Essentials of Engineering Mathematics

WORKED EXAMPLES AND PROBLEMS

Alan Jeffrey Professor of Engineering Mathematics

University of Newcastle upon Tyne

CHAPMAN & HALL University and Professional Division

London • Glasgow • New York • Tokyo • Melbourne • Madras Ei

Contents

Preface

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

22 23 24 25 26 27 28

29 30 31 32

Real numbers, inequalities and intervals 1 Function, domain and ränge 12 Basic coordinate geometry 17 Polar coordinates 35 Mathematical induction 40 Binomial theorem 44 Combination of functions 50 Symmetry in functions and graphs 55 Inverse functions 60 Compiex numbers: real and imaginary forms 66 Geometry of compiex numbers 75 Modulus-argument form of a compiex number 81 Roots of compiex numbers 87 Limits 92 One-sided limits: continuity 101 Derivatives 109 Leibniz's formula 123 Differentials 126 Differentiation of inverse trigonometric functions 133 Implicit differentiation 136 Parametrically defined curves and parametric differentiation 141 The exponential function 148 The logarithmic function 156 Hyperbolic functions 164 Inverse hyperbolic functions 169 Properties and applications of differentiability 174 Functions of two variables 191 Limits and continuity of functions of two real variables 197 Partial differentiation 203 The total differential 215 The chain rule 222 Change of variable in partial differentiation 228

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33 34 35 36 37

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40 41 42 43 44

45 46 47 48 49 50

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57 58 59 60 61 62 63 64 65 66 67

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Antidifferentiation (Integration) Integration by Substitution Some useful Standard forms Integration by parts Partial fractions and integration of rational functions The definite integral The fundamental theorem of integral calculus and the evaluation of definite integrals Improper integrals Numerical integration Geometrical applications of definite inegrals Centre of mass of a plane lamina (centroid) Applications of integration to the hydrostatic pressure on a plate Moments of inertia Sequences Infinite numerical series Power series Taylor and Maclaurin series Taylor's theorem for functions of two variables: stationary points and their identification Fourier series Determinants Matrices: equality, addition, subtraction, scaling and transposition Matrix multiplication The inverse matrix Solution of a System of linear equations: Gaussian elimination The Gauss-Seidel iterative method The algebraic eigenvalue problem Sealars, vectors and vector addition Vectors in component form The straight line The scalar produet (dot produet) The plane The vector produet (cross produet) Applications of the vector produet Differentiation and integration of vectors Dynamics of a particle and the motion of a particle in a plane Scalar and vector fields and the gradient of a scalar funetion Ordinary differential equations: order and degree, initial and boundary conditions

236 247 260 264

276 287

295 310 316 323 332

340 347 350 353 373 381

401 410 429

444 450 461

468 478 485 494 500 508 512 517 522 529 539

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Answers Referem Index

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71 72 73 74 75 76

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First order differential equations solvable by Separation of variables The method of isociines and Euler's methods Homogeneous and near homogeneous equations Exact differential equations The first order linear differential equation The Bernoulli equation The structure of Solutions of linear differential equations of any order Determining the complementary function for constant coefficient equations Determining particular integrals of constant coefficient equations Differential equations describing oscillations Simultaneous first order linear constant coefficient differential equations The Laplace transform and transform pairs The Laplace transform of derivatives The shift theorems and the Heaviside step function Solution of initial value problems Enlarging the list of Laplace transform pairs

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575 581 591 598 603 609

611

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626 635-

643 648 655

657 666 677

681 810 820