Elements of the Differential and Integral Calculus

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Elements of the Differential and Integral Calculus  (1911) 
by William Anthony Granville
ELEMENTS OF THE DIFFERENTIAL
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]

PREFACE

Differential Calculus[edit]

CHAPTER I
COLLECTION OF FORMULAS

  1. Formulas from Algebra, Trigonometry, and Analytic Geometry
  2. Greek alphabet
  3. Rules for signs in the four quadrants
  4. Natural values of the trigonometric functions
  5. Tables of logarithms

CHAPTER II
VARIABLES AND FUNCTIONS

  1. Variables and constants
  2. Interval of a variable
  3. Continuous variation
  4. Functions
  5. Independent and dependent variables
  6. Notation of functions
  7. Values of the independent variable for which a function is defined

CHAPTER III
THEORY OF LIMITS

  1. Limit of a variable
  2. Division by zero excluded
  3. Infinitesimals
  4. The concept of infinity (\infty)
  5. Limiting value of a function
  6. Continuous and discontinuous functions
  7. Continuity and discontinuity of functions illustrated by their graphs
  8. Fundamental theorems on limits
  9. Special limiting values
  10. The limit of \tfrac{\sin\ x}{x} as x \dot= 0
  11. The number e
  12. Expressions assuming the form \tfrac{\infty}{\infty}

CHAPTER IV
DIFFERENTIATION

  1. Introduction
  2. Increments
  3. Comparison of increments
  4. Derivative of a function of one variable
  5. Symbols for derivatives
  6. Differentiable functions
  7. General rule for differentiation
  8. Applications of the derivative to Geometry

CHAPTER V
RULES FOR DIFFERENTIATING STANDARD ELEMENTARY FORM

  1. Importance of General Rule
  2. Differentiation of a constant
  3. Differentiation of a variable with respect to itself
  4. Differentiation of a sum
  5. Differentiation of the product of a constant and a function
  6. Differentiation of the product of two functions
  7. Differentiation of the product of any finite number of functions
  8. Differentiation of a function with a constant exponent
  9. Differentiation of a quotient
  10. Differentiation of a function of a function
  11. Differentiation of inverse functions
  12. Differentiation of a logarithm
  13. Differentiation of the simple exponential function
  14. Differentiation of the general exponential function
  15. Logarithmic differentiation
  16. Differentiation of \sin\ v
  17. Differentiation of \cos\ v
  18. Differentiation of \tan\ v
  19. Differentiation of \cot\ v
  20. Differentiation of \sec\ v
  21. Differentiation of \csc\ v
  22. Differentiation of \operatorname{vers}\ v[1]
  23. Differentiation of \arcsin\ v
  24. Differentiation of \arccos\ v
  25. Differentiation of \arctan\ v
  26. Differentiation of \arccot\ v
  27. Differentiation of \arcsec\ v
  28. Differentiation of \arccsc\ v
  29. Differentiation of \operatorname{arcvers}\ v
  30. Implicit functions
  31. Differentiation of implicit functions

CHAPTER VI
SIMPLE APPLICATIONS OF THE DERIVATIVE

  1. Direction of a curve
  2. Equations of tangent and normal, lengths of subtangent and subnormal
  3. Rectangular coördinates
  4. Parametric equations of a curve
  5. Angle between the radius vector drawn to a point on a curve and the tangent to the curve at that point
  6. Lengths of polar subtangent and polar subnormal
  7. Solution of equations having multiple roots
  8. Applications of the derivative in mechanics. Velocity
  9. Component velocities
  10. Acceleration
  11. Component accelerations

CHAPTER VII
SUCCESSIVE DIFFERENTIATION

  1. Definition of successive derivatives
  2. Notation
  3. The nth derivative
  4. Leibnitz's formula for the nth derivative of a product
  5. Successive differentiation of implicit functions

CHAPTER VIII
MAXIMA AND MINIMA. POINTS OF INFLECTION. CURVE TRACING

  1. Introduction
  2. Increasing and decreasing functions
  3. Tests for determining when a function is increasing and when decreasing
  4. Maximum and minimum values of a function
  5. First method for examining a function for maximum and minimum values
  6. Second method for examining a function for maximum and minimum values
  7. Definition of points of inflection and rule for finding points of inflection
  8. Curve tracing

CHAPTER IX
DIFFERENTIALS

  1. Introduction
  2. Definitions
  3. Infinitesimals
  4. Derivative of the arc in rectangular coördinates
  5. Derivative of the arc in polar coördinates
  6. Formulas for finding the differentials of functions
  7. Successive differentials

CHAPTER X
RATES

  1. The derivative considered as the ratio of two rates

CHAPTER XI
CHANGE OF VARIABLE

  1. Interchange of dependent and independent variables
  2. Change of the dependent variable
  3. Change of the independent variable
  4. Simultaneous change of both independent and dependent variables

CHAPTER XII
CURVATURE. RADIUS OF CURVATURE

  1. Curvature
  2. Curvature of a circle
  3. Curvature at a point
  4. Formulas for curvature
  5. Radius of curvature
  6. Circle of curvature

CHAPTER XIII
THEOREM OF MEAN VALUE. INDETERMINATE FORMS

  1. Rolle's Theorem
  2. The Theorem of Mean Value
  3. The Extended Theorem of Mean Value
  4. Maxima and minima treated analytically
  5. Indeterminate forms
  6. Evaluation of a function taking on an indeterminate form
  7. Evaluation of the indeterminate form \tfrac{0}{0}
  8. Evaluation of the indeterminate form \tfrac{\infty}{\infty}
  9. Evaluation of the indeterminate form 0 \cdot \infty
  10. Evaluation of the indeterminate form \infty - \infty
  11. Evaluation of the indeterminate forms 0^0, 1^\infty, \infty^0

CHAPTER XIV
CIRCLE OF CURVATURE. CENTER OF CURVATURE

  1. Circle of curvature. Center of curvature
  2. Second method for finding center of curvature
  3. Center of curvature the limiting position of the intersection of normals at neighboring points
  4. Evolutes
  5. Properties of the evolute
  6. Involutes and their mechanical construction

CHAPTER XV
PARTIAL DIFFERENTIATION

  1. Continuous functions of two or more independent variables
  2. Partial derivatives
  3. Partial derivatives interpreted geometrically
  4. Total derivatives
  5. Total differentials
  6. Differentiation of implicit functions
  7. Successive partial derivatives
  8. Order of differentiation immaterial

CHAPTER XVI
ENVELOPES

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

CHAPTER XVII
SERIES

  1. Introduction
  2. Infinite series
  3. Existence of a limit
  4. Fundamental test for convergence
  5. Comparison test for convergence
  6. Cauchy's ratio test for convergence
  7. Alternating series
  8. Absolute convergence
  9. Power series

CHAPTER XVIII
EXPANSION OF FUNCTIONS

  1. Introduction
  2. Taylor's Theorem and Taylor's Series
  3. Maclaurin's Theorem and Maclaurin's Series
  4. Computation by series
  5. Approximate formulas derived from series. Interpolation
  6. Taylor's Theorem for functions of two or more variables
  7. Maxima and minima of functions of two independent variables

CHAPTER XIX
ASYMPTOTES. SINGULAR POINTS

  1. Rectilinear asymptotes
  2. Asymptotes found by method of limiting intercepts
  3. Method of determining asymptotes to algebraic curves
  4. Asymptotes in polar coördinates
  5. Singular points
  6. Determination of the tangent to an algebraic curve at a given point by inspection
  7. Nodes
  8. Cusps
  9. Conjugate or isolated points
  10. Transcendental singularities

CHAPTER XX
APPLICATIONS TO GEOMETRY OF SPACE

  1. Tangent line and normal plane to a skew curve whose equations are given in parametric form
  2. Tangent plane to a surface
  3. Normal line to a surface
  4. Another form of the equations of the tangent line to a skew curve
  5. 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

  1. Integration
  2. Constant of integration. Indefinite integral
  3. Rules for integrating standard elementary forms
  4. Trigonometric differentials
  5. Integration of expressions containing \sqrt{a^2-x^2} or \sqrt{x^2 \pm a^2} by a trigonometric substitution

CHAPTER XXIII
CONSTANT OF INTEGRATION

  1. Determination of the constant of integration by means of initial conditions
  2. Geometrical signification of the constant of integration
  3. Physical signification of the constant of integration

CHAPTER XXIV
THE DEFINITE INTEGRAL

  1. Differential of an area
  2. The definite integral
  3. Calculation of a definite integral
  4. Calculation of areas
  5. Geometrical representation of an integral
  6. Mean value of \Phi(x)
  7. Interchange of limits
  8. Decomposition of the interval
  9. The definite integral a function of its limits
  10. Infinite limits
  11. When y = \Phi(x) is discontinuous

CHAPTER XXV
INTEGRATION OF RATIONAL FRACTIONS

  1. Introduction
  2. Case I
  3. Case II
  4. Case III
  5. Case IV

CHAPTER XXVI
INTEGRATION BY SUBSTITUTION OF A NEW VARIABLE. RATIONALIZATION

  1. Introduction
  2. Differentials containing fractional powers of x only
  3. Differentials containing fractional powers of a + bx only
  4. Change in limits corresponding to change in variable
  5. Differentials containing no radical except \sqrt{a + bx + x^2}
  6. Differentials containing no radical except \sqrt{a + bx - x^2}
  7. Binomial differentials
  8. Conditions of integrability of binomial differentials
  9. Transformation of trigonometric differentials
  10. Miscellaneous substitutions

CHAPTER XXVII
INTEGRATION BY PARTS. REDUCTION FORMULAS

  1. Formula for integration by parts
  2. Reduction formulas for binomial differentials
  3. Reduction formulas for trigonometric differentials
  4. To find \int e^{ax}\sin{nx}dx and \int e^{ax} \cos{nx}dx

CHAPTER XXVIII
INTEGRATION A PROCESS OF SUMMATION

  1. Introduction
  2. The fundamental theorem of Integral Calculus
  3. Analytical proof of the Fundamental Theorem
  4. Areas of plane curves. Rectangular coördinates
  5. Area when curve is given in parametric form
  6. Areas of plane curves. Polar coördinates
  7. Length of a curve
  8. Lengths of plane curves. Rectangular coördinates
  9. Lengths of plane curves. Polar coördinates
  10. Volumes of solids of revolution
  11. Areas of surfaces of revolution
  12. Miscellaneous applications

CHAPTER XXIX
SUCCESSIVE AND PARTIAL INTEGRATION

  1. Successive integration
  2. Partial integration
  3. Definite double integral. Geometric interpretation
  4. Value of a definite double integral over a region
  5. Plane area as a definite double integral. Rectangular coördinates
  6. Plane area as a definite double integral. Polar coördinates
  7. Moment of area
  8. Center of area
  9. Moment of inertia. Plane areas
  10. Polar moment of inertia. Rectangular coördinates
  11. Polar moment of inertia. Polar coördinates
  12. General method for finding the areas of surfaces
  13. Volumes found by triple integration

CHAPTER XXX
ORDINARY DIFFERENTIAL EQUATIONS

  1. Differential equations. Order and degree
  2. Solutions of differential equations
  3. Verifications of solutions
  4. Differential equations of the first order and of the first degree
  5. Differential equations of the nth order and of the first degree

CHAPTER XXXI
INTEGRAPH. APPROXIMATE INTEGRATION. TABLE OF INTEGRALS

  1. Mechanical integration
  2. Integral curves
  3. The integraph
  4. Polar planimeter
  5. Area swept over by a line
  6. Approximate integration
  7. Trapezoidal rule
  8. Simpson's rule (parabolic rule)
  9. Integrals for reference

INDEX

Notes[edit]

  1. The function vers is an abbreviation for versine, which is equal to 1 - \cos.


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.