Kline Morris. Calculus. An Intuitive and Physical Approach

Second Edition. — Mineola, NY: Dover Publications, 1998. — 960 р. — ISBN 9780486404530, 0486404536.Application-oriented introduction relates the subject as closely as possible to science. In-depth explorations of the derivative, the differentiation and integration of the powers of x, and theorems on differentiation and antidifferentiation lead to a definition of the chain rule and examinations of trigonometric functions, logarithmic and exponential functions, techniques of integration, polar coordinates, much more. Clear-cut explanations, numerous drills, illustrative examples. 1967 edition. Solution guide available upon request.Why calculus?
The Historical Motivations for the Calculus
The Creators of the Calculus
The Nature of the Calculus
The derivative
The Concept of Function
The Graph or Curve of a Function
Average and Instantaneous Speed
The Method of Increments
A Matter of Notation
The Method of Increments Applied to y = ax2
The Derived Function
The Differentiation of Simple Monomials
The Differentiation of Simple Polynomials
The Second Derivative
The antiderived function or the integral
The Integral
Straight Line Motion in One Direction
Up and Down Motion
Motion Along an Inclined Plane
Appendix The Coordinate Geometry of Straight Lines
A1. The Need for Geometrical Interpretation
A2. The Distance Formula
A3. The Slope of a Straight Line
A4. The Inclination of a Line
A5. Slopes of Parallel and Perpendicular Lines
A6. The Angle Between Two Lines
A7. The Equation of a Straight Line
A8. The Distance from a Point to a Line
A9. Equation and Curve
The geometrical significance of the derivative
The Derivative as Slope
The Concept of Tangent to a Curve
Applications of the Derivative as the Slope
The Equation of the Parabola
Physical Applications of the Derivative as Slope
Further Discussion of the Derivative as the Slope
The differentiation and integration of powers of xThe Functions xn for Positive Integral n
A Calculus Method of Finding Roots
Differentiation and Integration of xn for Fractional Values of n
Some theorems on differentiation and antidifferentiationSome Remarks about Functions
The Differentiation of Sums and Differences of Functions
The Differentiation of Products and Quotients of Functions
The Integration of Combinations of Functions
All Integrals Differ by a Constant
The Power Rule for Negative Exponents
The Concept of Work and an Application
The Chain ruleThe Chain Rule
Application of the Chain Rule to Differentiation
The Differentiation of Implicit Functions
Equations of the Ellipse and Hyperbola
Differentiation of the Equations of Ellipse and Hyperbola
Integration Employing the Chain Rule
The Problem of Escape Velocity
Related Rates
Appendix Transformation of Coordinates
A1. Introduction
A2. Rotation of Axes
A3. Translation of Axes
A4. Invariants
MAXIMA and MinimaThe Geometrical Approach to MAXIMA and Minima
Analytical Treatment of MAXIMA and Minima
An Alternative Method of Determining Relative MAXIMA and Minima
Some Applications of the Method of MAXIMA and Minima
Some Applications to Economics
Curve Tracing
9 The definite integralArea as the Limit of a Sum
The Definite Integral
The Evaluation of Definite Integrals
Areas Below the x-Axis
Areas Between Curves
Some Additional Properties of the Definite Integral
Numerical Methods for Evaluating Definite Integrals
Appendix The Sum of the Squares of the First n Integers
The trigonometric functionsThe Sinusoidal Functions
Some Preliminaries on Limits
Differentiation of the Trigonometric Functions
Integration of the Trigonometric Functions
Application of the Trigonometric Functions to Periodic Phenomena
The inverse trigonometric functions
The Notion of an Inverse Function
The Inverse Trigonometric Functions
The Differentiation of the Inverse Trigonometric Functions
Integration Involving the Inverse Trigonometric Functions
Change of Variable in Integration
Time of Motion Under Gravitational Attraction
Logarithmic and exponential functionsA Review of Logarithms
The Derived Functions of Logarithmic Functions
Exponential Functions and Their Derived Functions
Problems of Growth and Decay
Motion in One Direction in a Resisting Medium
Up and Down Motion in Resisting Media
Hyperbolic Functions
Logarithmic Differentiation
Differentials and the law of the mean
Differentials
The Mean Value Theorem of the Differential Calculus
Indeterminate Forms
Further techniques of integrationIntegration by Parts
Reduction Formulas
Integration by Partial Fractions
Integration by Substitution and Change of Variable
The Use of Tables
Some geometric uses of the definite integralVolumes of Solids: The Cylindrical Element
Volumes of Solids: The Shell Game
Lengths of Arcs of Curves
Curvature
Areas of Surfaces of Revolution
Remarks on Approximating Figures
Some physical applications of the definite integralThe Calculation of Work
Applications to Economics
The Hanging Chain
Gravitational Attraction of Rods
Gravitational Attraction of Disks
Gravitational Attraction of Spheres
Polar coordinates
The Polar Coordinate System
The Polar Coordinate Equations of Curves
The Polar Coordinate Equations of the Conic Sections
The Relation Between Rectangular and Polar Coordinates
The Derivative of a Polar Coordinate Function
Areas in Polar Coordinates
Arc Length in Polar Coordinates
Curvature in Polar Coordinates
Rectangular parametric equations and curvilinear motionThe Parametric Equations of a Curve
Some Additional Examples of Parametric Equations
Projectile Motion in a Vacuum
Slope, Area, Arc Length, and Curvature Derived from Parametric Equations
An Application of Arc Length
Velocity and Acceleration in Curvilinear Motion
Tangential and Normal Acceleration in Curvilinear Motion
Polar parametric equations and curvilinear motion
Polar Parametric Equations
Velocity and Acceleration in the Polar Parametric Representation
Kepler’s Laws
Satellites and Projectiles
Taylor’s theorem and infinite series
The Need to Approximate Functions
The Approximation of Functions by Polynomials
Taylor’s Formula
Some Applications of Taylor’s Theorem
The Taylor Series
Infinite Series of Constant Terms
Tests for Convergence and Divergence
Absolute and Conditional Convergence
The Ratio Test
Power Series
Return to Taylor’s Series
Some Applications of Taylor’s Series
Series as Functions
Functions of two or more variables and their geometric representation
Functions of Two or More Variables
Basic Facts on Three-Dimensional Cartesian Coordinates
Equations of Planes
Equations of Straight Lines
Quadric or Second Degree Surfaces
Remarks on Further Work in Solid Analytic Geometry
Partial differentiation
Functions of Two or More Variables
Partial Differentiation
The Geometrical Meaning of the Partial Derivatives
The Directional Derivative
The Chain Rule
Implicit Functions
Differentials
MAXIMA and Minima
Envelopes
Multiple integralsVolume Under a Surface
Some Physical Applications of the Double Integral
The Double Integral
The Double Integral in Cylindrical Coordinates
Triple Integrals in Rectangular Coordinates
Triple Integrals in Cylindrical Coordinates
Triple Integrals in Spherical Coordinates
The Moment of Inertia of a Body
24 An introduction to differential equationsFirst-Order Ordinary Differential Equations
Second-Order Linear Homogeneous Differential Equations
Second-Order Linear Non-Homogeneous Differential Equations
A reconsideration of the foundationsThe Concept of a Function
The Concept of the Limit of a Function
Some Theorems on Limits of Functions
Continuity and Differentiability
The Limit of a Sequence
Some Theorems on Limits of Sequences
The Definite Integral
Improper Integrals
The Fundamental Theorem of the Calculus
The Directions of Future Work
Tables