Bird’s Comprehensive Engineering Mathematics

Description

Contents......Page 3 Preface......Page 14 Section A Number and algebra......Page 16 1.1 Introduction......Page 17 1.2 Revision of addition and subtraction......Page 18 1.3 Revision of multiplication and division......Page 19 1.4 Highest common factors and lowest common multiples......Page 21 1.5 Order of operation and brackets......Page 23 2.1 Introduction......Page 25 2.2 Adding and subtracting fractions......Page 26 2.3 Multiplication and division of fractions......Page 28 2.4 Order of operation with fractions......Page 30 Revision Test 1 Basic arithmetic and fractions......Page 32 3.2 Converting decimals to fractions and vice versa......Page 33 3.3 Significant figures and decimal places......Page 35 3.4 Adding and subtracting decimal numbers......Page 36 3.5 Multiplying and dividing decimal numbers......Page 37 4.2 Adding, subtracting, multiplying and dividing......Page 39 4.3 Further calculator functions......Page 41 4.4 Errors and approximations......Page 46 4.6 Evaluation of formulae......Page 47 5.1 Introduction......Page 52 5.2 Percentage calculations......Page 53 5.3 Further percentage calculations......Page 54 5.4 More percentage calculations......Page 56 Revision Test 2 Decimals, calculations and percentages......Page 59 6.1 Introduction......Page 60 6.2 Ratios......Page 61 6.3 Direct proportion......Page 63 6.4 Inverse proportion......Page 67 7.2 Powers and roots......Page 69 7.3 Laws of indices......Page 71 8.2 SI units......Page 76 8.3 Common prefixes......Page 77 8.4 Standard form......Page 80 8.5 Engineering notation......Page 82 8.6 Metric conversions......Page 84 8.7 Metric – US/Imperial conversions......Page 87 Revision Test 3 Ratio, proportion, powers, roots, indices and units......Page 92 9.1 Introduction......Page 93 9.2 Basic operations......Page 94 9.3 Laws of indices......Page 97 10.2 Brackets......Page 101 10.3 Factorisation......Page 103 10.4 Laws of precedence......Page 104 Multiple choice questions Test 1......Page 107 11.2 Solving equations......Page 109 11.3 Practical problems involving simple equations......Page 113 Revision Test 4 Algebra and simple equations......Page 118 12.2 Transposing formulae......Page 119 12.3 Further transposing of formulae......Page 121 12.4 More difficult transposing of formulae......Page 124 13.1 Introduction......Page 127 107.4 Using z-transforms to solve difference equations......Page 128 13.3 Further solving of simultaneous equations......Page 129 13.4 Solving more difficult simultaneous equations......Page 131 13.5 Practical problems involving simultaneous equations......Page 133 13.6 Solving simultaneous equations in three unknowns......Page 137 Revision Test 5 Transposition and simultaneous equations......Page 139 14.2 Solution of quadratic equations by factorisation......Page 140 14.3 Solution of quadratic equations by ‘completing the square......Page 143 14.4 Solution of quadratic equations by formula......Page 145 14.5 Practical problems involving quadratic equations......Page 146 14.6 Solution of linear and quadratic equations simultaneously......Page 148 Multiple choice questions Test 2......Page 150 15.1 Introduction to logarithms......Page 152 15.2 Laws of logarithms......Page 154 15.3 Indicial equations......Page 157 15.4 Graphs of logarithmic functions......Page 158 16.1 Introduction to exponential functions......Page 160 16.2 The power series for ex......Page 161 16.3 Graphs of exponential functions......Page 163 16.4 Napierian logarithms......Page 165 16.5 Laws of growth and decay......Page 168 17.1 Introduction to inequalities......Page 172 17.3 Inequalities involving a modulus......Page 173 17.4 Inequalities involving quotients......Page 174 17.5 Inequalities involving square functions......Page 175 17.6 Quadratic inequalities......Page 176 Revision Test 6 Quadratics, logarithms, exponentials and inequalities......Page 179 Formulae/revision hints for Section A......Page 180 Section B Further number and algebra......Page 182 18.1 Polynomial division......Page 183 18.2 The factor theorem......Page 185 18.3 The remainder theorem......Page 187 19.1 Introduction to partial fractions......Page 190 19.2 Worked problems on partial fractions with linear factors......Page 191 19.3 Worked problems on partial fractions with repeated linear factors......Page 193 19.4 Worked problems on partial fractions with quadratic factors......Page 195 20.1 Simple sequences......Page 197 20.2 The nth term of a series......Page 198 20.3 Arithmetic progressions......Page 199 20.4 Geometric progressions......Page 202 20.5 Combinations and permutations......Page 205 Revision Test 7 Further algebra, partial fractions and number sequences......Page 207 21.1 Pascal’s triangle......Page 208 21.3 Worked problems on the binomial series......Page 210 21.4 Further worked problems on the binomial series......Page 212 21.5 Practical problems involving the binomial theorem......Page 214 Chapter 22 Maclaurin’s series......Page 217 22.2 Derivation of Maclaurin’s theorem......Page 218 22.4 Worked problems on Maclaurin’s series......Page 219 22.5 Numerical integration using Maclaurin’s series......Page 222 22.6 Limiting values......Page 224 Revision Test 8 The binomial and Maclaurin’s series......Page 227 23.1 Introduction to iterative methods......Page 228 23.2 The bisection method......Page 229 23.3 An algebraic method of successive approximations......Page 233 23.4 The Newton–Raphson method......Page 236 24.1 Introduction to hyperbolic functions......Page 239 24.2 Graphs of hyperbolic functions......Page 241 24.3 Hyperbolic identities......Page 243 24.4 Solving equations involving hyperbolic functions......Page 245 24.5 Series expansions for cosh x and sinh x......Page 247 Revision Test 9 Iterative methods and hyperbolic functions......Page 249 25.1 Introduction......Page 250 25.2 Binary numbers......Page 251 25.3 Octal numbers......Page 255 25.4 Hexadecimal numbers......Page 257 Chapter 26 Boolean algebra and logic circuits......Page 261 26.1 Boolean algebra and switching circuits......Page 262 26.3 Laws and rules of Boolean algebra......Page 266 26.4 De Morgan’s laws......Page 268 26.5 Karnaugh maps......Page 270 26.6 Logic circuits......Page 274 26.7 Universal logic gates......Page 277 Revision Test 10 Numbering systems, Boolean algebra and logic circuits......Page 281 Multiple choice questions Test 3......Page 282 Formulae/revision hints for Section B......Page 284 Section C Areas and volumes......Page 286 27.2 Common shapes......Page 287 27.3 Areas of common shapes......Page 290 27.4 Areas of similar shapes......Page 298 28.2 Properties of circles......Page 299 28.3 Radians and degrees......Page 301 28.4 Arc length and area of circles and sectors......Page 302 28.5 The equation of a circle......Page 306 28.6 Linear and angular velocity......Page 307 28.7 Centripetal force......Page 309 Revision Test 11 Areas of common shapes and the circle......Page 311 29.1 Introduction......Page 313 29.2 Volumes and surface areas of common shapes......Page 314 29.3 Summary of volumes and surface areas of common solids......Page 320 29.4 More complex volumes and surface areas......Page 321 29.5 Volumes and surface areas of frusta of pyramids and cones......Page 326 29.6 The frustum and zone of a sphere......Page 329 29.7 Prismoidal rule......Page 332 29.8 Volumes of similar shapes......Page 334 30.1 Areas of irregular figures......Page 335 30.2 Volumes of irregular solids......Page 338 30.3 Mean or average values of waveforms......Page 339 Revision Test 12 Volumes, irregular areas and volumes, and mean values......Page 343 Formulae/revision hints for Section C......Page 345 Section D Graphs......Page 349 31.2 Axes, scales and co-ordinates......Page 350 31.3 Straight line graphs......Page 352 31.4 Gradients, intercepts and equations of graphs......Page 355 31.5 Practical problems involving straight line graphs......Page 362 32.2 Determination of law......Page 369 32.3 Revision of laws of logarithms......Page 372 32.4 Determination of laws involving logarithms......Page 373 33.1 Logarithmic scales and logarithmic graph paper......Page 378 33.2 Graphs of the form y = axn......Page 379 33.3 Graphs of the form y = abx......Page 382 33.4 Graphs of the form y = aekx......Page 383 Revision Test 13 Graphs......Page 386 34.2 Worked problems on polar curves......Page 388 35.1 Graphical solution of simultaneous equations......Page 392 35.2 Graphical solution of quadratic equations......Page 394 35.4 Graphical solution of cubic equations......Page 398 36.2 Standard curves......Page 401 36.3 Simple transformations......Page 404 36.4 Periodic functions......Page 409 36.6 Even and odd functions......Page 410 36.7 Inverse functions......Page 412 36.8 Asymptotes......Page 414 36.10 Worked problems on curve sketching......Page 420 Revision Test 14 Polar curves, graphical solution of equations and functions and their curves......Page 424 Multiple choice questions Test 4......Page 425 Formulae/revision hints for Section D......Page 429 Section E Geometry and trigonometry......Page 430 37.2 Angular measurement......Page 431 37.3 Triangles......Page 437 37.4 Congruent triangles......Page 441 37.5 Similar triangles......Page 443 37.6 Construction of triangles......Page 445 38.1 Introduction......Page 447 38.2 The theorem of Pythagoras......Page 448 38.3 Sines, cosines and tangents......Page 450 38.4 Evaluating trigonometric ratios of acute angles......Page 452 38.5 Reciprocal ratios......Page 455 38.6 Fractional and surd forms of trigonometric ratios......Page 457 38.7 Solving right-angled triangles......Page 458 38.8 Angles of elevation and depression......Page 461 38.9 Trigonometric approximations for small angles......Page 463 Revision Test 15 Angles, triangles and trigonometry......Page 464 39.1 Graphs of trigonometric functions......Page 467 39.2 Angles of any magnitude......Page 468 39.3 The production of sine and cosine waves......Page 471 39.4 Terminology involved with sine and cosine waves......Page 472 39.5 Sinusoidal form: A sin(.t ± a......Page 475 39.6 Complex waveforms......Page 476 40.2 Changing from Cartesian to polar co-ordinates......Page 482 40.3 Changing from polar to Cartesian co-ordinates......Page 484 40.4 Use of Pol/Rec functions on calculators......Page 485 41.1 The sine and cosine rules......Page 487 41.3 Worked problems on the solution of triangles and their areas......Page 488 41.4 Further worked problems on the solution of triangles and their areas......Page 490 41.5 Practical situations involving trigonometry......Page 491 41.6 Further practical situations involving trigonometry......Page 493 Revision Test 16 Trigonometric waveforms, Cartesian and polar co-ordinates and non-right-angled triangles......Page 496 42.1 Trigonometric identities......Page 497 42.2 Worked problems on trigonometric identities......Page 498 42.3 Trigonometric equations......Page 499 42.4 Worked problems (i) on trigonometric equations......Page 500 42.5 Worked problems (ii) on trigonometric equations......Page 501 42.7 Worked problems (iv) on trigonometric equations......Page 502 43.1 The relationship between trigonometric and hyperbolic functions......Page 504 43.2 Hyperbolic identities......Page 505 44.1 Compound angle formulae......Page 508 44.2 Conversion of a sin .t +b cos .t into R sin(.t +a......Page 510 44.3 Double angles......Page 514 44.4 Changing products of sines and cosines into sums or differences......Page 516 44.5 Changing sums or differences of sines and cosines into products......Page 517 44.6 Power waveforms in a.c. circuits......Page 518 Revision Test 17 Trigonometric identities and equations and compound angles......Page 522 Multiple choice questions Test 5......Page 523 Formulae/revision hints for Section E......Page 527 Section F Complex numbers......Page 529 45.1 Cartesian complex numbers......Page 530 45.3 Addition and subtraction of complex numbers......Page 532 45.4 Multiplication and division of complex numbers......Page 533 45.6 The polar form of a complex number......Page 535 45.7 Multiplication and division in polar form......Page 537 45.8 Applications of complex numbers......Page 538 45.9 Using a calculator with complex numbers......Page 541 Chapter 46 De Moivre’s theorem......Page 543 46.2 Powers of complex numbers......Page 544 46.3 Roots of complex numbers......Page 545 46.4 The exponential form of a complex number......Page 547 46.5 Introduction to locus problems......Page 548 Revision Test 18 Complex numbers and De Moivre’s theorem......Page 552 Formulae/revision hints for Section F......Page 553 Section G Matrices and determinants......Page 554 47.1 Matrix notation......Page 555 47.2 Addition, subtraction and multiplication of matrices......Page 556 47.4 The determinant of a 2 by 2 matrix......Page 559 47.5 The inverse or reciprocal of a 2 by 2 matrix......Page 560 47.6 The determinant of a 3 by 3 matrix......Page 561 47.7 The inverse or reciprocal of a 3 by 3 matrix......Page 562 48.1 Solution of simultaneous equations by matrices......Page 565 48.2 Solution of simultaneous equations by determinants......Page 568 48.3 Solution of simultaneous equations using Cramer’s rule......Page 571 48.4 Solution of simultaneous equations using the Gaussian elimination method......Page 572 48.5 Eigenvalues and eigenvectors......Page 574 Revision Test 19 Matrices and determinants......Page 579 Formulae/revision hints for Section G......Page 580 Section H Vector geometry......Page 581 49.2 Scalars and vectors......Page 582 49.4 Addition of vectors by drawing......Page 583 49.5 Resolving vectors into horizontal and vertical components......Page 586 49.6 Addition of vectors by calculation......Page 587 49.7 Vector subtraction......Page 591 49.8 Relative velocity......Page 593 49.9 i, j and k notation......Page 594 50.1 Combination of two periodic functions......Page 596 50.2 Plotting periodic functions......Page 597 50.3 Determining resultant phasors by drawing......Page 598 50.4 Determining resultant phasors by the sine and cosine rules......Page 600 50.5 Determining resultant phasors by horizontal and vertical components......Page 601 50.6 Determining resultant phasors by complex numbers......Page 603 51.1 The unit triad......Page 607 51.2 The scalar product of two vectors......Page 608 51.3 Vector products......Page 612 51.4 Vector equation of a line......Page 615 Revision Test 20 Vectors......Page 618 Multiple choice questions Test 6......Page 619 Formulae/revision hints for Section H......Page 622 Section I Differential calculus......Page 623 52.2 Functional notation......Page 624 52.3 The gradient of a curve......Page 625 52.4 Differentiation from first principles......Page 626 52.5 Differentiation of y = axn by the general rule......Page 629 52.6 Differentiation of sine and cosine functions......Page 630 52.7 Differentiation of eax and ln ax......Page 632 53.1 Differentiation of common functions......Page 634 53.2 Differentiation of a product......Page 636 53.3 Differentiation of a quotient......Page 638 53.4 Function of a function......Page 639 53.5 Successive differentiation......Page 640 54.1 Rates of change......Page 643 54.2 Velocity and acceleration......Page 645 54.3 Turning points......Page 647 54.4 Practical problems involving maximum and minimum values......Page 651 54.5 Points of inflexion......Page 654 54.6 Tangents and normals......Page 656 54.7 Small changes......Page 657 Revision Test 21 Methods of differentiation and some applications......Page 659 55.1 Introduction to parametric equations......Page 660 55.3 Differentiation in parameters......Page 661 55.4 Further worked problems on differentiation of parametric equations......Page 663 56.2 Differentiating implicit functions......Page 666 56.3 Differentiating implicit functions containing products and quotients......Page 667 56.4 Further implicit differentiation......Page 668 57.2 Laws of logarithms......Page 671 57.4 Differentiation of further logarithmic functions......Page 672 57.5 Differentiation of [f (x)] x......Page 674 Revision Test 22 Parametric, implicit and logarithmic differentiation......Page 676 58.1 Standard differential coefficients of hyperbolic functions......Page 677 58.2 Further worked problems on differentiation of hyperbolic functions......Page 678 59.1 Inverse functions......Page 680 59.2 Differentiation of inverse trigonometric functions......Page 682 59.3 Logarithmic forms of inverse hyperbolic functions......Page 685 59.4 Differentiation of inverse hyperbolic functions......Page 687 60.2 First-order partial derivatives......Page 691 60.3 Second-order partial derivatives......Page 694 61.1 Total differential......Page 697 61.2 Rates of change......Page 698 61.3 Small changes......Page 700 62.1 Functions of two independent variables......Page 703 62.2 Maxima, minima and saddle points......Page 704 62.4 Worked problems on maxima, minima and saddle points for functions of two variables......Page 705 62.5 Further worked problems on maxima, minima and saddle points for functions of two variables......Page 708 Revision Test 23 Further differentiation and applications......Page 713 Multiple choice questions Test 7......Page 714 Formulae/revision hints for Section I......Page 716 Section J Integral calculus......Page 719 63.1 The process of integration......Page 720 63.3 Standard integrals......Page 721 63.4 Definite integrals......Page 724 64.2 Algebraic substitutions......Page 727 64.3 Worked problems on integration using algebraic substitutions......Page 728 64.4 Further worked problems on integration using algebraic substitutions......Page 729 64.5 Change of limits......Page 730 65.2 Worked problems on integration of sin2 x, cos2 x, tan2 x and cot2 x......Page 732 65.3 Worked problems on integration of powers of sines and cosines......Page 734 65.4 Worked problems on integration of products of sines and cosines......Page 736 65.5 Worked problems on integration using the sin . substitution......Page 737 65.6 Worked problems on integration using the tan . substitution......Page 738 65.7 Worked problems on integration using the sinh . substitution......Page 739 65.8 Worked problems on integration using the cosh . substitution......Page 740 Revision Test 24 Standard integration and some substitution methods......Page 743 66.2 Worked problems on integration using partial fractions with linear factors......Page 744 66.3 Worked problems on integration using partial fractions with repeated linear factors......Page 746 66.4 Worked problems on integration using partial fractions with quadratic factors......Page 747 67.1 Introduction......Page 749 67.2 Worked problems on the t = tan . 2 substitution......Page 750 67.3 Further worked problems on the t = tan . 2 substitution......Page 751 68.2 Worked problems on integration by parts......Page 754 68.3 Further worked problems on integration by parts......Page 756 Revision Test 25 Integration using partial fractions, tan ./2 and ‘by parts......Page 760 69.2 Using reduction formulae for integrals of the form xnex dx......Page 761 69.3 Using reduction formulae for integrals of the form xn cosx dx and xn sin x dx......Page 762 69.4 Using reduction formulae for integrals of the form sinn x dx and cosn x dx......Page 765 69.5 Further reduction formulae......Page 768 70.1 Double integrals......Page 770 70.2 Triple integrals......Page 772 71.2 The trapezoidal rule......Page 774 71.3 The mid-ordinate rule......Page 777 71.4 Simpson’s rule......Page 778 71.5 Accuracy of numerical integration......Page 782 Revision Test 26 Reduction formulae, double and triple integrals, and numerical integration......Page 783 72.1 Area under a curve......Page 784 72.2 Worked problems on the area under a curve......Page 785 72.3 Further worked problems on the area under a curve......Page 788 72.4 The area between curves......Page 791 73.1 Mean or average values......Page 793 73.2 Root mean square values......Page 795 74.1 Introduction......Page 798 74.2 Worked problems on volumes of solids of revolution......Page 799 74.3 Further worked problems on volumes of solids of revolution......Page 801 75.2 The first moment of area......Page 803 75.5 Worked problems on centroids of simple shapes......Page 804 75.6 Further worked problems on centroids of simple shapes......Page 806 75.7 Theorem of Pappus......Page 808 76.1 Second moments of area and radius of gyration......Page 812 76.4 Perpendicular axis theorem......Page 813 76.5 Summary of derived results......Page 814 76.6 Worked problems on second moments of area of regular sections......Page 815 76.7 Worked problems on second moments of area of composite areas......Page 818 Revision Test 27 Applications of integration......Page 821 Multiple choice questions Test 8......Page 822 Formulae/revision hints for Section J......Page 824 Section K Differential equations......Page 828 77.1 Family of curves......Page 829 77.2 Differential equations......Page 830 77.3 The solution of equations of the form dy dx = f (x......Page 831 77.4 The solution of equations of the form dy dx = f (y......Page 832 77.5 The solution of equations of the form dy dx = f (x) · f (y......Page 834 78.2 Procedure to solve differential equations of the form P dy dx = Q......Page 837 78.3 Worked problems on homogeneous first-order differential equations......Page 838 78.4 Further worked problems on homogeneous first-order differential equations......Page 839 79.1 Introduction......Page 841 79.3 Worked problems on linear first-order differential equations......Page 842 79.4 Further worked problems on linear first-order differential equations......Page 843 80.1 Introduction......Page 846 80.2 Euler’s method......Page 847 80.3 Worked problems on Euler’s method......Page 848 80.4 The Euler–Cauchy method......Page 852 80.5 The Runge–Kutta method......Page 857 Revision Test 28 First-order differential equations......Page 863 81.1 Introduction......Page 864 81.3 Worked problems on differential equations of the form a d2y dx2 +b dy dx +cy = 0......Page 865 81.4 Further worked problems on practical differential equations of the form a d2y dx2 +b dy dx +cy = 0......Page 867 82.1 Complementary function and particular integral......Page 871 82.3 Worked problems on differential equations of the form a d2y dx2 +b dy dx +cy = f (x) where f (x) is a constant or polynomial......Page 872 82.4 Worked problems on differential equations of the form a d2y dx2 +b dy dx +cy = f (x) where f (x) is an exponential function......Page 874 82.5 Worked problems on differential equations of the form a d2y dx2 +b dy dx +cy = f (x) where f (x) is a sine or cosine function......Page 876 82.6 Worked problems on differential equations of the form a d2y dx2 +b dy dx +cy = f (x) where f (x) is a sum or a product......Page 878 Chapter 83 Power series methods of solving ordinary differential equations......Page 881 83.2 Higher-order differential coefficients as series......Page 882 83.3 Leibniz’s theorem......Page 884 83.4 Power series solution by the Leibniz–Maclaurin method......Page 885 83.5 Power series solution by the Frobenius method......Page 888 83.6 Bessel’s equation and Bessel’s functions......Page 895 83.7 Legendre’s equation and Legendre polynomials......Page 900 Chapter 84 An introduction to partial differential equations......Page 905 84.3 Solution of partial differential equations by direct partial integration......Page 906 84.4 Some important engineering partial differential equations......Page 908 84.5 Separating the variables......Page 909 84.6 The wave equation......Page 910 84.7 The heat conduction equation......Page 914 84.8 Laplace’s equation......Page 916 Revision Test 29 Second-order differential equations, power series methods and partial differential equations......Page 919 Formulae/revision hints for Section K......Page 920 Section L Statistics and probability......Page 923 Chapter 85 Presentation of statistical data......Page 924 85.1 Some statistical terminology......Page 925 85.2 Presentation of ungrouped data......Page 926 85.3 Presentation of grouped data......Page 929 86.1 Measures of central tendency......Page 936 86.2 Mean, median and mode for discrete data......Page 937 86.3 Mean, median and mode for grouped data......Page 938 86.4 Standard deviation......Page 939 86.5 Quartiles, deciles and percentiles......Page 941 Chapter 87 Probability......Page 943 87.1 Introduction to probability......Page 944 87.2 Laws of probability......Page 945 87.3 Permutations and combinations......Page 949 87.4 Bayes’ theorem......Page 950 Revision Test 30 Presentation of statistical data, mean, median, mode, standard deviation and probability......Page 953 88.1 The binomial distribution......Page 955 88.2 The Poisson distribution......Page 959 89.1 Introduction to the normal distribution......Page 962 89.2 Testing for a normal distribution......Page 967 90.2 The Pearson product-moment formula for determining the linear correlation coefficient......Page 970 90.4 Worked problems on linear correlation......Page 971 91.2 The least-squares regression lines......Page 975 91.3 Worked problems on linear regression......Page 976 Revision Test 31 Binomial, Poisson and normal distributions, correlation and regression......Page 981 92.2 Sampling distributions......Page 982 92.3 The sampling distribution of the means......Page 983 92.4 The estimation of population parameters based on a large sample size......Page 986 92.5 Estimating the mean of a population based on a small sample size......Page 991 93.1 Hypotheses......Page 995 93.2 Type I and type II errors......Page 996 93.3 Significance tests for population means......Page 1002 93.4 Comparing two sample means......Page 1007 94.1 Chi-square values......Page 1012 94.2 Fitting data to theoretical distributions......Page 1014 94.4 The sign test......Page 1021 94.5 Wilcoxon signed-rank test......Page 1024 94.6 The Mann–Whitney test......Page 1027 Revision Test 32 Sampling and estimation theories, significance testing, chi-square and distribution-free tests......Page 1034 Multiple choice questions Test 9......Page 1036 Formulae/revision hints for Section L......Page 1038 Section M Laplace transforms......Page 1040 95.2 Definition of a Laplace transform......Page 1041 95.4 Laplace transforms of elementary functions......Page 1042 95.5 Worked problems on standard Laplace transforms......Page 1043 96.2 Laplace transforms of the form eatf (t......Page 1046 96.3 The Laplace transforms of derivatives......Page 1048 96.4 The initial and final value theorems......Page 1050 97.2 Inverse Laplace transforms of simple functions......Page 1052 97.3 Inverse Laplace transforms using partial fractions......Page 1055 97.4 Poles and zeros......Page 1057 98.1 Heaviside unit step function......Page 1059 98.3 Laplace transform of H(t -c).f (t -c......Page 1063 98.4 Inverse Laplace transforms of Heaviside functions......Page 1064 99.2 Procedure to solve differential equations by using Laplace transforms......Page 1066 99.3 Worked problems on solving differential equations using Laplace transforms......Page 1067 100.2 Procedure to solve simultaneous differential equations using Laplace transforms......Page 1071 100.3 Worked problems on solving simultaneous differential equations by using Laplace transforms......Page 1072 Revision Test 33 Laplace transforms......Page 1077 Formulae/revision hints for Section M......Page 1078 Section N Fourier series......Page 1079 Chapter 101 Fourier series for periodic functions of period 2p......Page 1080 101.3 Fourier series......Page 1081 101.4 Worked problems on Fourier series of periodic functions of period 2p......Page 1082 102.1 Expansion of non-periodic functions......Page 1087 102.2 Worked problems on Fourier series of non-periodic functions over a range of 2p......Page 1088 103.2 Fourier cosine and Fourier sine series......Page 1093 103.3 Half-range Fourier series......Page 1097 104.1 Expansion of a periodic function of period L......Page 1100 104.2 Half-range Fourier series for functions defined over range L......Page 1104 105.2 Harmonic analysis on data given in tabular or graphical form......Page 1106 105.3 Complex waveform considerations......Page 1110 106.2 Exponential or complex notation......Page 1113 106.3 The complex coefficients......Page 1114 106.4 Symmetry relationships......Page 1118 106.5 The frequency spectrum......Page 1121 106.6 Phasors......Page 1122 Revision Test 34 Fourier series......Page 1126 Formulae/revision hints for Section N......Page 1127 Section O Z-transforms......Page 1128 Chapter 107 An introduction to z-transforms......Page 1129 107.1 Sequences......Page 1130 107.2 Some properties of z-transforms......Page 1133 107.3 Inverse z-transforms......Page 1136 Revision Test 35 Z-transforms......Page 1142 Formulae/revision hints for Section O......Page 1143 Answers to practice exercises......Page 1144 Answers to multiple choice questions......Page 1196 Index......Page 1197