x[/itex] for large [itex]x[/itex]
[*] Exercises
[/LIST]
[*] Introduction to Differential Equations
[LIST]
[*] Introduction
[*] Terminology and notation
[*] A first-order differential equation for the exponential function
[*] First-order linear differential equations
[*] Exercises
[*] Some physical problems leading to first-order linear differential equations
[*] Exercises
[*] Linear equations of second order with constant coefficients
[*] Existence of solutions of the equation [itex]y^{\prime\prime}+by=0[/itex]
[*] Reduction of the general equation to the special case [itex]y^{\prime\prime}+by=0[/itex]
[*] Uniqueness theorem for the equation [itex]y^{\prime\prime}+by=0[/itex]
[*] Complete solution of the equation [itex]y^{\prime\prime}+by=0[/itex]
[*] Complete solution of the equation [itex]y^{\prime\prime}+ay^\prime+by=0[/itex]
[*] Exercises
[*] Nonhomogeneous linear equations of second order with constant coefficients
[*] Special methods for determining a particular solution of the nonhomogeneous equation [itex]y^{\prime\prime}+ay^\prime+by=R[/itex]
[*] Exercises
[*] Examples of physical problems leading to linear second-order equations with constant coefficients
[*] Exercises
[*] Remarks concerning nonlinear differential equations
[*] Integral curves and direction fields
[*] Exercises
[*] First-order separable equations
[*] Exercises
[*] Homogeneous first-order equations
[*] Exercises
[*] Some geometrical and physical problems leading to first-order equations
[*] Miscellaneous review exercises
[/LIST]
[*] Complex Numbers
[LIST]
[*] Historical introduction
[*] Definitions and field properties
[*] The complex numbers as an extension of the real numbers
[*] The imaginary unit [itex]i[/itex]
[*] Geometric interpretation. Modulus and argument
[*] Exercises
[*] Complex exponentials
[*] Complex-valued functions
[*] Examples of differentiation and integration formulas
[*] Exercises
[/LIST]
[*] Sequences, Infinite Series, Improper Integrals
[LIST]
[*] Zeno’s paradox
[*] Sequences
[*] Monotonic sequences of real numbers
[*] Exercises
[*] Infinite series
[*] The linearity property of convergent series
[*] Telescoping series
[*] The geometric series
[*] Exercises
[*] Exercises on decimal expansions
[*] Tests for convergence
[*] Comparison tests for series of nonnegative terms
[*] The integral test
[*] Exercises
[*] The root test and the ratio test for series of nonnegative terms
[*] Exercises
[*] Alternating series
[*] Conditional and absolute convergence
[*] The convergence tests of Dirichlet and Abel
[*] Exercises
[*] Rearrangements of series
[*] Miscellaneous review exercises
[*] Improper integrals
[*] Exercises
[/LIST]
[*] Sequences and Series of Functions
[LIST]
[*] Pointwise convergence of sequences of functions
[*] Uniform convergence of sequences of functions
[*] Uniform convergence and continuity
[*] Uniform convergence and integration
[*] A sufficient condition for uniform convergence
[*] Power series. Circle of convergence
[*] Exercises
[*] Properties of functions represented by real power series
[*] The Taylor’s series generated by a function
[*] A sufficient condition for convergence of a Taylor’s series
[*] Power-series expansions for the exponential and trigonometric functions
[*] Bernstein’s theorem
[*] Exercises
[*] Power series and differential equations
[*] The binomial series
[*] Exercises
[/LIST]
[*] Vector Algebra
[LIST]
[*] Historical introduction
[*] The vector space of n-tuples of real numbers
[*] Geometric interpretation for [itex]n \leq 3[/itex]
[*] Exercises
[*] The dot product
[*] Length or norm of a vector
[*] Orthogonality of vectors
[*] Exercises
[*] Projections. Angle between vectors in n-space
[*] The unit coordinate vectors
[*] Exercises
[*] The linear span of a finite set of vectors
[*] Linear independence
[*] Bases
[*] Exercises
[*] The vector space [itex]V_n(\mathbb{C})[/itex] of n-tuples of complex numbers
[*] Exercises
[/LIST]
[*] Applications of Vector Algebra to Analytic Geometry
[LIST]
[*] Introduction
[*] Lines in n-space
[*] Some simple properties of straight lines
[*] Lines and vector-valued functions
[*] Exercises
[*] Planes in Euclidean n-space
[*] Planes and vector-valued functions
[*] Exercises
[*] The cross product
[*] The cross product expressed as a determinant
[*] Exercises
[*] The scalar triple product
[*] Cramer’s rule for solving a system of three linear equations
[*] Exercises
[*] Normal vectors to planes
[*] Linear Cartesian equations for planes
[*] Exercises
[*] The conic sections
[*] Eccentricity of conic sections
[*] Polar equations for conic sections
[*] Exercises
[*] Conic sections symmetric about the origin
[*] Cartesian equations for the conic sections
[*] Exercises
[*] Miscellaneous exercises on conic sections
[/LIST]
[*] Calculus of Vector-values Functions
[LIST]
[*] Vector-valued functions of a real variable
[*] Algebraic operations. Components
[*] Limits, derivatives, and integrals
[*] Exercises
[*] Applications to curves. Tangency
[*] Applications to curvilinear motion. Velocity, speed, and acceleration
[*] Exercises
[*] The unit tangent, the principal normal, and the osculating plane of a curve
[*] Exercises
[*] The definition of arc length
[*] Additivity of arc length
[*] The arc-length function
[*] Exercises
[*] Curvature of a curve
[*] Exercises
[*] Velocity and acceleration in polar coordinates
[*] Plane motion with radial acceleration
[*] Cylindrical coordinates
[*] Exercises
[*] Applications to planetary motion
[*] Miscellaneous review exercises
[/LIST]
[*] Linear Spaces
[LIST]
[*] Introduction
[*] The definition of a linear space
[*] Examples of linear spaces
[*] Elementary consequences of the axioms
[*] Exercises
[*] Subspaces of a linear space
[*] Dependent and independent sets in a linear space
[*] Bases and dimension
[*] Exercises
[*] Inner products, Euclidean norms spaces,
[*] Orthogonality in a Euclidean space
[*] Exercises
[*] Construction of orthogonal sets. The Gram-Schmidt process
[*] Orthogonal complements. Projections
[*] Best approximation of elements in a Euclidean space by elements in a finite-dimensional subspace
[*] Exercises
[/LIST]
[*] Linear Transformations and Matrices
[LIST]
[*] Linear transformations
[*] Null space and range
[*] Nullity and rank
[*] Exercises
[*] Algebraic operations on linear transformations
[*] Inverses
[*] One-to-one linear transformations
[*] Exercises
[*] Linear transformations with prescribed values
[*] Matrix representations of linear transformations
[*] Construction of a matrix representation in diagonal form
[*] Exercises
[*] Linear spaces of matrices
[*] Isomorphism between linear transformations and matrices
[*] Multiplication of matrices
[*] Exercises
[*] Systems of linear equations
[*] Computation techniques
[*] Inverses of matrices square
[*] Exercises
[*] Miscellaneous exercises on matrices
[/LIST]
[*] Answers to exercises
[*] Index
[/LIST]
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