# Mathematics for Engineers and Scientists

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## Book Description

Since its original publication in 1969, Mathematics for Engineers and Scientists has built a solid foundation in mathematics for legions of undergraduate science and engineering students. It continues to do so, but as the influence of computers has grown and syllabi have evolved, once again the time has come for a new edition.

Thoroughly revised to meet the needs of today's curricula, Mathematics for Engineers and Scientists, Sixth Edition covers all of the topics typically introduced to first- or second-year engineering students, from number systems, functions, and vectors to series, differential equations, and numerical analysis. Among the most significant revisions to this edition are:

Although designed as a textbook with problem sets in each chapter and selected answers at the end of the book, Mathematics for Engineers and Scientists, Sixth Edition serves equally well as a supplemental text and for self-study. The author strongly encourages readers to make use of computer algebra software, to experiment with it, and to learn more about mathematical functions and the operations that it can perform.

## Table of Contents

NUMBERS, TRIGONOMETRIC FUNCTIONS AND COORDINATE GEOMETRY

Sets and numbers

Integers, rationals and arithmetic laws

Absolute value of a real number

Mathematical induction

Review of trigonometric properties

Cartesian geometry

Polar coordinates

Completing the square

Logarithmic functions

Greek symbols used in mathematics

VARIABLES, FUNCTIONS AND MAPPINGS

Variables and functions

Inverse functions

Some special functions

Curves and parameters

Functions of several real variables

SEQUENCES, LIMITS AND CONTINUITY

Sequences

Limits of sequences

The number e

Limits of functions -/ continuity

Functions of several variables -/ limits, continuity

A useful connecting theorem

Asymptotes

COMPLEX NUMBERS AND VECTORS

Introductory ideas

Basic algebraic rules for complex numbers

Complex numbers as vectors

Modulus -/ argument form of complex numbers

Roots of complex numbers

Introduction to space vectors

Scalar and vector products

Geometrical applications

Applications to mechanics

Problems

DIFFERENTIATION OF FUNCTIONS OF ONE OR MORE REAL VARIABLES

The derivative

Rules of differentiation

Some important consequences of differentiability

Higher derivatives _/ applications

Partial differentiation

Total differentials

Envelopes

The chain rule and its consequences

Change of variable

Some applications of dy/dx=1/ dx/dy

Higher-order partial derivatives

EXPONENTIAL, LOGARITHMIC AND HYPERBOLIC FUNCTIONS AND AN INTRODUCTION TO COMPLEX FUNCTIONS

The exponential function

Differentiation of functions involving the exponential function

The logarithmic function

Hyperbolic functions

Exponential function with a complex argument

Functions of a complex variable, limits, continuity and differentiability

FUNDAMENTALS OF INTEGRATION

Definite integrals and areas

Integration of arbitrary continuous functions

Integral inequalities

The definite integral as a function of its upper limit -/ the indefinite integral

Differentiation of an integral containing a parameter

Other geometrical applications of definite integrals

Centre of mass and moment of inertia

Line integrals

SYSTEMATIC INTEGRATION

Integration of elementary functions

Integration by substitution

Integration by parts

Reduction formulae

Integration of rational functions - partial fractions

Other special techniques of integration

Integration by means of tables

Problems

DOUBLE INTEGRALS IN CARTESIAN AND PLANE POLAR COORDINATES

Double integrals in Cartesian coordinates

Double integrals using polar coordinates

Problems

MATRICES AND LINEAR TRANSFORMATIONS

Matrix algebra

Determinants

Linear dependence and linear independence

Inverse and adjoint matrices

Matrix functions of a single variable

Solution of systems of linear equations

Eigenvalues and eigenvectors

Matrix interpretation of change of variables in partial differentiation

Linear transformations

Applications of matrices and linear transformations

Problems

SCALARS, VECTORS AND FIELDS

Curves in space

Antiderivatives and integrals of vector functions

Some applications

Fields, gradient and directional derivative

Divergence and curl of a vector

Conservative fields and potential functions

Problems

SERIES, TAYLOR'S THEOREM AND ITS USES

Series

Power series

Taylor's theorem

Applications of Taylor's theorem

Applications of the generalized mean value theorem

DIFFERENTIAL EQUATIONS AND GEOMETRY

Introductory ideas

Possible physical origin of some equations

Arbitrary constants and initial conditions

First-order equations - direction fields and isoclines

Orthogonal trajectories

First-order differential equations

Equations with separable variables

Homogeneous equations

Exact equations

The linear equation of first order

Direct deductions

HIGHER-ORDER LINEAR DIFFERENTIAL EQUATIONS

Linear equations with constant coefficients _/ homogeneous case

Linear equations with constant coefficients _/ inhomogeneous case

Variation of parameters

Oscillatory solutions

Coupled oscillations and normal modes

Systems of first-order equations

Two-point boundary value problems

The Laplace transform

The Delta function

Applications of the Laplace transform

FOURIER SERIES

Introductory ideas

Convergence of Fourier series

Different forms of Fourier series

Differentiation and integration

NUMERICAL ANALYSIS

Errors and efficient methods of calculation

Solution of linear equations

Interpolation

Numerical integration

Solution of polynomial and transcendental equations

Numerical solutions of differential equations

Determination of eigenvalues and eigenvectors

PROBABILITY AND STATISTICS

The elements of set theory for use in probability and statistics

Probability, discrete distributions and moments

Continuous distributions and the normal distribution

Mean and variance of a sum of random variables

Statistics - inference drawn from observations

Linear regression

SYMBOLIC ALGEBRAIC MANIPULATION BY COMPUTER SOFTWARE

Maple

MATLAB

ANSWERS

REFERENCE LISTS:

Useful identities and constants

Basic derivatives and rules

Laplace transform pairs

Short table of integrals

INDEX