Mathematics for Engineers and ScientistsBy Alan Jeffrey

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