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

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

$\S 4.$ 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

Selected Answers

Index of Symbols

Index

Next

Further Editions

• 1947: David V. Widder: Advanced Calculus

Source work progress

• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 3$. Functions of Several Variables

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