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Mathematics Reference Book for Scientists and Engineers

J. H. Heinbockel
Publication Date: 
Number of Pages: 
[Reviewed by
H. Ric Blacksten
, on

This handbook, authored by a Professor Emeritus of Mathematics and Statistics at Old Dominion University, Norfolk, Virgina, has the feel of a collection of well used and perfected class notes covering those topics in undergraduate graduate mathematics that undergird much of physics and engineering.

The book consists of nine chapters: (1) Preliminaries; (2) Selected Fundamental Concepts; (3) Geometry; (4) Calculus; (5) Vector Calculus; (6) Ordinary Differential Equations; (7) Special Functions; (8) Probability and Statistics; and (9) Selected Applied Mathematical Topics. Appendices include tables of units of measurement and integrals. The book has a ten-page index.

The Preliminaries chapter provides a nice overview of the history of mathematics, some basic arithmetic, including alternative algorithms for adding, subtracting and dividing numbers. The chapter on Selected Fundamental Concepts includes fundamental topics in algebra and trigonometry. The Geometry chapter reviews fundamental concepts and tools from Euclidean Geometry and Analytic Geometry. Calculus provides a solid review of the fundamental concepts, theorems and techniques typically mastered by an undergraduate majoring or minoring in mathematics. The Vector Calculus chapter delves into the mathematics essential for undergraduate and graduate work in physics and engineering. The Ordinary Differential Equations chapter is just that. (There is no discussion of partial differentiation beyond a simple definition in the Calculus chapter.) The Special Functions chapter covers a broad range of functions and transforms found in physics, probability and statistics, and engineering. The Probability and Statistics chapter is largely limited to probability basics, not statistics. The concluding Selected Applied Mathematical Topics chapter contains brief discussions of classical topics in Newtonian dynamics, vibrations, resonance, electrical circuits, and even chemical kinetics.

Covering this broad range of topics in less than six hundred pages, the exposition is necessarily often rather terse, though always meticulous. As such, the handbook is indeed a reference text, not a textbook.

The handbook has approximately two dozen tables and three hundred numbered figures or smaller embedded illustrations. The tables, figures and illustrations are black and white with shading achieved through dots and lines. I found the layout of the handbook refreshingly old-fashioned without the distraction of numerous sidebars.

This is a nice book to have on your desk. I enjoyed reviewing topics I’d learned as a physics and mathematics major and professional, but have rarely used in my chosen career of operations research analysis. I enjoyed finding things I did not know, such as a simple subtraction algorithm based on adding rather than borrowing.

You can find more comprehensive mathematical handbooks at triple the size and triple the price of Heinbockel’s. But this paperback book’s physical size (6"x9"x1.25") is convenient to take on a walk or slip in a briefcase. The price, less than $25.00, is appealing for an ancillary handbook.

H. Ric Blacksten is a Principal Analyst with Innovative Decisions, Inc. He has worked as a physicist, mathematician and, for the latter part of his career, an operations research analyst specializing in mathematical modeling and simulation. His email address is


Chapter 1 Preliminaries

Introduction, History of Mathematics, Growth of Mathematics, Selected Contributors to Mathematical Development, Mathematical Specialty Areas, More Information, Mathematical Notation, Number Representation, Sufficient and Necessary Conditions, Indirect Proofs, Mathematical Induction, Arithmetic, Rule of Signs, The Rule of Nine, Arithmetic Operations, Calculators, Number Theory, The Greek Alphabet, Elementary Business Mathematics, Solutions of Linear and Quadratic Equations, Cubic Equation, Polynomial Equation, Special Products and Factors, Group, Algebraic Field

Chapter 2 Selected Fundamental Concepts

Permutations and Combinations, Elementary Probability, The Binomial Formula, Multinomial Expansion, Theory of Proportions, Sum of Arithmetic Progression, Sum of Geometric Progression, Equations Containing Radicals, Matrix Algebra, Special Matrices and Properties, Linear Dependence and Independence, Properties of Matrix Multiplication, Vector Norms, System of Linear Equations, Determinant of Order n, Pythagorean Theorem, Trigonometry, Trigonometric Functions, Cofunctions, Trigonometric Functions Defined for Other Angles, Sign Variation of the Trigonometric Functions, Graphs of the Trigonometric Functions, Trigonometric Functions of Sums and Differences, Double-angle Formulas, Half-angle Formulas, Product, Sum and Difference Formula, Simple Harmonic Motion, Inverse Functions, Inverse Trigonometric Functions, Principal Value Properties, Hyperbolic Functions, Hyperbolic Identities, Properties of Hyperbolic Functions, Inverse Hyperbolic Functions, Complex Numbers, Summary of Properties of Trigonometric and Hyperbolic Functions

Chapter 3 Geometry

Classification of Triangles, Similar Triangles, Congruent Triangles, Golden Ratio, Medians and Perpendicular Bisectors, Law of sines for Triangle, Law of Cosines for Triangle, Law of Tangents for Triangle, Area of Triangle, Miscellaneous Properties of a Triangle, Two-dimensional Rectangular coordinates, Translation and Rotation of Axes, Vectors, Vector Components, Direction Cosines, Properties of Vectors, Curve Sketching, Special Graph Paper, Straight Lines, Use of Determinants, Polar Coordinates, Curve Sketching in Polar Coordinates, Geometric Shapes, Polyhedron, Conic Sections, Determinants and Conic Sections, Conic Sections in Polar Coordinates, Functions and Graphs, Plane Curves, Plane Curves in Polar and Parametric Form, Spirals, Cycloids, Epicycloids, Hypocycloid, The Ovals of Cassini, Solid Analytic Geometry, Geometry and Graphics

Chapter 4 Calculus 

Limits, Properties of Limits, Continuity,ε−δ Definition of Continuity, Intermediate Value Theorem, Derivatives, Basic differentiation Rules, Differentials, Properties of Differentials, Higher Derivatives, Parametric Functions, Leibnitz Rule for Differentiating Products, Partial Derivatives, Implicit Differentiation, Mean Value Theorem for Derivatives, Rolle’s Theorem, Cauchy’s Generalized Mean Value Theorem, Indeterminate Forms 0/0, ∞/∞,0·∞,∞−∞,00, ∞0, 1, Maximum and Minimum Values, Taylor Series for Functions of One Variable, Examples of Series Expansions, Taylor Series for Functions of Two Variables, Summary of Differentiation Rules, The Indefinite Integral, Rules for Integration, Differentiation and Integration, The Exponential and Logarithmic Functions, The Definite Integral, Improper Integrals, Bliss’s Theorem, Arc Length Formula, Area Formulas, Average Value of a Function, Volumes, Surface Area, Double Integrals, Double Integrals in Polar Coordinates, Triple Integrals, Evaluation of Integrals ∫ f(x)dx, Substitutions, Reduction Formula, Differentiation and Integration of Arrays, Inequalities Involving Integrals, General Series, Weierstrass Approximation Theorems, Infinite Series of Functions, Numerical Integration

Chapter 5 Vector Calculus 

Introduction, Vector Addition and Subtraction, Unit Vectors, Scalar or Dot Product (inner product), Direction Cosines Associated With Vectors, The Cross Product or Outer Product, Properties of the Cross Product, Scalar and Vector Fields, The Gradient, Integration of Vectors, Line Integrals of Scalar and Vector Functions, Representation of Line Integrals, Surface and Volume Integrals, Surface Placed in a Scalar Field, Surface Placed in a Vector Field, Surface Area from Parametric Form, Volume Integrals, Divergence, Gauss Divergence Theorem, Physical Interpretation of Divergence, Green’s Theorem in the Plane, Solution of Differential Equations by Line Integrals, Change of Variable in Integration, The Curl of a Vector Field, Physical Interpretation of Curl, Stokes’ Theorem, Related Integral Theorems, Region of Integration, Green’s First and Second Identities, Additional Operators, Properties of the Del Operator, Curvilinear Coordinates, Left and Right-handed Coordinate Systems, Gradient, Divergence, Curl and Laplacian

Chapter 6 Ordinary Differential Equations 

Linear Differential Operators, Solutions to Differential Equations, Existence and Uniqueness of Solutions, Solutions Containing a Complex Number, Differential Equations Easily Solved, Exact Differential Equations, Linear First Order Differential Equations, Dependent Variable Absent, Independent Variable Absent, Parametric Solutions to Differential Equations, Linear nth Order Differential Equations , Solutions to nth Order Linear Homogeneous Differential Equations, Wronskian Determinant, Characteristic Equation, The Phase Plane, Boundary Value Problems, Solution of Nonhomogeneous Linear Differential Equations, Method of Undetermined Coefficients, Variation of Parameters, Differential Equations with Variable Coefficients, Cauchy or Euler Equations, Second Order Exact Differential Equations, Adjoint Operators, Forms Associated with Second Order Differential Equations, Abel’s Formula, Self Adjoint Form, Nonhomogeneous Equations, Series Representation, Solution of Differential Equations by Series Methods, Numerical Solution to First Order Differential Equations

Chapter 7 Special Functions 

Integer Function, Heaviside Step Function, Impulse Function and Dirac Delta Function, Inverse Trigonometric Functions, The Gamma Function, The Beta Function, Bessel functions, Modified Bessel’s Equation, The hypergeometric function, Recursion Formula, Generalized hypergeometric function, The Riemann Differential Equation, Generating Functions, Orthogonal Functions, Sturm-Liouville systems, Orthogonality, Fourier Trigonometric Series, Sturm- Liouville Theorem, Fourier Integral Theorem, The Fresnel Integrals, Integral Equations, The Error Function, The Sine, Cosine and Exponential Integrals, Elliptic Integral of the First Kind, Elliptic Integral of the Second Kind, Elliptic Integral of the Third Kind, Riemann Zeta Function, The Laplace Transform, The Fourier Transforms

Chapter 8 Probability and Statistics 

Introduction, The Representation of Data, Tabular Representation of Data, Arithmetic Mean or Sample Mean, Median, Mode and Percentiles, The Geometric and Harmonic Mean, The Root Mean Square (RMS), Mean Deviation and Sample Variance, Probability, Probability Fundamentals, Probability of an Event, Conditional Probability, Discrete and Continuous Probability Distributions, Scaling, The Normal Distribution, Standardization, The Binomial Distribution, The Multinomial Distribution, The Poisson Distribution, The Hypergeometric Distribution, The Exponential Distribution, The Gamma Distribution, Chi-Square χ2 Distribution, Student’s t-Distribution, The F-Distribution, The Uniform Distribution, Confidence Intervals, Least Squares Curve Fitting, Linear Regression, Monte Carlo Methods, Linear Interpolation

Chapter 9 Selected Applied Mathematics Topics 

Motion of a Particle (Dynamics), Kepler’s Laws, Moment of a Force, Center of Mass for System of Particles, Center of Mass for Plane Areas, Centroids and Volumes, Second Moments or Moments of Inertia, Angular Velocity, Angular Momentum, Moments and Newton’s Second Law, Impulse-Momentum Laws, Euler Angles, Space Curves, Curvature, and Torsion, Velocity and Acceleration, Relative Motion, Mechanical Vibrations, Phenomenon of Beats, Vibrations of a Spring Mass System, Mechanical Resonance, Torsional Vibrations, Solid Angles, Laplace’s Equation, The Periodic Table of Elements, Modeling of Chemical Kinetics, Thermodynamics, Electrical Circuits, Fourterminal networks, Prisms


Appendix A Units of Measurement 

Appendix B Table of Integrals 

Appendix C Miscellaneous Topics