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

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