# Book:David V. Widder/Advanced Calculus/Second Edition

## David V. Widder: Advanced Calculus (2nd Edition)

Published $\text {1961}$, Dover Publications, Inc.

ISBN 0-486-66103-2

### Contents[1]

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
$\S 9.$ 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
2.3: 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
$\S 6.$ 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
3.3: Least squares
$\S 4.$ Functions of Three Variables
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
$\S 4.$ 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

Index of Symbols

Index

Next

### Notes

1. The source work from which this contents list was taken (the 1989 edition from Dover Publications Inc.) appears to have typographical mistakes in its contents list. These have been corrected in this transcription as and when they have been detected.

## Source work progress

Exercises for Chapter $1$ section $\S 2$ have been ignored because they are tedious and repetitive.