Calculus: An Intuitive and Physical Approach by Kline

[micromass]

• Author: Morris Kline
• Title: Calculus: An Intuitive and Physical Approach
• Amazon link: https://www.amazon.com/dp/0486404536/?tag=pfamazon01-20
• Prerequisities: High-School Mathematics
• Level: Undergrad

Table of Contents:

[LIST]
[*] Why Calculus?
[LIST]
[*] The Historical Motivations for the Calculus
[*] The Creators of the Calculus
[*] The Nature of the Calculus
[/LIST]
[*] The Derivative
[LIST]
[*] 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=ax^2
[*] The Derived Function
[*] The Differentiation of Simple Monomials
[*] The Differentiation of Simple Polynomials
[*] The Second Derivative
[/LIST]
[*] The Antiderived Function or the Integral
[LIST]
[*] The Integral
[*] Straight Line Motion in One Direction
[*] Up and Down Motion
[*] Motion Along an Inclined Plane
[*] Appendix: The Coordinate Geometry of Straight Lines
[LIST]
[*] The Need for Geometrical Interpretation
[*] The Distance Formula
[*] The Slope of a Straight Line
[*] The Inclination of a Line
[*] Slopes of Parallel and Perpendicular Lines
[*] The Angle Between Two Lines
[*] The Equation of a Straight Lines
[*] The Distance from a Point to a Line
[*] Equation and Curve
[/LIST]
[/LIST]
[*] The Geometrical Significance of the Derivative
[LIST]
[*] The Derivative as Slope
[*] The Concept of Tangent to a Curve
[*] Applications of the Derivative as the Slope
[*] The Equation of the Paraboa
[*] Physical Applications of the Derivative as Slope
[*] Further Discussion of the Derivative as the Slope
[/LIST]
[*] The Differentiation and Integration of Powers of x
[LIST]
[*] Introduction
[*] The Functions x^n for Positive Integral n
[*] A Calculus Method of Finding Roots
[*] Differentiation and Integration of x^n for Fractional Values of n
[/LIST]
[*] Some Theorems on Differentiation and Antidifferentiation
[LIST]
[*] Introduction
[*] Some 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
[/LIST]
[*] The Chain Rule
[LIST]
[*] Introduction
[*] The 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
[*] Appendix: Transformation of Coordinates
[LIST]
[*] Introduction
[*] Rotation of Axes
[*] Translation of Axes
[*] Invariants
[/LIST]
[/LIST]
[*] Maxima and Minima
[LIST]
[*] Introduction
[*] The 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
[/LIST]
[*] The Definite Integral
[LIST]
[*] Introduction
[*] Area as the Limit of a Sum
[*] The Definite Integral
[*] The Evaluation of Definite Integrals
[*] Area Below the x-Axis
[*] Areas Between Curves
[*] Some Additional Properties of the Definite Integrals
[*] Numerical Methods for Evaluating Definite Integrals
[*] Appendix: The Sum of the Squares of the First n Integers
[/LIST]
[*] The Trigonometric Functions
[LIST]
[*] Introduction
[*] The Sinusoidal Function
[*] Some Preliminaries on Limits
[*] Differentiation of the Trigonometric Functions
[*] Integration of the Trigonometric Functions
[*] Application of the Trigonometric Functions to Periodic Phenomena
[/LIST]
[*] The Inverse Trigonometric Functions
[LIST]
[*] 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
[/LIST]
[*] Logarithmic and Exponential Functions
[LIST]
[*] Introduction
[*] A 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
[/LIST]
[*] Differentials and the Law of the Mean
[LIST]
[*] Differentiation
[*] The Mean Value Theorem of the Differential Calculus
[*] Indeterminate Forms
[/LIST]
[*] Further Techniques of Integration
[LIST]
[*] Introduction
[*] Integration by Parts
[*] Reduction Formulas
[*] Integration by Partial Fractions
[*] Integration by Substitution and Change of Variable
[*] The Use of Tables
[/LIST]
[*] Some Geometric Uses of the Definite Integral
[LIST]
[*] Introduction
[*] Volumes 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
[/LIST]
[*] Some Physical Applications of the Definite Integral
[LIST]
[*] Introduction
[*] The calculation of Work
[*] Applications to Economics
[*] The Hanging Chain
[*] Gravitational Attraction of Rods
[*] Gravitational Attraction of Disks
[*] Gravitational Attraction of Spheres
[/LIST]
[*] Polar Coordinates
[LIST]
[*] 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
[/LIST]
[*] Rectangular Parametric Equations and Curvilinear Motion
[LIST]
[*] Introduction
[*] The Parametric Equations of a Curve
[*] Some Additional Examples of Parametric Equations
[*] Projective 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
[/LIST]
[*] Polar Parametric Equations and Curvilinear Motion
[LIST]
[*] Polar Parametric Equations
[*] Velocity and Acceleration in the Polar Parametric Representation
[*] Kepler's Laws
[*] Statellites and Projectiles
[/LIST]
[*] Taylor's Theorem and Infinite Series
[LIST]
[*] 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
[/LIST]
[*] Functions of Two or More Variables and Their Geometric Representation
[LIST]
[*] 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
[/LIST]
[*] Partial Differentiation
[LIST]
[*] 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
[/LIST]
[*] Multiple Integrals
[LIST]
[*] Introduction
[*] Volume 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
[/LIST]
[*] An Introduction to Differential Equations
[LIST]
[*] Introduction
[*] First-Order Ordinary Differential Equations
[*] Second-Order Linear Homogeneous Differential Equations
[*] Second-Order Linear Non-Homogeneous Differential Equations
[/LIST]
[*] A Reconsideration of the Foundations
[LIST]
[*] Introduction
[*] The Concept of a Function
[*] The Concept of a 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
[/LIST]
[*] Tables
[*] Index
[/LIST]

 

[fourier jr]

The thing that I like about this book is that it doesn't just briefly mention applications, it actually does them. So in the chapter on polar coordinates he has a complete section on Kepler's laws, in the max/min sections he has Fermat's principle of least time, etc.