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Mathematics for Electrical Engineering and Computing

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TLFeBOOK

Mathematics for Electrical Engineering and Computing Mary Attenborough

AMSTERDAM BOSTON LONDON HEIDELBERG NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO TLFeBOOK

Chapter 20 20.1.

L = {a n bc|n  2}

20.3. (a) S ⇒ aSa ⇒ aaa (b) S ⇒ aSa ⇒ abSba ⇒ abbSbba ⇒ abbcbba 20.4. S ::= A“b”

A ::= aC∗

C ::= “c”|“a”

20.6.

40 58

35 38

75 65

66 54

54 45

72 22

58 60

44 55

541

Chapter 21 21.1.

99.5625, 8.9496

21.2.

1/3

21.3.

2/3

21.4.

2/3

21.5.

4/13

21.6.

1/12

21.7.

1/6

21.8.

3/51

21.9.

1/(10000)

21.10.

0.9362

21.11.

(a) 3/5, (b)3/5, (c) 3/10

21.12.

0.091

21.13.

(a) 0.06681, (b) 0.02275, (c) 0.00135, (d) 0.30854

21.14.

(a) 0.30854, (b) 0.02275, (c) 0.15866, (d) 0.22663

21.15.

(a) 0.99018, (b) 0.74751, (c) 0.63056, (d) 0.63056, (e) 0.26112, (f) 0.73769

21.16.

17.8%

21.17.

(a) 0.699, (b) 0.139, (c) 0.192, (d) 0.273

21.18.

8.858, the councillor is correct.

21.19.

(a) 0.512, (b) 0.2048, (c) 0.879

21.20.

Model probabilities: 0.2231, 0.3347, 0.2510, 0.1255, 0.0471, 0.0141, 0.0045; Model frequencies: 22, 33, 25, 13, 5, 1, 1 There is good agreement between the model and the actual number of incidents.

21.21.

0.5402

21.22.

(a) 0.4823, (b) 0.3614, (c) 0.1563, (d) 0.6936

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Index

0 and 1 Laws: Boolean Algebra, 78–80

A Acceleration, 238 circular motion, 219–23 Absorption Laws: Boolean Algebra, 78–80, 82 Addition: of complex numbers, 209 of matrices, 296–7 of vectors, 191 Adjacency matrix, 465 Adjoint matrix, 333–4 Admittance, 218 Algebra, 76–7 Aliasing, 257 Amplitude, 94, 97, 100–1, 102–3 Amplitude modulation, 114 AND gate, 81 And, operation on propositions and predicates, 62–3 Angle between two vectors, 199 Angular frequency, 418 Anti-differentiation, 133 Approximations, 441–2 Area function, 150–1 Area of a parallelogram, 202 Argand diagram, 208 Argument, of a complex number, 215–17, 222–6, 229–31 Arithmetic progressions, 259–62 common difference, 260–1 general term, 261 sum of n terms, 261–2 Array, 295 Arrow diagrams, 8–9 Assignment operator, 283 Associative laws, Boolean Algebra, 78–80 Asymptotes, vertical, 242, 245 Augmented matrix, 325, 327 Auxiliary equation, 359, 364, 376 Average: rate of change, 117 speed, 116 velocity, 117

B Back-substitution, 328 Backus Naur Form, 483 Basis vectors, 198–9 Bayes’s theorem, 556–7 Bernoulli trial, 524 Best fit line, 341 Bias, 418, 423 Binary operations, 482 Binomial: distribution, 524, 533–4 mean, 525 single trial: mean, 525; variance, 525 expansion, 271–2, 275 series, 267–72, 275–6 theorem, 270–2, 275–6 Bison, 487 Boolean: Algebra, 76–88 operators, 79 Set, 77 Bound Laws: Boolean Algebra, 78–80 Boundary conditions, 352

C Capacity of a network, 471–3 Cardinality of a finite set, 7 Cartesian basis vectors, 198–9 Cartesian form: conversion of complex numbers: from polar form, 216 to exponential form, 223, 224 of vectors, 189 Central limit theorem, 517 Central tendency, 496 Centripetal force, 222 Chain rule, 124–6, 438–40 Characteristic equation, 360 Chord, gradient of, 117–18 Circuits, in a graph, 464 Circular motion, 220–5 Class: frequency, 496, 498 interval, 494, 499 midpoint, 494, 496, 499 Codomain, 8 Cofactors, 333–4 Column vector, 303

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Index Commutative Laws: Boolean Algebra, 78–80 Complement, 5 Complement Laws: Boolean Algebra, 78–80 Complementary function, 363–5, 368–70 Complex conjugate, 210–12, 214, 226 Complex equations, 229–31 Complex numbers, 206–36 addition, 209 application to a.c linear circuits, 218–20 argument, 215–17, 222–6, 229–31 division, 211–12, 217, 226 equality, 208 exponential form, 223–31 imaginary part, 208–11, 218, 221, 224, 225, 229 modulus, 215–16, 222–30 multiplication, 209, 217, 225 polar form, 215–7, 226–7 powers, 225–6, 228 real part, 208–11, 218, 221, 224–5 Complex plane, 208 Complex variable, 403 Composite function, derivative of, 124–6 Composite functions, 18 Composition of functions, 18 Compound angle identities, 104–5 Computer graphics, 306 Conditional probability, 511–13 Conditions, 72 Conjugate, 210–12 Consistent system, 315, 318–20, 322 Constant of integration, 133 Continuous function, 47–8 Convergence, 282, 284–5 criterion, 285 Convolution, 387, 391, 398–400, 405–6, 412–13 Cosecant (cosec), 90 cosech, hyperbolic cosecant, 174 cosh, hyperbolic cosine, 173 Cosine (cos), 90 graph, 91 relationship with sine, 92, 93 symmetry of, 92 Cotangent (cotan), 90 coth, hyperbolic cotangent, 174 Cover up rule, 393–4, 396, 399, 401, 408, 409, 410, 413 Cross product, 198 Cumulative distribution function, 517, 522 Cumulative frequency, 496 Cumulative relative frequency, 496 Curl, 449–50 Curve fitting, 51–3, 341–2 Cycle, 418 length, 96, 99 rate, 97, 100

D De Moivre’s theorem, 230–1 De Morgan’s Laws: Boolean Algebra, 78–80

543

Decay of charge on a capacitor, 164 Decibels, 102–3 Decomposing functions, 20–1 Degrees, 89–90 Del operator, 448–50 Delta, 117, 261 Delta function, 385–6, 405 Dependent variable, 8 Derivation: of symbols, 481 tree, 483 Derivative, 118 function, 118–20 of a composite function, 122–4 of a product, 126–7 of a quotient, 127 of a sum, 123 of af(x), 122 of inverse trigonometric functions, 125–6 of simple functions, 120–3, 126, 181 of the derivative, 220, 239 partial, 436–45 second, 239, 240 total, 440 Determinants, 304–5, 321 Determined system, 322, 336 Deviation, 499 Diagonal matrices, 301 Difference equations, 163, 376–80 general solution, 376 homogeneous, 376–8 linear with constant coefficients, solution of, 376–8, 408–10 particular solution, 377 trial solutions, 377 Difference operator, 261, 264 Differential equations, 133, 153–66, 346–75 boundary conditions, 364 complementary function, 363–5, 368–70 d2 x/dt2 = −ω2 x, 220 dy/dt = ky, 170 homogeneous, 358–60, 362, 363–5, 367, 371–2 initial conditions, 352 linear, 354–5 linear with constant coefficients, 356 solution of, 358–68, 394–6 general, 353, 357–8, 360–1, 364, 366, 368 particular, 133, 353, 355, 358, 360–5, 367 order of, 352 steady state solution, 369–70 systems of, 347–51, 372–5 solving, 372–5 time invariance, 356–8 transient solution, 369–70 trial solutions, 360 Differential operator, 354 Differentiating vector fields, 449–50 Differentiation, 110–33, 180–1 applications, 134–6

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544

Index Digital: circuits, 81 signals, 257–9 ramp, 11 signals, 10, 12, 14 square wave, 11 wave, 11 Digraphs, 462 underlying graph, 462 see also arcs, vertices Dimension of matrices, 298 Direct current (dc) component of a signal, 423 Discontinuities, infinite, 245–6 Discrete functions, 10, 16 Discrete systems, 375 Distributive Law: Boolean Algebra, 78–80 Divergence, 286 Divide by zero, 246 Division of complex numbers, 211–12, 217, 226 Document Type Definition (DTD), 488 Domain: of a function, 9, 13 of a predicate, 62

E e, 162, 166–9 Echelon form, 325 Edges, incidence with a vertex, 461 Edges, multiple, 461 Eigenvalues and eigenvectors, 335–8, 372–3 Electrical circuits, 349–61 Elements, 4 Empty set, 4 Equality: of complex numbers, 208 Equations: matrix, 305–6 quadratic, 32 trigonometric, 110–11 see also Differential equations, Systems of linear equations Equivalence, of predicates, 64–5 Even functions, 42, 92, 172, 424–5 Even permutation, 332 Events, 501–2, 504–5 disjoint, 504 non-disjoint, 505 EXOR gate, 81 exp, 167 Exponent, 226 Exponential: conversion to rectangular, 225 derivative of, 170–1 distribution, 521–3 form of complex numbers, 223–31 function, The, 166–72 functions, 32–4, 162–89

growth and decay, 162–6 mean, 523 standard deviation, 523 relationships, 51–2 Extended Backus Naur Form (EBNF), 485–7 Extensible Markup Language (XML), 487–8

F Failure rate, 521, 523 Fibonacci sequence, 256 Finite impulse response (FIR), 377 Finite state machine, 474 Finite state recognisers, 476 Flow augmenting path, 473 Flow function of a network, 473 For all, 70 Forcing function, 347, 348, 360, 367, 370–3 Fourier: amplitude and phase form, 426–7 analysis, 418 complex form, 428–30 partial sums, 421 series, 418–32 sine and cosine form, 419–21 Frequency analysis, 418 Frequency distribution, 500–2, 517 Frequency response, 401–2, 414 Function of a function, 123 Functions, 1, 8–23 codomain of, 7 combining, 17–22 composite: derivative of, 123–5 composition, 18 decomposing, 20–1 difference, 117 domain of, 8, 13 exponential, 33–4, 162–89 image under, 8 inverse, 21–3 existence of, 40–1 linear, 22–3 many-to-one, 40–1 mean value of, 155–6 modulus, 41 of more than one variable, 435–5 one-to-one, 40–1 periodic, 418–19 product of, 17 derivative of, 126 integral of, 139–43 quadratic, 31–3 quotient of, 17 derivative of, 127 range, 23 real, 9 r.m.s. value of, 155 sum, 17 derivative of, 123 integral of, 135

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Index symmetry, 41–3 trigonometric, 88–117 unit step, 384–5, 405 Fundamental period, 91–2, 94, 96, 97, 99

G Gauss-Jordan elimination, 332–4 Gaussian elimination, 326–31, 344–5 partial pivoting, 331 pivotal equation, 326–31 Generalized functions, 385 Geometric growth, 163 Geometric progressions, 263–9 common ratio, 264–5 general term, 264 sum of n terms, 265–7 sum to infinity, 267–9 Geometrical vectors, 198 Gibb’s phenomenon, 421–2 Gradient, 26–30 of a chord, 117–18 of the tangent, 238 Grammar, 480 classification, 483 context free, 484 Graph sketching: by analysing the function’s behaviour, 244–52 linear functions, 30–1 using transformations, 34–40, 94–6 Graphical user interface (GUI), 306 Graphs of functions, 8–13, 26–43 area bounded by, 154–5 area under, 147–51 hyperbolic functions, 274 inverse hyperbolic functions, 177 Graphs, 461–78 bipartite, 463 complete, 463 connected, 464 definition, 461 directed, see digraphs isomorphism of, 462 matrix representation of, 465 planar, 465 simple, 461 subgraph, 463 weighted, 462 see also Edges; Trees; Vertices Greedy algorithm for the minimum spanning tree, 467–8 Growth: constant, 163 exponential, 162–6 geometric, 163

H Half-wave symmetry, 424–5 Harmonics, 418 Hermitian matrix, 302–3 Homogeneous equation, 358–60, 362, 363–5, 367, 371–2, 376–8 HTML, 488 Hyperbola, 33, 173

545

Hyperbolic functions, 172–4 cosech, 174 cosh, 173 coth, 174 graphs of, 175 identities, 175 relationship with Trigonometric functions, 230 sech, 174 sinh, 173 tanh, 174

I Idempotent Law, 78–80 Identity Laws: Boolean Algebra, 78–80 Image, under a function, 8 Imaginary part, of a complex number, 208–11, 218, 221, 224, 225, 229 Impedance, 219–21 Implication, 67–9 Impulse function, 384–5, 404 Impulse response function, 397–400, 411–14 Incidence matrix, 461 Inconsistent system, 319–20, 322–3, 329 Independent variable, 8 Indeterminate point, 279 Indeterminate system, 314, 316, 324 Inequalities, 43–8 combining, 45–6 solving, 43–4, 47–8 Infinite impulse response (IIR), 376, 412 Infinity: sum to, 267–9 tending to, 245–6, 286 Initial conditions, 352, 358, 360, 361, 364, 366, 375–6, 378 Instantaneous velocity, 118 Integers, 4 Integral operator, 354 Integral: round a closed curve, 453–4 scalar line, 451–3 surface, 454–6 Integrals: definite, 147–53 indefinite, 132 of the form ∫ f(ax + b)dx, 138 of the form ∫ f(u)du/dx dx, 139 standard, 134, 181 Integrating vector fields, 451–6 Integration, 132–61, 181–5, applications of, 145–7 as the inverse of differentiation, 133 by parts, 142, 183, 383 changing the variable of, 135–8 constant of, 133 numerical, 156–9 of a composite function, 136–42 of a product, 139–43 of a sum, 135 of af(x), 135 using partial fractions, 183–5

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546

Index Integration (continued) using substitution, 135–41, 182–4 using trigonometric identities, 143–4 Intersection, 6 Inverse differentiation, 132–3 Inverse function, 21–3 existence of, 40–2 derivatives of, 125–6 graphs, 109, 110 of linear functions, 22 of hyperbolic functions, 176 graphs, 176 logarithmic equivalences, 176 of trigonometric functions, 109–11 Inverse: of a matrix, 303–5, 322–3, 330–1, 333–5 of a power, 51

J j, 206–7 JavaCC (Java Compiler Compiler), 487

K Karnaugh maps, 82–7 Kirchhoff’s current law (KCL), 315 Kirchhoff’s voltage law (KVL), 314

L L’Hopital’s rule, 279–81 Languages, 479–90 context-free, 483–5 Laplace transforms, 382–402 and systems theory, 397–402 application to solving differential equations, 394–6 convolution property, 387, 391, 398–9 definition, 382–3 existence, 383 inverse, 386–9 poles of, 394 properties of, 386–91 table of, 386 using partial fractions, 392–4 Least squares data fitting, 338–42 LIFO, 485 Limits, 259, 266–7, 282–3 finite, 287 Line fitting, 40–1, 338 Linear differential equations, 353–4 Linear relationships, 40–1 Linear time invariant systems (LTI), 356 Linearity: of Laplace transforms, 386–7, 390 of z transforms, 405–6 Linear functions, 22–3 Logarithms: Napierian, 169 natural, 169 Logarithmic functions, derivative of, 172–3

Logic circuits, 81 Logic gates, 81–2 Loop, 462 LRC circuits, 349–51

M Maclaurin series, 273–5 definition, 273–5 Many-to-one functions, 38, 42 Mapping, 306 Markup, 487 Mathematical model, 57–8 Matrices, 295–314 addition, 296 adjoint, 334 diagonal, 301 dimension of, 296–7 eigenvalues and eigenvectors, 338–41 Hermitian, 302–3 inverse of, 303–5, 322–3, 330–1, 333–5 a 2x2 matrix, 303–5 existence of, 303 finding: by elimination, 330–1 by using the adjoint, 333–5 lower triangular, 302 multiplication, 297–9 notation, 296 skew-symmetric, 302 solving equations, 305–6 square, 301 subtraction, 296 symmetric, 302 to represent graphs, 466 transpose of, 302 unit, 300 upper triangular, 302 used for transformations, 306–16 Max-flow, min-cut theorem, 472 Maxima, 237–44 Maximum power, 243 Mean value, 155 Mean, 496–9, 501, 516–17, 519, 520, 523, 525, 528 of a continuous distribution, 523 of a single trial, 525 of binomial distribution, 525 of Poisson distribution, 528 Mechanical system: damped oscillations, 370–2 rotational, 351–2 Minima, 237–44 Minimization, 82 Minimum spanning tree, 467 Minimum squared error, 338 Minors, 332–4 Models of growth, 163 Modulus: function, 41 of a complex number, 212, 216, 225 Multiplication: of complex numbers, 210, 219, 228 of matrices, 297–9

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Index N NAND gate, 81–2 Napierian logarithm, 169 Natural logarithm, 169 Natural numbers, 4 Networks, 468–72 capacity of, 471 flow function, 471 maximum flow, 472–4 shortest path, 468–71 Newton’s law of cooling, 165 Newton’s second law, 224, 347 Newton-Raphson method of solving equations, 283–6 Non-terminal symbol, 480 NOR gate, 81 Normal distribution, 516–21 standardized, 519 area in the tail, 518, 519 Normal equations, 343 NOT gate, 81 Null set, 2 Number line, 44–7, 49–50 Numbers: sets of: integers, 4 natural, 4 rationals, 4 reals, 4 see also Complex numbers Numerical methods: for solving equations, 282–5 of integration, 156–7

O Odd functions, 42–3, 92–3, 173, 424 Odd permutation, 334 One-to-one functions, 40–1 Operations: propositions and predicates, 62–4 sets, 5–7 Operators: Boolean, 78–81 linear, 354–6 OR gate, 81 Or: exclusive, 63, 81 non-exclusive, 63 operation on propositions and predicates, 63 Orthogonal axes, 198 Oscillating sequence, 287 Outcome, 502

547

Permutations, 332 Phase, 98 Phasors, 195–6, 207 rotation by π/2, 206–8 Piecewise continuous functions 10 Point of inflexion, 238 Polar coordinates, conversion to rectangular, 194–5 Polar form of complex numbers, 216–19 conversion to rectangular, 217–18 Poles of the Laplace transform, 392–3 Poisson distribution, 526–8 Population growth, 163–4 Population, 494 Position vectors, 203 Power relationships, 53–4 Power series, 274–9 Powers of complex numbers, 225–6, 228 Predicates, 61–72 applications, 72 domain of, 62 Principal root, 232 Probability, 501–15 addition law of, 504–6 of failure of an electrical circuit, 514–16 multiplication law of, 512 independent events, 512 using a probability density function, 504 using the cumulative distribution function, 518 where outcomes are equally likely, 504–5 Probability density function, 504, 518 Probability distribution, 502–3 Probability function, 502 Probability trees, 508–11 Problem solving, 57–61 Productions, 480 Products of vectors, 198–201 Progressive waves, 100–1 velocity, 100–1 Proper subsets, 5 Propositions, 61–72 as a Boolean Algebra, 78–9 Pushdown recognizer, 484 Pythagoras’s theorem, 90

Q Quadratic equations, 32–3 formula, 32 complex solutions, 207–8, 212–15 roots of, 32–3 Quadratic functions, 32–3

P Parallel vectors, 202 Parsers, 484 Partial differentiation, 339, 436–45 Partial fractions, 184–6 Particular solution, 134 Pascal’s triangle, 269–72 Paths, in a graph, 465 Period, fundamental, 418 Periodic functions, 418

R Radians, 88–9 Radioactive decay, 164 Ramp functions, 10–11 Rationals, 4 Reactance, 218 Real part, of a complex number, 208–11, 218, 221, 224, 225 Real numbers, 4

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548

Index Rectangular coordinates, conversion to polar, 193 Rectangular form: conversion to exponential, 223–4 conversion to polar, 215–16 of vectors, 190 Rectangular pulse functions, 386 Recurrence relations, 16, 255–7, 261, 264 Recursion, 485 Reflection, 309 Reflection, of graphs, 36, 37 Relations, 7 Relative frequency, 496 Resonance, 370 Resultant: admittance, 218 impedance, 218 of vectors, 191 Rewriting rules, 480 Right-angles, 198–9 Root mean squared (r.m.s.) value, 156 Roots: of a quadratic equation, 32–3 of unity, 229–30 Rotation: of axes, 307, 310–11, 314 of coordinates, 307 Row vector, 303

S Sample, 494 Sample mean, 498 Sample space, 500 Sampling: interval, 10, 258 theorem, 258 Scalar: product, 198–9 quantities, 188, 191 Scalar line integral, 451–4 Scaling, 310, 312–14, 335–7 of graphs, 38–9 Scatter diagram, 341 Secant (sec), 90 sech, hyperbolic secant, 174 Sentence, 480–3 grammatical, 481–3 Sequences, 11, 254–9 general term, 255 see also Arithmetic progressions and Geometric progressions Series, 259–60 binomial, 267–72 see also Fourier series Sets, 4–7 as a Boolean Algebra, 77–8 cardinality, 7 complement, 5 intersection, 6 union, 6 see also Algebra and Functions Shortest path problem, 468–71 Sigma notation, 259 Signals, 9 Simple harmonic motion, 222

Simpson’s rule, 156–7 Sine: graph, 87 relationship with cosine, 87, 89 symmetry of, 88, 89 Single input, single output system (SISO), 343 sinh, hyperbolic sine, 173 Sinusoidal functions, 100–3 of distance, 100–1 of time, 100 Sine (sin), 90 Slope: direction of maximum, 447 of a curve, 240 of a surface, 447 Spring, damped forced motion of, 248–9, 347–9 Square matrices, 301 Square numbers, 255 Square wave, 10–11 Stack, 485 Standard deviation, 496, 499 of a continuous distribution, 523 Standing waves, 107–8 Start symbol, 480 State transition diagrams, 474–6 State variables, 347–53 Stationary points, 238–42, 249–51 Statistical modelling, 516, 517 see also Binomial, Exponential, Normal and Poisson distribution Straight line, 26–30, 50–3 String, 480 Subsets, 5 Subtraction: of matrices, 296–7 of vectors, 191 Sum of products, Boolean algebra, 82 Superposition of solutions, 354–5 Superposition of waves, 107–9 Surface integrals, 454–6 Surfaces, 435–6 Symmetry: of functions, 41–3, 423–4 of matrices, 302 System, 346–7 resonance, 370 response, 346–7 stability, 368–9 Systems of linear equations, 314–29 consistent, 318–22 in three unknowns, 323 in two unknowns, 323 inconsistent, 318–22 indeterminate, 318–22 matrix form, 320 solving using elimination, 324–6 solving using substitution, 316–17 see also Gaussian elimination

T Tangent, gradient of, 118 Tangent function (tan), 90 symmetry of, 92

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Index tanh, hyperbolic tangent, 174 Taylor series, 278–80 definition, 278 Terminal symbol, 480 There exists, 72 Three dimensional (3d) vectors, 197–8, 200 Time to failure, 522, 524 Total derivative, 440 Transfer function, 397–9, 401–2, 411–14 Transformations: combined, 310–14 of axes, 307, 310–14 of coordinates, 307 of graphs, 34–40, 94–6 of the plane, 307 Translation: of axes, 309 of coordinates, 309 of graphs, 35–6 vectors, 190 Transpose of a matrix, 300, 302–3, 334 Trapezoidal rule, 156–7 Trees, 465–7 parsing, 466, 487 properties of, 465 spanning, 466–7 Trials, 546 independent, 509 not independent, 510 repeated, 508–10 Triangular wave, 258 Trigonometric equations, 110–11 Trigonometric functions, 88–115 complex expressions, 227–8 inverses of, 109–12 relationship with Hyperbolic functions, 228 Trigonometric identities, 103–5 compound angle, 105 use in integration, 143–4 Truth table, 63, 79 Turning points, 239–40, 246–7 Two dimensional (2d) vectors, 189–90

U Undefined function values, 13, 279 Undersampling, 257 Union, 6 Unit matrix, 300, 322, 330, 335 Unit step function, 384–5, 389, 404, 409 Unit vectors, 197 Universal Set, 6 Upper triangular matrix, 302, 325, 330

549

Vector equation of a line, 202–3 Vector field: differentiating, 449–50 divergence of, 449 Vector product, 201–2 Vector quantities, 188–9 Vectors, 188–205 addition of, 191 angle between, 199 at right angles, 199–200 basis, 198–9 Cartesian form, 189 dimension of, 188–9 direction cosines, 200 direction of, 192–3 geometrical, 189, 198 magnitude of, 192–3 multiplication by a scalar, 197 parallel, 202 polar coordinates, 192 rectangular form, 189 scalar product, 198–9 subtraction of, 191 unit, 197 Vector product, 201–2 Velocity, 237 average, 117 circular motion, 219–23 instantaneous, 116 of a progressive wave, 101 Venn diagrams, 4, 76–8 Vertices, 461 adjacent, 461 degree of, 462 in-degree, 462 neighbours, 461 out-degree, 462

W Walks, in a graph, 464–5 Waves, 97–103, 107–9 functions of time, 100 functions of distance, 100 progressive, 101 standing, 107 superposition, 107–8 triangular, 258 Well-formed document, 488 wff (well formed formula), 480

X XML, 487–9

V Valid document, 488 Variable, 61 Variance, 494, 497–9, 501, 525, 528 Vector calculus, 447–58

Y YACC (Yet Another Compiler Compiler), 487

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550

Index Z z transforms, 403–17 and systems theory, 411–15 application to solving difference equations, 408–10 convolution property, 405–6, 412–13

definition, 403–4 existence of, 405–8 properties, 405–8 table of, 404

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