Hamming R.W. Methods of Mathematics Applied to Calculus, Probability, and Statistics

New York: Dover Publications, 2004. — 880 p.This text focuses on the most widely used applications of mathematical methods, including those related to probability and statistics. The 4-part treatment begins with algebra and analytic geometry and proceeds to an exploration of the calculus of algebraic functions and transcendental functions and applications. 1985 edition. Includes 310 figures and 18 tables.Algebra and Analytic Geometry
Prologue
The Importance of Mathematics
The Uniqueness of Mathematics
The Unreasonable Effectiveness of Mathematics
Mathematics as a Language
What is Mathematics?
Mathematical Rigor
Advice to You
Remarks on Learning the CourseThe Integers
The Integers
On Proving Theorems
Mathematical Induction
The Binomial Theorem
Mathematical Induction Using Undetermined Coefficients
The Ellipsis Method
Review and Fallacies in AlgebraFractions-Rational Numbers
Rational Numbers
Euclid's Algorithm
The Rational Number System
Irrational Numbers
On Finding Irrational Numbers
Decimal Representation of a Rational Number
Inequalities
Exponents-An Application of Rational Numbers
Summary and Further Remarks
Real Numbers, Functions, and Philosophy
The Real Line
Philosophy
The Idea of a Function
The Absolute Value Function
Assumptions About Continuity
Polynomials and Integers
Linear Independence.
Complex Numbers
More PhilosophyAnalytic Geometry
Cartesian Coordinates
The Pythagorean Distance
Curves
Linear Equations-Straight Lines
Slope
Special Forms of the Straight Line
On Proving Geometric Theorems in Analytic Geometry
*The Normal Form of the Straight Line
Translation of the Coordinate Axes
*The Area of a Triangle
*A Problem in Computer Graphics
The Complex PlaneCurves of Second Degree-Conics
Strategy
Circles
Completing the Square
A More General Form of the Second-Degree Equation
Ellipses
Hyperbolas
Parabolas
Miscellaneous Cases
*Rotation of the Coordinate Axes
*The General Analysis
Symmetry
Nongeometric Graphing
Summary of Analytic Geometry
The Calculus of Algebraic Functions
Derivatives in Geometry
A History of the Calculus
The Idea of a Limit
Rules for Using Limits
Limits of Functions-Missing Values
The "Process
Composite Functions
Sums of Powers of x
Products and Quotients
An Abstraction of Differentiation
On the Formal Differentiation of FunctionsGeometric Applications
Tangent and Normal Lines
Higher Derivatives-Notation
Implicit Differentiation
Curvature
MAXIMA and Minima
Inflection Points
Curve Tracing
Functions, Equations, and CurvesNongeometric Applications
Scaling Geometry
Equivalent Ideas
Velocity
Acceleration
Simple Rate Problems
More Rate Problems
Newton's Method for Finding Zeros
Multiple Zeros
The Summation Notation
Generating Identities
Generating Functions-Place HoldersRichard Hamming introduces students to calculus and more in Methods of Mathematics. His work contains many interesting mathematical gems. At the same time, it will not replace a standard calculus textbook for most students.In 800 pages and 25 chapters, Methods covers all the standard material contained within a first year calculus sequence. It deals with differential and integral calculus and includes interesting applications. It visits the basic theory of infinite series and differential equations as well. Moreover, it attends to functions of several variables, but not in a way that would substitute for a multivariable calculus course.Hamming’s text is noteworthy for its treatment of probability along with some unusual and interesting mathematics. This begins with his discussions of induction (2.3) and the method of undetermined coefficients (2.5). Later chapters address probability, especially in the context of calculus. Near the end, he devotes a full chapter to Fourier series, including the discrete Fourier transform and the Gibbs phenomenon. The book sprinkles additional tidbits along the way such as Buffon’s needle problem, liberal use of Wallis integrals and linear difference equations.Why not use such a lovely and endearing text in lieu of a current calculus textbook? On reason is that its treatment of topics is somewhat unorthodox. While it eventually covers a year of calculus, Hamming’s ordering of the material may be foreign to the average calculus instructor. In addition, despite its title, little specific discussion of statistics takes place other than in reference to the least squares approximation. Even the discussion of probability, notable for its treatment of moment generating functions, omits important well-known discrete distributions or fails to mention them by name (uniform, geometric, negative binomial, and hypergeometric).Besides functioning as a mere math textbook, Methods offers advice to a novice scientist. The author directs to the reader to issues in the text that are more important than others and also shares his conception of “how mathematics is really done.” He wisely informs the reader that mathematical results are rarely discovered in the final form that students see.