# 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

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