# 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 **public domain** in countries and areas where the copyright term is the author's **life plus 70 years or less**. This work may also be in the **public domain** in countries and areas with longer native copyright terms that apply the **rule of the shorter term** to foreign works.