Book:David V. Widder/Advanced Calculus

Subject Matter

 * Calculus

=== Contents ===

Preface to the First Edition

Preface to the Second Edition

1 Partial Differentiation
$$\S 1.$$ Introduction
 * 1.1: Partial derivatives
 * 1.2: Implicit functions
 * 1.3: Higher order derivatives

$$\S 2.$$ Functions of One Variable
 * 2.1: Limits and continuity
 * 2.2: Derivatives
 * 2.3: Rolle's theorem
 * 2.4: Law of the mean

$$\S 3.$$ Functions of Several Variables
 * 3.1: Limits and continuity
 * 3.2: Derivatives
 * 3.3: A basic mean value theorem
 * 3.4: Composite functions
 * 3.5: Further cases
 * 3.6: Differentiable functions

$$\S 4.$$ Homogeneous Functions. Higher Derivatives
 * 4.1: Definition of homogeneous functions
 * 4.2: Euler's theorem
 * 4.3: Higher derivatives

$$\S 5.$$ Implicit Functions
 * 5.1: Differentiation of implicit functions
 * 5.2: Other cases
 * 5.3: Higher derivatives

$$\S 6.$$ Simultaneous Equations. Jacobians
 * 6.1: Two equations in two unknowns
 * 6.2: Jacobians
 * 6.3: Further cases
 * 6.4: The inverse of a transformation

$$\S 7.$$ Dependent and Independent Variables
 * 7.1: First illustration
 * 7.2: Second illustration
 * 7.3: Third illustration

$$\S 8.$$ Differentials. Directional Derivatives
 * 8.1: The differential
 * 8.2: Meaning of the differential
 * 8.3: Directional derivatives
 * 8.4: The gradient

$$\S 9.$$ Taylor's Theorem
 * 9.1: Functions of a single variable
 * 9.2: Functions of two variables

$$\S 10.$$ Jacobians
 * 10.1: Implicit functions
 * 10.2: The inverse of a transformation
 * 10.3: Change of variable

$$\S 11.$$ Equality of Cross Derivatives
 * 11.1: A preliminary result
 * 11.2: The principal result
 * 11.3: An example

$$\S 12.$$ Implicit Functions
 * 12.1: The existence theorem
 * 12.2: Functional dependence
 * 12.3: A criterion for functional dependence
 * 12.4: Simultaneous equations

2 Vectors
$$\S 1.$$ Introduction
 * 1.1: Definition of a vector
 * 1.2: Algebra of vectors
 * 1.3: Properties of the operations
 * 1.4: Sample vector calculations

$$\S 2.$$ Solid Analytic Geometry
 * 2.1: Syllabus for solid geometry
 * 2.2: Comments on the syllabus
 * 2.3: Vector applications

$$\S 3.$$ Space Curves
 * 3.1: Examples of curves
 * 3.2: Specialized curves

$$\S 4.$$ Surfaces
 * 4.1: Examples of surfaces
 * 4.2: Specialized surfaces

$$\S 5.$$ A Symbolic Vector
 * 5.1: Definition of $$\vec \nabla$$
 * 5.2: Directional derivatives
 * 5.3: Meaning of the gradient

$$\S 6.$$ Invariants
 * 6.1: Change of axes
 * 6.2: Invariance of inner product
 * 6.3: Invariance of outer product

3 Differential Geometry
$$\S 1.$$ Arc Length of a Space Curve
 * 1.1: An integral formula for arc length
 * 1.2: Tangent to a curve

$$\S 2.$$ Osculating Plane
 * 2.1: Zeros. Order of contact
 * 2.2: Equation of the osculating plane
 * 2.3: Trihedral at a point

$$\S 3.$$ Curvature and Torsion
 * 3.1: Curvature
 * 3.2: Torsion

$$\S 4.$$ Frenet-Serret Formulas
 * 4.1: Derivation of the formulas
 * 4.2: An application

$$\S 5.$$ Surface Theory
 * 5.1: The normal vector
 * 5.2: Tangent plane
 * 5.3: Normal line
 * 5.4: An example

$$\S 6.$$ Fundamental Differential Forms
 * 6.1: First fundamental form
 * 6.2: Arc length and angle
 * 6.3: Second fundamental form
 * 6.4: Curvature of a normal section of a surface

$$\S 7.$$ Mercator Maps
 * 7.1: Curves on a sphere
 * 7.2: Curves on a cylinder
 * 7.3: Mercator maps

4 Applications of Partial Differentiation
$$\S 1.$$ Maxima and minima
 * 1.1: Necessary conditions
 * 1.2: Sufficient conditions
 * 1.3: Points of inflection

$$\S 2.$$ Functions of Two Variables
 * 2.1: Absolute maximum or minimum
 * 2.2: Illustrative examples
 * 2.3: Critical treatment of an elementary problem

$$\S 3.$$ Sufficient Conditions
 * 3.1: Relative extrema
 * 3.2: Saddle-points
 * 3.3: Least squares

$$\S 4.$$ Functions of Three Variables
 * 4.1: Quadratic forms
 * 4.2: Relative extrema

$$\S 5.$$ Lagrange's Multipliers
 * 5.1: One relation between two variables
 * 5.2: One relation among three variables
 * 5.3: Two relations among three variables

$$\S 6.$$ Families of Plane Curves
 * 6.1: Envelopes
 * 6.2: Curve as envelope of its tangents
 * 6.3: Evolute as envelope of normals

$$\S 7.$$ Families of Surfaces
 * 7.1: Envelopes of families of surfaces
 * 7.2: Developable surfaces

5 Stieltjes Integral
$$\S 1.$$ Introduction
 * 1.1: Definitions
 * 1.2: Existence of the integral

$$\S 2.$$ Properties of the Integral
 * 2.1: A table of properties
 * 2.2: Sums
 * 2.3: Riemann integrals
 * 2.4: Extensions

$$\S 3.$$ Integration by Parts
 * 3.1: Partial summation
 * 3.2: The formula

$$\S 4.$$ Laws of the Mean
 * 4.1: First mean-value theorem
 * 4.2: Second mean-value theorem

$$\S 5.$$ Physical Applications
 * 5.1: Mass of a material wire
 * 5.2: Moment of inertia

$$\S 6.$$ Continuous Functions
 * 6.1: The Heine-Borel theorem
 * 6.2: Bounds of continuous functions
 * 6.3: Maxima and minima of continuous functions
 * 6.4: Uniform continuity
 * 6.5: Duhamel's theorem
 * 6.6: Another property of continuous function
 * 6.7: Critical remarks

$$\S 7.$$ Existence of Stieltjes Integrals
 * 7.1: Preliminary results
 * 7.2: Proof of theorem I
 * 7.3: The Riemann integral

6 Multiple Integrals
$$\S 1.$$ Introduction
 * 1.1: Regions
 * 1.2: Definitions
 * 1.3: Existence of the integral

$$\S 2.$$ Properties of Double Integrals
 * 2.1: A table of properties
 * 2.2: Iterated integrals
 * 2.3: Volume of a solid

$$\S 3.$$ Evaluation of Double Integrals
 * 3.1: The fundamental theorem
 * 3.2: Illustrations

$$\S 4.$$ Polar Coordinates
 * 4.1: Region $$R_\theta$$ and $$R_r$$
 * 4.2: The fundamental theorem
 * 4.3: Illustrations

$$\S 5.$$ Change in Order of Integration
 * 5.1: Rectangular coordinates
 * 5.2: Polar coordinates

$$\S 6.$$ Applications
 * 6.1: Duhamel's theorem
 * 6.2: Center of gravity of a plane lamina
 * 6.3: Moments of inertia

$$\S 7.$$ Further Applications
 * 7.1: Definition of area
 * 7.2: A preliminary result
 * 7.3: The integral formula
 * 7.4: Critique of the definition
 * 7.5: Attraction

$$\S 8.$$ Triple Integrals
 * 8.1: Definition of the integral
 * 8.2: Iterated integral
 * 8.3: Applications

$$\S 9.$$ Other Coordinates
 * 9.1: Cylindrical coordinates
 * 9.2: Spherical coordinates

$$\S 10.$$ Existence of Double Integrals
 * 10.1: Uniform continuity
 * 10.2: Preliminary results
 * 10.3: Proof of theorem I
 * 10.4: Area

7 Line and Surface Integrals
$$\S 1.$$ Introduction
 * 1.1: Curves
 * 1.2: Definition of line integrals
 * 1.3: Work

$$\S 2.$$ Green's Theorem
 * 2.1: A first form
 * 2.2: A second form
 * 2.3: Remarks
 * 2.4: Area

$$\S 3.$$ Application
 * 3.1: Existence of exact differentials
 * 3.2: Exact differential equations
 * 3.3: A further result
 * 3.4: Multiply connected regions

$$\S 4.$$ Surface Integrals
 * 4.1: Definition of surface integrals
 * 4.2: Green's or Gauss's theorem
 * 4.3: Extensions

$$\S 5.$$ Change of Variable in Multiple Integrals
 * 5.1: Transformations
 * 5.2: Double integrals
 * 5.3: An application
 * 5.4: Remarks
 * 5.5: An auxiliary result

$$\S 6.$$ Line Integrals in Space
 * 6.1: Definition of the line integral
 * 6.2: Stokes's theorem
 * 6.3: Remarks
 * 6.4: Exact differentials
 * 6.5: Vector considerations

8 Limits and Indeterminate Forms
$$\S 1.$$ The Indeterminate Form $$0/0$$
 * 1.1: The law of the mean
 * 1.2: Generalized law of the mean
 * 1.3: L'Hospital's rule

$$\S 3.$$ The Indeterminate Form $$\infty / \infty$$
 * 2.1: L'Hospital's rule

$$\S 3.$$ Other Indeterminate Forms
 * 3.1: The form $$0 \cdot \infty$$
 * 3.2: The form $$\infty - \infty$$
 * 3.3: The forms $$0^0$$, $$0^\infty$$, $$\infty^\infty$$, $$1^\infty$$

$$\S 4.$$ Other Methods. Orders of Infinity
 * 4.1: The method of series
 * 4.2: Change of variable
 * 4.3: Orders of infinity

$$\S 5.$$ Superior and Inferior Limits
 * 5.1: Limit points of a sequence
 * 5.2: Properties of superior and inferior limits
 * 5.3: Cauchy's criterion
 * 5.4: L'Hospital's rule (concluded)

9 Infinite Series
$$\S 1.$$ Convergence of Series. Comparison Tests
 * 1.1: Convergence and divergence
 * 1.2: Comparison tests

$$\S 2.$$ Convergence Tests
 * 2.1: D'Alembert's ratio test
 * 2.2: Cauchy's test
 * 2.3: Maclaurin's integral test

$$\S 3.$$ Absolute Convergence. Altering Series
 * 3.1: Absolute and conditional convergence
 * 3.2: Leibniz's theorem on alternating series

$$\S 4.$$ Limit Tests
 * 4.1: Limit test for convergence
 * 4.2: Limit test for divergence

$$\S 5.$$ Uniform Convergence
 * 5.1: Definition of uniform convergence
 * 5.2: Weierstrass's $$M$$-test
 * 5.3: Relation to absolute convergence

$$\S 6.$$ Applications
 * 6.1: Continuity of the sum of a series
 * 6.2: Integration of series
 * 6.3: Differentiation of series

$$\S 7.$$ Divergent Series
 * 7.1: Precaution
 * 7.2: Cesàro summability
 * 7.3: Regularity
 * 7.4: Other methods of summability

$$\S 8.$$ Miscellaneous Methods
 * 8.1: Cauchy's inequality
 * 8.2: Hölder and Minkowski inequalities
 * 8.3: Partial summation

$$\S 9.$$ Power Series
 * 9.1: Region of convergence
 * 9.2: Uniform convergence
 * 9.3: Abel's theorem

10 Convergence of Improper Integrals
$$\S 1.$$ Introduction
 * 1.1: Classification of improper integrals
 * 1.2: Type I., Convergence
 * 1.3: Comparison tests
 * 1.4: Absolute convergence

$$\S 2.$$ Type I. Limit Tests
 * 2.1: Limit test for convergence
 * 2.2: Limit test for divergence

$$\S 3.$$ Type I. Conditional Convergence
 * 3.1: Integrand with oscillating sign
 * 3.2: Sufficient conditions for conditional convergence

$$\S 4.$$ Type III
 * 4.1: Convergence
 * 4.2: Comparison tests
 * 4.3: Absolute convergence
 * 4.4: Limit tests
 * 4.5: Oscillating integrands

$$\S 5.$$ Combination of Types
 * 5.1: Type II
 * 5.2: Type IV
 * 5.3: Summary of limit tests
 * 5.4: Combinations of integrals

$$\S 6.$$ Uniform Convergence
 * 6.1: The Weierstrass $$M$$-test

$$\S 7.$$ Properties of Proper Integrals
 * 7.1: Integral as a function of its limits of integration
 * 7.2: Integral as a function of a parameter
 * 7.3: Integrals as composite functions
 * 7.4: Application to Taylor's formula

$$\S 8.$$ Application of Uniform Convergence
 * 8.1: Continuity
 * 8.2: Integration
 * 8.3: Differentiation

$$\S 9.$$ Divergent Integrals
 * 9.1: Cesàro summability
 * 9.2: Regularity
 * 9.3: Other methods of summability

$$\S 10.$$ Integral Inequalities
 * 10.1: The Schwarz inequality
 * 10.2: The Hölder inequality
 * 10.3: The Minkowski inequality

11 The Gamma Function. Evaluation of Definite Integrals
$$\S 1.$$ Introduction
 * 1.1: The gamma function
 * 1.2: Extension of definition
 * 1.3: Certain constants related to $$\Gamma \left({x}\right)$$
 * 1.4: Other expressions for $$\Gamma \left({x}\right)$$

$$\S 2.$$ The Beta Function
 * 2.1: Definition and convergence
 * 2.2: Other integral expressions
 * 2.3: Relation to $$\Gamma \left({x}\right)$$
 * 2.4: Wallis's product

$$\S 3.$$ Evaluation of Definite Integrals
 * 3.1: Differentiation with respect to a parameter
 * 3.2: Use of special Laplace transforms
 * 3.3: The method of infinite series

$$\S 4.$$ Stirling's Formula
 * 4.1: Preliminary results
 * 4.2: Proof of Stirling's formula
 * 4.3: Existence of Euler's constant
 * 4.4: Infinite products
 * 4.5: An infinite product for $$\Gamma \left({x}\right)$$

12 Fourier Series
$$\S 1.$$ Introduction
 * 1.1: Definitions
 * 1.2: Orthogonality relation
 * 1.3: Further examples of Fourier series

$$\S 2.$$ Several Classes of Functions
 * 2.1: The classes $$P$$, $$D$$, $$D^1$$
 * 2.2: Relation among the classes
 * 2.3: Abbreviations

$$\S 3.$$ Convergence of a Fourier Series to Its Defining Function
 * 3.1: Bessel's inequality
 * 3.2: The Riemann-Lebesgue theorem
 * 3.3: The remainder of a Fourier series
 * 3.4: The convergence theorem

$$\S 4.$$ Extensions and Applications
 * 4.1: Points of discontinuity
 * 4.2: Riemann's theorem
 * 4.3: Applications

$$\S 5.$$ Vibrating String
 * 5.1: Fourier series for an arbitrary interval
 * 5.2: Differential equation of vibrating string
 * 5.3: A boundary-value problem
 * 5.4: Solution of the problem
 * 5.5: Uniqueness of solution
 * 5.6: Special cases

$$\S 6.$$ Summability of Fourier Series
 * 6.1: Preliminary results
 * 6.2: Fejer's theorem
 * 6.3: Uniformity

$$\S 7.$$ Applications
 * 7.1: Trigonometric approximation
 * 7.2: Weierstrass'a theorem on polynomial approximation
 * 7.3: Least square approximation
 * 7.4: Parseval's theorem
 * 7.5: Uniqueness

$$\S 8.$$ Fourier Integral
 * 8.1: Analogies with Fourier series
 * 8.2: Definition of a Fourier integral
 * 8.3: A preliminary result
 * 8.4: The convergence theorem
 * 8.5: Fourier transform

13 The Laplace Transform
$$\S 1.$$ Introduction
 * 1.1: Relation to power series
 * 1.2: Definitions

$$\S 2.$$ Region of Convergence
 * 2.1: Power series
 * 2.2: Convergence theorem
 * 2.3: Examples

$$\S 3.$$ Absolute and Uniform Convergence
 * 3.1: Absolute convergence
 * 3.2: Uniform convergence
 * 3.3: Differentiation of generating functions

$$\S 5.$$ Operational Properties of the Transform
 * 4.1: Linear operations
 * 4.2: Linear change of variable
 * 4.3: Differentiation
 * 4.4: Integration
 * 4.5: Illustrations

$$\S 5.$$ Resultant
 * 5.1: Definition of resultant
 * 5.2: Product of generating functions
 * 5.3: Application

$$\S 6.$$ Tables of Transforms
 * 6.1: Some new functions
 * 6.2: Transforms of the functions

$$\S 7.$$ Uniqueness
 * 7.1: A preliminary result
 * 7.2: The principal result

$$\S 8.$$ Inversion
 * 8.1: Preliminary results
 * 8.2: The inversion formula

$$\S 9.$$ Representation
 * 9.1: Rational functions
 * 9.2: Power series in $$1/s$$
 * 9.3: Illustrations

$$\S 10.$$ Related Transforms
 * 10.1: The bilateral Laplace transform
 * 10.2: Laplace-Stieltjes transform
 * 10.3: The Stieltjes transform

14 Applications of the Laplace Transform
$$\S 1.$$ Introduction
 * 1.1: Integrands that are generating functions
 * 1.2 Integrands that are determining functions

$$\S 2.$$ Linear Differential Equation
 * 2.1: First order equations
 * 2.2: Uniqueness of solution
 * 2.3: Equations of higher order

$$\S 3.$$ The General Homogeneous Case
 * 3.1: The problem
 * 3.2: The class $$E$$
 * 3.3: Rational functions
 * 3.4: Solution of the problem

$$\S 4.$$ The Nonhomogeneous Case
 * 4.1: The problem
 * 4.2: Solution of the problem
 * 4.3: Uniqueness of solution

$$\S 5.$$ Difference Equations
 * 5.1: The problem
 * 5.2: The power series transform
 * 5.3: A property of the transform
 * 5.4: Solution of difference equations

$$\S 6.$$ Partial differential Equations
 * 6.1: The first transformation
 * 6.2: The second transformation
 * 6.3: The plucked string

Selected Answers

Index of Symbols

Index