# Elements of the Differential and Integral Calculus

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AND INTEGRAL CALCULUS

*(REVISED EDITION)*

BY

WILLIAM ANTHONY GRANVILLE, PH.D.. LL.D.

FORMERLY PRESIDENT OF PENNSYLYANIA COLLEGE

WITH THE EDITORIAL COOPERATION OF

PERCEY F. SMITH, PH.D.

PROFESSOR OF MATHEMATICS IN THE SHEFFIELD SCIENTIFIC SCHOOL

YALE UNIVERSITY

## Table of Contents[edit]

### Differential Calculus[edit]

CHAPTER I

COLLECTION OF FORMULAS

- Formulas from Algebra, Trigonometry, and Analytic Geometry
- Greek alphabet
- Rules for signs in the four quadrants
- Natural values of the trigonometric functions
- Tables of logarithms

CHAPTER II

VARIABLES AND FUNCTIONS

- Variables and constants
- Interval of a variable
- Continuous variation
- Functions
- Independent and dependent variables
- Notation of functions
- Values of the independent variable for which a function is defined

CHAPTER III

THEORY OF LIMITS

- Limit of a variable
- Division by zero excluded
- Infinitesimals
- The concept of infinity ()
- Limiting value of a function
- Continuous and discontinuous functions
- Continuity and discontinuity of functions illustrated by their graphs
- Fundamental theorems on limits
- Special limiting values
- The limit of as
- The number
- Expressions assuming the form

CHAPTER IV

DIFFERENTIATION

- Introduction
- Increments
- Comparison of increments
- Derivative of a function of one variable
- Symbols for derivatives
- Differentiable functions
- General rule for differentiation
- Applications of the derivative to Geometry

CHAPTER V

RULES FOR DIFFERENTIATING STANDARD ELEMENTARY FORM

- Importance of General Rule
- Differentiation of a constant
- Differentiation of a variable with respect to itself
- Differentiation of a sum
- Differentiation of the product of a constant and a function
- Differentiation of the product of two functions
- Differentiation of the product of any finite number of functions
- Differentiation of a function with a constant exponent
- Differentiation of a quotient
- Differentiation of a function of a function
- Differentiation of inverse functions
- Differentiation of a logarithm
- Differentiation of the simple exponential function
- Differentiation of the general exponential function
- Logarithmic differentiation
- Differentiation of
- Differentiation of
- Differentiation of
- Differentiation of
- Differentiation of
- Differentiation of
- Differentiation of
^{[1]} - Differentiation of
- Differentiation of
- Differentiation of
- Differentiation of
- Differentiation of
- Differentiation of
- Differentiation of
- Implicit functions
- Differentiation of implicit functions

CHAPTER VI

SIMPLE APPLICATIONS OF THE DERIVATIVE

- Direction of a curve
- Equations of tangent and normal, lengths of subtangent and subnormal
- Rectangular coördinates
- Parametric equations of a curve
- Angle between the radius vector drawn to a point on a curve and the tangent to the curve at that point
- Lengths of polar subtangent and polar subnormal
- Solution of equations having multiple roots
- Applications of the derivative in mechanics. Velocity
- Component velocities
- Acceleration
- Component accelerations

CHAPTER VII

SUCCESSIVE DIFFERENTIATION

- Definition of successive derivatives
- Notation
- The
*n*th derivative - Leibnitz's formula for the
*n*th derivative of a product - Successive differentiation of implicit functions

CHAPTER VIII

MAXIMA AND MINIMA. POINTS OF INFLECTION. CURVE TRACING

- Introduction
- Increasing and decreasing functions
- Tests for determining when a function is increasing and when decreasing
- Maximum and minimum values of a function
- First method for examining a function for maximum and minimum values
- Second method for examining a function for maximum and minimum values
- Definition of points of inflection and rule for finding points of inflection
- Curve tracing

CHAPTER IX

DIFFERENTIALS

- Introduction
- Definitions
- Infinitesimals
- Derivative of the arc in rectangular coördinates
- Derivative of the arc in polar coördinates
- Formulas for finding the differentials of functions
- Successive differentials

CHAPTER X

RATES

CHAPTER XI

CHANGE OF VARIABLE

- Interchange of dependent and independent variables
- Change of the dependent variable
- Change of the independent variable
- Simultaneous change of both independent and dependent variables

CHAPTER XII

CURVATURE. RADIUS OF CURVATURE

- Curvature
- Curvature of a circle
- Curvature at a point
- Formulas for curvature
- Radius of curvature
- Circle of curvature

CHAPTER XIII

THEOREM OF MEAN VALUE. INDETERMINATE FORMS

- Rolle's Theorem
- The Theorem of Mean Value
- The Extended Theorem of Mean Value
- Maxima and minima treated analytically
- Indeterminate forms
- Evaluation of a function taking on an indeterminate form
- Evaluation of the indeterminate form
- Evaluation of the indeterminate form
- Evaluation of the indeterminate form
- Evaluation of the indeterminate form
- Evaluation of the indeterminate forms

CHAPTER XIV

CIRCLE OF CURVATURE. CENTER OF CURVATURE

- Circle of curvature. Center of curvature
- Second method for finding center of curvature
- Center of curvature the limiting position of the intersection of normals at neighboring points
- Evolutes
- Properties of the evolute
- Involutes and their mechanical construction

CHAPTER XV

PARTIAL DIFFERENTIATION

- Continuous functions of two or more independent variables
- Partial derivatives
- Partial derivatives interpreted geometrically
- Total derivatives
- Total differentials
- Differentiation of implicit functions
- Successive partial derivatives
- Order of differentiation immaterial

CHAPTER XVI

ENVELOPES

- Family of curves. Variable parameter
- Envelope of a family of curves depending on one parameter
- The evolute of a given curve considered as the envelope of its normals
- Two parameters connected by one equation of condition

CHAPTER XVII

SERIES

- Introduction
- Infinite series
- Existence of a limit
- Fundamental test for convergence
- Comparison test for convergence
- Cauchy's ratio test for convergence
- Alternating series
- Absolute convergence
- Power series

CHAPTER XVIII

EXPANSION OF FUNCTIONS

- Introduction
- Taylor's Theorem and Taylor's Series
- Maclaurin's Theorem and Maclaurin's Series
- Computation by series
- Approximate formulas derived from series. Interpolation
- Taylor's Theorem for functions of two or more variables
- Maxima and minima of functions of two independent variables

CHAPTER XIX

ASYMPTOTES. SINGULAR POINTS

- Rectilinear asymptotes
- Asymptotes found by method of limiting intercepts
- Method of determining asymptotes to algebraic curves
- Asymptotes in polar coördinates
- Singular points
- Determination of the tangent to an algebraic curve at a given point by inspection
- Nodes
- Cusps
- Conjugate or isolated points
- Transcendental singularities

CHAPTER XX

APPLICATIONS TO GEOMETRY OF SPACE

- Tangent line and normal plane to a skew curve whose equations are given in parametric form
- Tangent plane to a surface
- Normal line to a surface
- Another form of the equations of the tangent line to a skew curve
- Another form of the equation of the normal plane to a skew curve

CHAPTER XXI

CURVES FOR REFERENCE

### Integral Calculus[edit]

CHAPTER XXII

INTEGRATION. RULES FOR INTEGRATING STANDARD ELEMENTARY FORMS

- Integration
- Constant of integration. Indefinite integral
- Rules for integrating standard elementary forms
- Trigonometric differentials
- Integration of expressions containing or by a trigonometric substitution

CHAPTER XXIII

CONSTANT OF INTEGRATION

- Determination of the constant of integration by means of initial conditions
- Geometrical signification of the constant of integration
- Physical signification of the constant of integration

CHAPTER XXIV

THE DEFINITE INTEGRAL

- Differential of an area
- The definite integral
- Calculation of a definite integral
- Calculation of areas
- Geometrical representation of an integral
- Mean value of
- Interchange of limits
- Decomposition of the interval
- The definite integral a function of its limits
- Infinite limits
- When is discontinuous

CHAPTER XXV

INTEGRATION OF RATIONAL FRACTIONS

- Introduction
- Case I
- Case II
- Case III
- Case IV

CHAPTER XXVI

INTEGRATION BY SUBSTITUTION OF A NEW VARIABLE. RATIONALIZATION

- Introduction
- Differentials containing fractional powers of only
- Differentials containing fractional powers of only
- Change in limits corresponding to change in variable
- Differentials containing no radical except
- Differentials containing no radical except
- Binomial differentials
- Conditions of integrability of binomial differentials
- Transformation of trigonometric differentials
- Miscellaneous substitutions

CHAPTER XXVII

INTEGRATION BY PARTS. REDUCTION FORMULAS

- Formula for integration by parts
- Reduction formulas for binomial differentials
- Reduction formulas for trigonometric differentials
- To find and

CHAPTER XXVIII

INTEGRATION A PROCESS OF SUMMATION

- Introduction
- The fundamental theorem of Integral Calculus
- Analytical proof of the Fundamental Theorem
- Areas of plane curves. Rectangular coördinates
- Area when curve is given in parametric form
- Areas of plane curves. Polar coördinates
- Length of a curve
- Lengths of plane curves. Rectangular coördinates
- Lengths of plane curves. Polar coördinates
- Volumes of solids of revolution
- Areas of surfaces of revolution
- Miscellaneous applications

CHAPTER XXIX

SUCCESSIVE AND PARTIAL INTEGRATION

- Successive integration
- Partial integration
- Definite double integral. Geometric interpretation
- Value of a definite double integral over a region
- Plane area as a definite double integral. Rectangular coördinates
- Plane area as a definite double integral. Polar coördinates
- Moment of area
- Center of area
- Moment of inertia. Plane areas
- Polar moment of inertia. Rectangular coördinates
- Polar moment of inertia. Polar coördinates
- General method for finding the areas of surfaces
- Volumes found by triple integration

CHAPTER XXX

ORDINARY DIFFERENTIAL EQUATIONS

- Differential equations. Order and degree
- Solutions of differential equations
- Verifications of solutions
- Differential equations of the first order and of the first degree
- Differential equations of the
*n*th order and of the first degree

CHAPTER XXXI

INTEGRAPH. APPROXIMATE INTEGRATION. TABLE OF INTEGRALS

- Mechanical integration
- Integral curves
- The integraph
- Polar planimeter
- Area swept over by a line
- Approximate integration
- Trapezoidal rule
- Simpson's rule (parabolic rule)
- Integrals for reference

## Notes[edit]

This work is in the
public domain in the United States because it was published before January 1, 1923.
The author died in 1943, so this work is also in the |