Apostol Calculus: Historical Intro, Concepts & Applications

[micromass]

• 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:

[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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Table of Contents of Volume II:

[LIST]
[*] Linear Analysis
[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
[*] Components
[*] Exercises
[*] Inner products, Euclidean spaces. Norms
[*] 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
[*] 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 square matrices
[*] Exercises
[*] Miscellaneous exercises on matrices
[/LIST]
[*] Determinants
[LIST]
[*] Introduction
[*] Motivation for the choice of axioms for a determinant function
[*] A set of axioms for a determinant function
[*] Computation of determinants
[*] The uniqueness theorem
[*] Exercises
[*] The product formula for determinants
[*] The determinant of the inverse of a nonsingular matrix
[*] Determinants and independence of vectors
[*] The determinant of a block-diagonal matrix
[*] Exercise
[*] Expansion formulas for determinants. Minors and cofactors
[*] Existence of the determinant function
[*] The determinant of a transpose
[*] The cofactor matrix
[*] Cramer's rule
[*] Exercises
[/LIST]
[*] Eigenvalues and Eigenvectors
[LIST]
[*] Linear transformations with diagonal matrix representations
[*] Eigenvectors and eigenvalues of a linear transformation
[*] Linear independence of eigenvectors corresponding to distinct eigenvalues
[*] Exercises
[*] The finite-dimensional case. Characteristic polynomials
[*] Calculation of eigenvalues and eigenvectors in the finite-dimensional case
[*] Trace of a matrix
[*] Exercises
[*] Matrices representing the same linear transformation. Similar matrices
[*] Exercises
[/LIST]
[*] Eigenvalues of Operators Acting on Euclidean Spaces
[LIST]
[*] Eigenvalues and inner products
[*] Hermitian and skew-Hermitian transformations
[*] Eigenvalues and eigenvectors of Hermitian and skew-Hermitian operators
[*] Orthogonality of eigenvectors corresponding to distinct eigenvalues
[*] Exercises
[*] Existence of an orthonormal set of eigenvectors for Hermitian and skew-Hermitian operators acting on finite-dimensional spaces
[*] Matrix representations for Hermitian and skew-Hermitian operators
[*] Hermitian and skew-Hermitian matrices. The adjoint of a matrix
[*] Diagonalization of a Hermitian or skew-Hermitian matrix
[*] Unitary matrices. Orthogonal matrices
[*] Exercise
[*] Quadratic forms
[*] Reduction of a real quadratic form to a diagonal form
[*] Applications to analytic geometry
[*] Exercises
[*] Eigenvalues of a symmetric transformation obtained as values of its quadratic form
[*] Extremal properties of eigenvalues of a symmetric transformation
[*] The finite-dimensional case
[*] Unitary transformations
[*] Exercises
[/LIST]
[*] Linear Differential Equations
[LIST]
[*] Historical introduction
[*] Review of results concerning linear equations of first and second orders
[*] Exercises
[*] Linear differential equations of order n
[*] The existence-uniqueness theorem
[*] The dimension of the solution space of a homogeneous linear equation
[*] The algebra of constant-coefficient operators
[*] Determination of a basis of solutions for linear equations with constant coefficients by factorization of operators
[*] Exercises
[*] The relation between the homogeneous and nonhomogeneous equations
[*] Determination of a particular solution of the nonhomogeneous equation. The method of variation of parameters
[*] Nonsingularity of the Wronskian matrix of n independent solutions of a homogeneous linear equation
[*] Special methods for determining a particular solution of the nonhomogeneous equation. Reduction to a system of first-order linear equations
[*] The annihilator method for determining a particular solution of the nonhomogeneous equation
[*] Exercises
[*] Miscellaneous exercises on linear differential equations
[*] Linear equations of second order with analytic coefficients
[*] The Legendre equation 
[*] The Legendre polynomials
[*] Rodrigues' formula for the Legendre polynomials
[*] Exercises
[*] The method of Frobenius
[*] The Bessel equation
[*] Exercises
[/LIST]
[*] Systems of Differential Equations
[LIST]
[*] Introduction
[*] Calculus of matrix functions
[*] Infinite series of matrices. Norms of matrices
[*] Exercises
[*] The exponential matrix
[*] The differential equation satisfied by [itex]e^{tA}[/itex]
[*] Uniqueness theorem for the matrix differential equation F'{t) = AF(t)
[*] The law of exponents for exponential matrices
[*] Existence and uniqueness theorems for homogeneous linear systems with constant coefficients
[*] The problem of calculating [itex]e^{tA}[/itex]
[*] The Cayley-Hamilton theorem
[*] Exercises
[*] Putzer's method for calculating [itex]e^{tA}[/itex]
[*] Alternate methods for calculating [itex]e^{tA}[/itex] in special cases
[*] Exercises
[*] Nonhomogeneous linear systems with constant coefficients
[*] Exercises
[*] The general linear system Y'(t) = P(t)Y(t) + Q(t)
[*] A power-series method for solving homogeneous linear systems
[*] Exercises
[*] Proof of the existence theorem by the method of successive approximations
[*] The method of successive approximations applied to first-order nonlinear systems
[*] Proof of an existence-uniqueness theorem for first-order nonlinear systems
[*] Exercises
[*] Successive approximations and fixed points of operators
[*] Normed linear spaces
[*] Contraction operators
[*] Fixed-point theorem for contraction operators
[*] Applications of the fixed-point theorem
[/LIST]
[/LIST]
[*] Nonlinear Analysis
[LIST]
[*] Differential Calculus of Scalar and Vector Fields
[LIST]
[*] Functions from Rn to Rm. Scalar and vector fields
[*] Open balls and open sets
[*] Exercises
[*] Limits and continuity
[*] Exercises
[*] The derivative of a scalar field with respect to a vector
[*] Directional derivatives and partial derivatives
[*] Partial derivatives of higher order
[*] Exercises
[*] Directional derivatives and continuity
[*] The total derivative
[*] The gradient of a scalar field
[*] A sufficient condition for differentiability
[*] Exercises
[*] A chain rule for derivatives of scalar fields
[*] Applications to geometry. Level sets. Tangent planes
[*] Exercises
[*] Derivatives of vector fields
[*] Differentiability implies continuity
[*] The chain rule for derivatives of vector fields
[*] Matrix form of the chain rule
[*] Exercises
[*] Sufficient conditions for the equality of mixed partial derivatives
[*] Miscellaneous exercises
[/LIST]
[*] Applications of the Differential Calculus
[LIST]
[*] Partial differential equations
[*] A first-order partial differential equation with constant coefficients
[*] Exercises
[*] The one-dimensional wave equation
[*] Exercises
[*] Derivatives of functions defined implicitly
[*] Worked examples
[*] Exercises
[*] Maxima, minima, and saddle points
[*] Second-order Taylor formula for scalar fields
[*] The nature of a stationary point determined by the eigenvalues of the Hessian matrix
[*] Second-derivative test for extrema of functions of two variables
[*] Exercises
[*] Extrema with constraints. Lagrange's multipliers
[*] Exercises
[*] The extreme-value theorem for continuous scalar fields
[*] The small-span theorem for continuous scalar fields (uniform continuity)
[/LIST]
[*] Line Integrals
[LIST]
[*] Introduction
[*] Paths and line integrals
[*] Other notations for line integrals
[*] Basic properties of line integrals
[*] Exercises
[*] The concept of work as a line integral
[*] Line integrals with respect to arc length 
[*] Further applications of line integrals
[*] Exercises
[*] Open connected sets. Independence of the path
[*] The second fundamental theorem of calculus for line integrals
[*] Applications to mechanics
[*] Exercises
[*] The first fundamental theorem of calculus for line integrals
[*] Necessary and sufficient conditions for a vector field to be a gradient
[*] Necessary conditions for a vector field to be a gradient
[*] Special methods for constructing potential functions
[*] Exercises
[*] Applications to exact differential equations of first order
[*] Exercises
[*] Potential functions on convex sets
[/LIST]
[*] Multiple Integrals
[LIST]
[*] Introduction
[*] Partitions of rectangles. Step functions
[*] The double integral of a step function
[*] The definition of the double integral of a function defined and bounded on a rectangle
[*] Upper and lower double integrals
[*] Evaluation of a double integral by repeated one-dimensional integration
[*] Geometric interpretation of the double integral as a volume
[*] Worked examples
[*] Exercises
[*] Integrability of continuous functions
[*] Integrability of bounded functions with discontinuities
[*] Double integrals extended over more general regions
[*] Applications to area and volume
[*] Worked examples
[*] Exercises
[*] Further applications of double integrals
[*] Two theorems of Pappus
[*] Exercises
[*] Green's theorem in the plane
[*] Some applications of Green's theorem
[*] A necessary and sufficient condition for a two-dimensional vector field to be a gradient
[*] Exercises
[*] Green's theorem for multiply connected regions
[*] The winding number
[*] Exercises
[*] Change of variables in a double integral
[*] Special cases of the transformation formula
[*] Exercises
[*] Proof of the transformation formula in a special case
[*] Proof of the transformation formula in the general case
[*] Extensions to higher dimensions
[*] Change of variables in an n-fold integral
[*] Worked examples
[*] Exercises
[/LIST]
[*] Surface Integrals
[LIST]
[*] Parametric representation of a surface
[*] The fundamental vector product
[*] The fundamental vector product as a normal to the surface
[*] Exercises
[*] Area of a parametric surface
[*] Exercises
[*] Surface integrals
[*] Change of parametric representation
[*] Other notations for surface integrals
[*] Exercises
[*] The theorem of Stokes
[*] The curl and divergence of a vector field
[*] Exercises
[*] Further properties of the curl and divergence
[*] Exercises
[*] Reconstruction of a vector field from its curl
[*] Exercises
[*] Extensions of Stokes' theorem
[*] The divergence theorem (Gauss' theorem)
[*] Applications of the divergence theorem
[*] Exercises 
[/LIST]
[/LIST]
[*] Special Topics
[LIST]
[*] Set Functions and Elementary Probability
[LIST]
[*] Historical introduction
[*] Finitely additive set functions
[*] Finitely additive measures
[*] Exercises
[*] The definition of probability for finite sample spaces
[*] Special terminology peculiar to probability theory
[*] Exercises
[*] Worked examples
[*] Exercises
[*] Some basic principles of combinatorial analysis
[*] Exercises
[*] Conditional probability
[*] Independence
[*] Exercises
[*] Compound experiments
[*] Bernoulli trials
[*] The most probable number of successes in n Bernoulli trials
[*] Exercises
[*] Countable and uncountable sets
[*] Exercises
[*] The definition of probability for countably infinite sample spaces
[*] Exercises
[*] Miscellaneous exercises on probability
[/LIST]
[*] Calculus of Probabilities
[LIST]
[*] The definition of probability for uncountable sample spaces
[*] Countability of the set of points with positive probability
[*] Random variables
[*] Exercises
[*] Distribution functions
[*] Discontinuities of distribution functions
[*] Discrete distributions. Probability mass functions
[*] Exercises
[*] Continuous distributions. Density functions
[*] Uniform distribution over an interval
[*] Cauchy's distribution
[*] Exercises
[*] Exponential distributions
[*] Normal distributions
[*] Remarks on more general distributions
[*] Exercises
[*] Distributions of functions of random variables
[*] Exercises
[*] Distributions of two-dimensional random variables
[*] Two-dimensional discrete distributions
[*] Two-dimensional continuous distributions. Density functions
[*] Exercises
[*] Distributions of functions of two random variables
[*] Exercises
[*] Expectation and variance
[*] Expectation of a function of a random variable
[*] Exercises
[*] Chebyshev's inequality
[*] Laws of large numbers
[*] The central limit theorem of the calculus of probabilities
[*] Exercises
[*] Suggested References
[/LIST]
[*] Introduction to Numerical Analysis
[LIST]
[*] Historical introduction
[*] Approximations by polynomials
[*] Polynomial approximation and normed linear spaces
[*] Fundamental problems in polynomial approximation
[*] Exercises
[*] Interpolating polynomials
[*] Equally spaced interpolation points
[*] Error analysis in polynomial interpolation
[*] Exercises
[*] Newton's interpolation formula
[*] Equally spaced interpolation points. The forward difference operator
[*] Factorial polynomials
[*] Exercises
[*] A minimum problem relative to the max norm
[*] Chebyshev polynomials
[*] A minimal property of Chebyshev polynomials
[*] Application to the error formula for interpolation
[*] Exercises
[*] Approximate integration. The trapezoidal rule
[*] Simpson's rule
[*] Exercises
[*] The Euler summation formula
[*] Exercises
[/LIST]
[/LIST]
[*] Suggested References
[*] Answers to exercises
[*] Index
[/LIST]

 

[micromass]

This book is a very good treatise on calculus. It is as rigorous as Spivak on many accounts. The main difference between Apostol and Spivak seems to be that Apostol's exercises are more computational (but still quite hard), while Spivak is more theoretical in many aspects. A good grasp of high-school mathematics is a must and I also advise having seen a bit of calculus already.

 

[mathwonk]

I found Apostol more scholarly and Spivak more fun. Both are excellent.

 

[Vargo]

Endurance required.

 

[IGU]

My kid self-studied calculus using Apostol. He had not been exposed to the subject before, so he didn't have anything to compare it to. He loved it, especially for the rigor of the proofs. Volume I took him about nine months, working all the interesting problems and some subset of the ones meant for practice. Volume II took another six months.

Highly recommended for people who love math and are willing to work at it.

-IGU-

 

[mathwonk]

that is fast.

 

[mathnerd15]

I really love Apostol, it is at a higher level than Stewart

 

[mathwonk]

if you think you have found one ask specifically about it. otherwise your question has little value, no matter what the answer is.

 

[mathnerd15]

I was just wondering about these indices in the Apostol copy I have. so it includes an introduction to metric spaces and topology?

 

[NathanaelNolk]

Hey guys I was wondering, is the second volume of Calculus a good book ? I really enjoyed the first one, and I was wondering if the second book was as good as the first one. Btw, does somebody know if the lectures from Edward Frenkel (UC Berkeley) on multivariable calculus could go well with the book ?

 

[variation32]

Nine months for Apostol Book 1? Gulp.

I started last June and am only on page 110 (or thereabouts). But then again, I am slogging through it from an evidently much lower maths basis, and very carefully writing up the entire experience in wat is so far 3 volumes of needlessly beautiful workbooks, black ink for Tom and his explanations/ exercises, blue ink for my interruptions, sideways wanders into related (or not) territory and exercise answers).

I mark my own work and award stars sparingly. I was once so impressed by myself that I gave myself a star and... a Saturn! What a lovely surprise that was.

I imagine the full two volumes might end up taking me five years. But it will have been worth every minute - almost every Sunday I can feel assured that I understand something that had me uttterly baffled on the Monday just gone.

 

[IGU]

Kudos! Sounds like you're learning it right.