1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Calculus Mathematical Analysis by Apostol

  1. Strongly Recommend

  2. Lightly Recommend

  3. Lightly don't Recommend

    0 vote(s)
  4. Strongly don't Recommend

    0 vote(s)
  1. Jan 22, 2013 #1

    Table of Contents:
    Code (Text):

    [*] The Real and Complex Number Systems
    [*] Introduction
    [*] The field axioms
    [*] The order axioms
    [*] Geometric representation of real numbers
    [*] Intervals
    [*] Integers
    [*] Tim unique factorization theorem for integers
    [*] Rational numbers
    [*] Irrational numbers
    [*] Upper bounds, maximum element, least upper bound (supremum)
    [*] The completeness
    [*] Some properties of the supremum
    [*] Properties of the integer deduced from the completeness axiom
    [*] The Archimedean property of the real-number system
    [*] Rational numbers with finite decimal representation
    [*] Finite decimal approximations to real mumbers
    [*] Infinite decimal representation of real numbers
    [*] Absolute values and the triangle inequality
    [*] The Cauchy-Schwarz inequality
    [*] Plus and minus infinity and the extended real number system R*
    [*] Complex numbers
    [*] Geometric representation of complex numbers
    [*] The imaginary unit
    [*] Absolute value of a complex number
    [*] Impossibility of ordering the complex numbers
    [*] Complex exponentials
    [*] Further properties of complex exponentials
    [*] The argument of a complex number
    [*] Integral powers and roots of complex numbers
    [*] Complex logarithms
    [*] Complex powers
    [*] Complex sines and cosines
    [*] Infinity and the extended complex plane C*
    [*] Exercises
    [*] Some Basic Notions of Set Theory
    [*] Introduction
    [*] Notations
    [*] Ordered pairs
    [*] Certesian product of two sets
    [*] Relations and functions
    [*] Further terminology concerning functions
    [*] One-to-one functions and inverses
    [*] Composite functions
    [*] Sequences
    [*] Similar (equinumerous) sets
    [*] Finite and infinite sets
    [*] Countable and uncountable sets
    [*] Uncountability of the real-number system
    [*] Set algebra
    [*] Countable collection of countable sets
    [*] Exercises
    [*] Elements of Point Set Topology
    [*] Introduction
    [*] Euclidean space [itex]R^n[/itex]
    [*] Open balls and open sets in [itex]R^n[/itex]
    [*] The structure of open sets in [itex]R^1[/itex]
    [*] Closed sets
    [*] Adherent points. Accumulation points
    [*] Closed sets and adherent points
    [*] Bolzano-Weierstrass theorem
    [*] The Cantor intersection theorem
    [*] The Lindelof covering theorem
    [*] The Heine-Borel covering theorem
    [*] Compactness in [itex]R^n[/itex]
    [*] Metric spaces
    [*] Point set topology in metric spaces
    [*] Compact subsets of a metric space
    [*] Boundary of a set
    [*] Exercises
    [*] Limits and Continuity
    [*] Introduction
    [*] Convergent sequences in a metric space
    [*] Cauchy sequences
    [*] Complete metric spaces
    [*] Limit of a function
    [*] Limits of complex-valued functions
    [*] Limits of vector-valued functions
    [*] Continuous functions
    [*] Continuity of composite functions
    [*] Continuous complex-valued and vector-valued functions
    [*] Examples of continuous functions
    [*] Continuity and inverse images of open or closed sets
    [*] Functions continuous on compact sets
    [*] Topological mappings (homeomorphisms)
    [*] Bolzano's theorem
    [*] Connectedness
    [*] Components of a metric space
    [*] Arcwise connectedness
    [*] Uniform continuity
    [*] Uniform continuity and compact sets
    [*] Fixed-point theorem for contractions
    [*] Discontinuities of real-valued functions
    [*] Monotonic functions
    [*] Exercises
    [*] Derivatives
    [*] Introduction
    [*] Definition of derivative
    [*] Derivatives and continuity
    [*] Algebra of derivatives
    [*] The chain rule
    [*] One-sided derivatives and infinite derivatives
    [*] Functions with nonzero derivative
    [*] Zero derivatives and local extrema
    [*] Rolle's theorem
    [*] The Mean-Value Theorem for derivatives
    [*] Intermediate-value theorem for derivatives
    [*] Taylor's formula with remainder
    [*] Derivatives of vector-valued functions
    [*] Partial derivatives
    [*] Differentiation of functions of a complex variable
    [*] The Cauchy-Riemann equations
    [*] Exercises
    [*] Functions of Bounded Variation and Rectifiable Curves
    [*] Introduction
    [*] Properties of monotonic functions
    [*] Functions of bounded variation
    [*] Total variation
    [*] Additive property of total variation
    [*] Total variation on [a,x] as a function of x
    [*] Functions of bounded variation expressed as the difference of bounded functions
    [*] Continuous functions of bounded variation
    [*] Curves and paths
    [*] Rectifiable paths and arc length
    [*] Additive and continuity properties of arc length
    [*] Equivalence of paths. Change of parameter
    [*] Exercises
    [*] The Riemann-Stieltjes Integral
    [*] Introduction
    [*] Notation
    [*] The definition of the Riemann-Stieltjes integral
    [*] Linear properties
    [*] Integration by parts
    [*] Change of variable in a Riemann-Stieltjes integral
    [*] Reduction to a Riemann integral
    [*] Step functions as integrators
    [*] Reduction of a Riemann-Stieltjes integral to a finite sum
    [*] Euler's summation formula
    [*] Monotonically increasing integrators. Upper and lower integrals
    [*] Additive and linearity properties of upper and lower integrals
    [*] Riemann's condition
    [*] Comparison theorems
    [*] Integrators of bounded variation
    [*] Sufficient conditions for existence of Riemann-Stieltjes integrals
    [*] Necessary conditions for existence of Riemann-Stieltjes integrals
    [*] Mean Value Theorems for Riemann-Stieltjes integrals
    [*] The integral as a function of the interval
    [*] Second fundamental theorem of integral calculus
    [*] Change of variable in a Riemann integral
    [*] Second Mean-Value Theorem for Riemann integrals
    [*] Riemann-Stieltjes integrals depending on a parameter
    [*] Differentiation under the integral sign
    [*] Interchanging the order of integration
    [*] Lebesgue's criterion for existence of Riemann integrals
    [*] Complex-valued Riemann-Stieltjes integrals
    [*] Exercises
    [*] Infinite Series and Infinite Products
    [*] Introduction
    [*] Convergent and divergent sequences of complex numbers
    [*] Limit superior and limit inferior of a real-valued sequence
    [*] Monotonic sequences of real numbers
    [*] Infinite series
    [*] Inserting and removing parentheses
    [*] Alternating series
    [*] Absolute and conditional convergence
    [*] Real and imaginary parts of a complex series
    [*] Tests for convergence of series with positive terms
    [*] The geometric series
    [*] The integral test
    [*] The big oh and little oh notation
    [*] The ratio test and the root test
    [*] Dirichlet's test and Abel's test
    [*] Partial sums of the geometric series [itex]\sum z^n[/itex] on the unit circle |z|=1
    [*] Rearrangements of series
    [*] Riemann's theorem on conditionally convergent series
    [*] Subseries
    [*] Double sequences
    [*] Double series
    [*] Rearrangement theorem for double series
    [*] A sufficient condition for equality of iterated series
    [*] Multiplication of series
    [*] Cesaro summability
    [*] Infinite products
    [*] Euler's product for the Riemann zeta function
    [*] Exercises
    [*] Sequences of Functions
    [*] Pointwise convergence of sequences of functions
    [*] Examples of sequences of real-values functions
    [*] Definition of uniform convergence
    [*] Uniform concergence and continuity
    [*] The Cauchy condition for uniform convergene
    [*] Uniform convergence of infinite series of functions
    [*] A space-filling curve
    [*] Uniform convergence and Riemann-Stieltjes integration
    [*] Nonuniformly convergent sequences that can be integrated term by
    [*] Uniform convergence and differentiation
    [*] Sufficient conditions for uniform convergence of a series
    [*] Uniform convergence and double sequences
    [*] Mean convergence
    [*] Power series
    [*] Multiplication of power series
    [*] The substitution theorem
    [*] Reciprocal of a power series
    [*] Real power series
    [*] The Taylor's series generated by a function
    [*] Bernstein's theorem
    [*] The binomial series
    [*] Abel's limit theorem
    [*] Tauber's theorem
    [*] Exercises
    [*] The Lebesgue Integral
    [*] Introduction
    [*] The integral of a step function
    [*] Monotonic sequences of step function
    [*] Upper function and their integrals
    [*] Riemann-integrable functions as eexamples of upper functions
    [*] The class of Lebesgue-integrable functions on a general interval
    [*] Basic properties of the Lebesgue integral
    [*] Lebesgue integration and sets of measure zero
    [*] The Levi monotone convergence theorems
    [*] Applicatiom of Lebesgue's dominated convergence theorem
    [*] Lebesgue integrals on unbounded intervals as limits of integrals on bounded intervals
    [*] Improper Riemann integrals
    [*] Measurable functions
    [*] Continuity of functions defined by Lebesgue integrals
    [*] Differentiation under the integral sign
    [*] Interchanging the order of integration
    [*] Measurable sets on the real line
    [*] The Lebesgue integral over arbitrary subsets of R
    [*] Lebesgue integrals of complex-valued functions
    [*] Inner products and norms
    [*] The set [itex]L^2(I)[/itex] of square-integrable functions
    [*] The set [itex]L^2(I)[/itex] as a semimetric space
    [*] A convergence theorem for series of functions in [itex]L^2(I)[/itex]
    [*] The Riesz-Fischer theorem
    [*] Exercises
    [*] Fourier Series and Fourier Integrals
    [*] Introduction
    [*] Orthogonal systems of functions
    [*] The theorem on best approximation
    [*] The Fourier series of a function relative to an orthonormal system
    [*] Properties of the Fourier coefficients
    [*] The Riesz-Fischer theorem
    [*] The convergence and representation problems for trigonometric series
    [*] The Riemann-Lebesgue lemma
    [*] The Dirichlet integrals
    [*] An integral representation for the partial sums of a Fourier series
    [*] Riemann's localization theorem
    [*] Sufficient conditions for convergence of a Fourier series at a particular point
    [*] Cesaro summability of Fourier series
    [*] Consequence, of Fejer's theorem
    [*] The Weierstrass approximation theorem
    [*] Other forms of Fourier series
    [*] The Fourier integral theorem
    [*] The exponential form of the Fourier integral theorem
    [*] Integral transforms
    [*] Convolutions
    [*] The convolution theorem for Fourier transforms
    [*] The Poisson summation formula
    [*] Exercises
    [*] Multivariable Differential Calculus
    [*] Introduction
    [*] The directional derivative
    [*] Directional derivatives and continuity
    [*] The total derivative
    [*] The total derivative expressed in terms of partial derivatives
    [*] An application to complex-values functions
    [*] The matrix of linear function
    [*] The Jacobian matrix
    [*] The chain rule
    [*] Matrix form of the chain rule
    [*] The Mean-Value Theorem for differentiable functions
    [*] A sufficient condition for differentiability
    [*] A sufficient condition for equality of mixed partial derivatives
    [*] Taylor's formula for functions from [itex]R^n[/itex] to [itex]R^1[/itex]
    [*] Exercises
    [*] Implicit Functions and Extremum Problems
    [*] Introduction
    [*] Functions with nonzero Jacobian determinant
    [*] The inverse function theorem
    [*] The implicit function theorem
    [*] Extrema of real-valued functions of one variable
    [*] Extrema of real-valued functions of several variables
    [*] Extremum problems with side conditions
    [*] Exercises
    [*] Multiple Riemann Integrals
    [*] Introduction
    [*] The measure of a bounded interval in [itex]R^n[/itex]
    [*] The Riemann integral of a bounded function defined on a compact interval in [itex]R^n[/itex]
    [*] Sets of measure zero and Lebesgue's criterion for existence of a multiple Riemann integral
    [*] Evaluation of a multiple integral by iterated integration
    [*] Jordan-measurable sets in [itex]R^n[/itex]
    [*] Multiple integration over Jordan-measurable sets
    [*] Additive property of the Riemann integral
    [*] Mean-Value Theorem for multiple integrals
    [*] Exercises
    [*] Multiple Lebesgue Integrals
    [*] Introduction
    [*] Step functions and their integrals
    [*] Upper functions end Lebesgue-integrable functions
    [*] Measurable functions and measurable sets in [itex]R^n[/itex]
    [*] Fubini's reduction theorem for double integrals
    [*] Some properties of sets of measure zero
    [*] Fubini's reduction theorem for double integrals
    [*] The Tonelli-Hobson test for integrability
    [*] Coordinate transformations
    [*] The transformation formula for multiple integrals
    [*] Proof of the transformation formula for linear coordinate transformations
    [*] Proof of the transformation formula for the characteristic function of a compact cube
    [*] Completion of the proof of the transformation formula
    [*] Exercises
    [*] Cauchy's Theorem and the Residue Calculus
    [*] Analytic functions
    [*] Paths and curves in the complex plane
    [*] Contour integrals
    [*] The integral along a circular path as a function of the radius
    [*] Cauchy's integral theorem for a circle
    [*] Homotopic curves
    [*] Invariance of contour integrals under homotopy
    [*] General form of Cauchy's integral theorem
    [*] Cauchy's integral formula
    [*] The winding number of a circuit with respect to a point
    [*] The unboundedness of the set of points with winding number zero
    [*] Analytic functions defined by contour integrals
    [*] Power-series expansions for analytic functions
    [*] Cauchy's inequalities. Liouville's theorem
    [*] Isolation of the zeros of an analytic function
    [*] The identity theorem for analytic functions
    [*] The maximum and minimum modulus of an analytic function
    [*] The open mapping theorem
    [*] Laurent expansions for functions analytic in an annulus
    [*] Isolated singularities
    [*] The residue of a function at an isolated singular point
    [*] The Cauchy residue theorem
    [*] Counting zeros and poles in a region
    [*] Evaluation of real-valued integrals by means of residues
    [*] Evaluation of Gauss's sum by residue calculus
    [*] Application of the residue theorem to the inversion formula for Laplace transforms
    [*] Conformal mappings
    [*] Exercises
    [*] Index of Special Symbols
    [*] Index
    Last edited by a moderator: May 6, 2017
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted