Higher Engineering Mathematics

John Bird's approach, based on numerous worked examples and interactive problems, is ideal for students from a wide range of academic backgrounds, and can be worked through at the student's own pace. Basic mathematical theories are explained in a straightforward manner, being supported by practical engineering examples and applications in order to ensure that readers can relate theory to practice. The extensive and thorough topic coverage makes this an ideal text for a range of university degree modules, foundation degrees, and HNC/D units. Now in its sixth edition, Higher Engineering Mathematics is an established textbook that has helped many thousands of students to gain exam success. It has been updated to maximise the book's suitability for first year engineering degree students and those following foundation degrees. This book also caters specifically for the engineering mathematics units of the Higher National Engineering schemes from Edexcel. As such it includes the core unit, Analytical Methods for Engineers, and two specialist units, Further Analytical Methods for Engineers and Engineering Mathematics, both of which are common to the electrical/electronic engineering and mechanical engineering pathways. For ease of reference a mapping grid is included that shows precisely which topics are required for the learning outcomes of each unit. The book is supported by a suite of free web downloads: . Introductory-level algebra: To enable students to revise the basic algebra needed for engineering courses - available at http://books.elsevier.com/companions/XXXXXXXXX . Instructor's Manual: Featuring full worked solutions and mark schemes for all of the assignments in the book and the remedial algebra assignment - available at http://www.textbooks.elsevier.com (for lecturers only) . Extensive Solutions Manual: 640 pages featuring worked solutions for 1,000 of the further problems and exercises in the book - available on http://www.textbooks.elsevier.com (for lecturers only) . Unique in introducing higher mathematical concepts from an engineering perspective, ensuring that readers understand what they need to do in order to turn theory into practice . Fully mapped to BTEC Higher National Engineering and Foundation Degree unit specifications . Free instructor's manual available online - contains worked solutions and a suggested mark scheme Title Page......Page 4 Copyright Page......Page 5 Contents......Page 6 Preface......Page 14 Syllabus Guidance......Page 16 1.2 Revision of basic laws......Page 20 1.3 Revision of equations......Page 22 1.4 Polynomial division......Page 25 1.5 The factor theorem......Page 27 1.6 The remainder theorem......Page 29 2.2 Worked problems on partial fractions with linear factors......Page 32 2.3 Worked problems on partial fractions with repeated linear factors......Page 35 2.4 Worked problems on partial fractions with quadratic factors......Page 36 3.1 Introduction to logarithms......Page 39 3.2 Laws of logarithms......Page 41 3.3 Indicial equations......Page 43 3.4 Graphs of logarithmic functions......Page 44 4.1 Introduction to exponential functions......Page 46 4.2 The power series for ex......Page 47 4.3 Graphs of exponential functions......Page 48 4.4 Napierian logarithms......Page 50 4.5 Laws of growth and decay......Page 53 4.6 Reduction of exponential laws to linear form......Page 56 Revision Test 1......Page 59 5.1 Introduction to hyperbolic functions......Page 60 5.2 Graphs of hyperbolic functions......Page 62 5.3 Hyperbolic identities......Page 64 5.4 Solving equations involving hyperbolic functions......Page 66 5.5 Series expansions for cosh x and sinh x......Page 68 6.2 Worked problems on arithmetic progressions......Page 70 6.3 Further worked problems on arithmetic progressions......Page 71 6.4 Geometric progressions......Page 73 6.5 Worked problems on geometric progressions......Page 74 6.6 Further worked problems on geometric progressions......Page 75 7.1 Pascal’s triangle......Page 77 7.3 Worked problems on the binomial series......Page 78 7.4 Further worked problems on the binomial series......Page 81 7.5 Practical problems involving the binomial theorem......Page 83 Revision Test 2......Page 86 8.2 Derivationof Maclaurin’s theorem......Page 87 8.4 Worked problems on Maclaurin’s series......Page 88 8.5 Numerical integration using Maclaurin’s series......Page 92 8.6 Limiting values......Page 93 9.2 The bisection method......Page 96 9.3 An algebraic method of successive approximations......Page 100 9.4 The Newton-Raphson method......Page 103 10.2 Binary numbers......Page 106 10.3 Octal numbers......Page 109 10.4 Hexadecimal numbers......Page 111 Revision Test 3......Page 115 11.2 The theorem of Pythagoras......Page 116 11.3 Trigonometric ratios of acute angles......Page 117 11.4 Evaluating trigonometric ratios......Page 119 11.5 Solution of right-angled triangles......Page 124 11.6 Angles of elevation and depression......Page 125 11.8 Area of any triangle......Page 127 11.9 Worked problems on the solution of triangles and finding their areas......Page 128 11.10 Further worked problems on solving triangles and finding their areas......Page 129 11.11 Practical situations involving trigonometry......Page 130 11.12 Further practical situation sinvolving trigonometry......Page 132 12.2 Changing from Cartesian into polar co-ordinates......Page 136 12.3 Changing from polar into Cartesian co-ordinates......Page 138 12.4 Use of Pol/Rec functions on calculators......Page 139 13.2 Properties of circles......Page 141 13.3 Radians and degrees......Page 142 13.4 Arc length and area of circles and sectors......Page 143 13.5 The equation of a circle......Page 146 13.6 Linear and angular velocity......Page 148 13.7 Centripetal force......Page 149 Revision Test 4......Page 152 14.1 Graphs of trigonometric functions......Page 153 14.2 Angles of any magnitude......Page 154 14.3 The production of a sine and cosine wave......Page 156 14.4 Sine and cosine curves......Page 157 14.5 Sinusoidal form A sin(ωt ± α)......Page 162 14.6 Harmonic synthesis with complex waveforms......Page 165 15.2 Worked problems on trigonometric identities......Page 171 15.4 Worked problems (i) on trigonometric equations......Page 173 15.5 Worked problems (ii) on trigonometric equations......Page 175 15.7 Worked problems (iv) on trigonometric equations......Page 176 16.1 The relationship between trigonometric and hyperbolic functions......Page 178 16.2 Hyperbolic identities......Page 179 17.1 Compound angle formulae......Page 182 17.2 Conversion of a sin ωt + b cos ωt into R sin(ωt + α)......Page 184 17.3 Double angles......Page 188 17.4 Changing products of sines and cosines into sums or differences......Page 189 17.5 Changing sums or differences of sines and cosines into products......Page 190 17.6 Power waveforms in a.c. circuits......Page 192 Revision Test 5......Page 196 18.1 Standard curves......Page 197 18.2 Simple transformations......Page 200 18.5 Even and odd functions......Page 205 18.6 Inverse functions......Page 207 18.7 Asymptotes......Page 209 18.8 Brief guide to curve sketching......Page 215 18.9 Worked problems on curve sketching......Page 216 19.1 Areas of irregular figures......Page 222 19.2 Volumes of irregular solids......Page 224 19.3 The mean or average value of a waveform......Page 225 Revision Test 6......Page 231 20.1 Cartesian complex numbers......Page 232 20.3 Addition and subtraction of complex numbers......Page 233 20.4 Multiplication and division of complex numbers......Page 235 20.5 Complex equations......Page 236 20.6 The polar form of a complex number......Page 237 20.7 Multiplication and division in polar form......Page 239 20.8 Applications of complex numbers......Page 240 21.2 Powers of complex numbers......Page 244 21.3 Roots of complex numbers......Page 245 21.4 The exponential formof a complex number......Page 247 22.2 Addition, subtraction and multiplication of matrices......Page 250 22.4 The determinant of a 2 by 2 matrix......Page 254 22.5 The inverse or reciprocal of a 2 by 2 matrix......Page 255 22.6 The determinant of a 3 by 3 matrix......Page 256 22.7 The inverse or reciprocal of a 3 by 3 matrix......Page 258 23.1 Solution of simultaneous equations by matrices......Page 260 23.2 Solution of simultaneous equations by determinants......Page 262 23.3 Solution of simultaneous equations using Cramers rule......Page 266 23.4 Solution of simultaneous equations using the Gaussian elimination method......Page 267 Revision Test 7......Page 269 24.3 Drawing a vector......Page 270 24.4 Addition of vectors by drawing......Page 271 24.5 Resolving vectors into horizontal and vertical components......Page 273 24.6 Addition of vectors by calculation......Page 274 24.7 Vector subtraction......Page 279 24.8 Relative velocity......Page 281 24.9 i, j and k notation......Page 282 25.2 Plotting periodic functions......Page 284 25.3 Determining resultant phasors by drawing......Page 286 25.4 Determining resultant phasors by the sine and cosine rules......Page 287 25.5 Determining resultant phasors by horizontal and vertical components......Page 289 25.6 Determining resultant phasors by complex numbers......Page 291 26.1 The unit triad......Page 294 26.2 The scalar product of two vectors......Page 295 26.3 Vector products......Page 299 26.4 Vector equation of a line......Page 302 Revision Test 8......Page 305 27.2 The gradient of a curve......Page 306 27.3 Differentiation from first principles......Page 307 27.4 Differentiation of common functions......Page 308 27.5 Differentiation of a product......Page 311 27.6 Differentiation of a quotient......Page 312 27.7 Function of a function......Page 314 27.8 Successive differentiation......Page 315 28.1 Rates of change......Page 318 28.2 Velocity and acceleration......Page 319 28.3 Turning points......Page 322 28.4 Practical problems involving maximum and minimum values......Page 326 28.5 Tangents and normals......Page 330 28.6 Small changes......Page 331 29.3 Differentiation in parameters......Page 334 29.4 Further worked problems on differentiation of parametric equations......Page 337 30.2 Differentiating implicit functions......Page 339 30.3 Differentiating implicit functions containing products and quotients......Page 340 30.4 Further implicit differentiation......Page 341 31.3 Differentiation of logarithmic functions......Page 344 31.4 Differentiation of further logarithmic functions......Page 345 31.5 Differentiation of [f(x)]x ......Page 347 Revision Test 9......Page 349 32.1 Standard differential coefficients of hyperbolic functions......Page 350 32.2 Further worked problems on differentiation of hyperbolic functions......Page 351 33.2 Differentiation of inverse trigonometric functions......Page 353 33.3 Logarithmic forms of inverse hyperbolic functions......Page 358 33.4 Differentiation of inverse hyperbolic functions......Page 360 34.2 First order partial derivatives......Page 364 34.3 Second order partial derivatives......Page 367 35.1 Total differential......Page 370 35.2 Rates of change......Page 371 35.3 Small changes......Page 373 36.1 Functions of two independent variables......Page 376 36.2 Maxima,minima and saddle points......Page 377 36.4 Worked problems on maxima, minima and saddle points for functions of two variables......Page 378 36.5 Further worked problems on maxima, minima and saddle points for functions of two variables......Page 380 Revision Test 10......Page 386 37.2 The general solution of integrals of the form axn......Page 387 37.3 Standard integrals......Page 388 37.4 Definite integrals......Page 391 38.2 Areas under and between curves......Page 394 38.3 Mean and r.m.s. values......Page 396 38.4 Volumes of solids of revolution......Page 397 38.5 Centroids......Page 399 38.6 Theorem of Pappus......Page 400 38.7 Second moments of area of regular sections......Page 402 39.3 Worked problems on integration using algebraic substitutions......Page 411 39.4 Further worked problems on integration using algebraic substitutions......Page 413 39.5 Change of limits......Page 414 Revision Test 11......Page 416 40.2 Worked problems on integration of sin2 x, cos2 x, tan2 x and cot2 x ......Page 417 40.3 Worked problems on powers of sines and cosines......Page 419 40.4 Worked problems on integration of products of sines and cosines......Page 420 40.5 Worked problems on integration using the sin θ substitution......Page 421 40.7 Worked problems on integration using the sinh θ substitution......Page 423 40.8 Worked problems on integration using the cosh θ substitution......Page 425 41.2 Worked problems on integration using partial fractions with linear factors......Page 428 41.3 Worked problems on integration using partial fractions with repeated linear factors......Page 430 41.4 Worked problems on integration using partial fractions with quadratic factors......Page 431 42.1 Introduction......Page 433 42.2 Worked problems on the t = tan θ/2 substitution......Page 434 42.3 Further worked problems on the t = tan θ/2 substitution......Page 435 Revision Test 12......Page 438 43.2 Worked problems on integration by parts......Page 439 43.3 Further worked problems on integration by parts......Page 441 44.2 Using reduction formulae for integrals of the form ?xn ex dx......Page 445 44.3 Using reduction formulae for integrals of the form ?xn cosx dx and ? xn sinx dx......Page 446 44.4 Using reduction formulae for integrals of the form ?sinn x dx and ?cosn x dx......Page 448 44.5 Further reduction formulae......Page 451 45.2 The trapezoidal rule......Page 454 45.3 The mid-ordinate rule......Page 456 45.4 Simpson’s rule ......Page 458 Revision Test 13......Page 462 46.1 Family of curves......Page 463 46.3 The solution of equations of the form dy/dx = f(x)......Page 464 46.4 The solution of equations of the form dy/dx = f(y)......Page 466 46.5 The solution of equations of the form dy/dx = f(x).f(y)......Page 468 47.3 Worked problems on homogeneous first order differential equations......Page 471 47.4 Further worked problems on homogeneous first order differential equations......Page 473 48.1 Introduction......Page 475 48.3 Worked problems on linear first order differential equations......Page 476 48.4 Further worked problems on linear first order differential equations......Page 477 49.2 Euler’s method......Page 480 49.3 Worked problems on Euler’s method......Page 481 49.4 An improved Euler method......Page 485 49.5 The Runge-Kutta method......Page 490 Revision Test 14......Page 495 50.1 Introduction......Page 496 50.3 Worked problems on differential equations of the form a d2y/dx2 + b dy/dx + cy = 0......Page 497 50.4 Further worked problems on practical differential equations of the form a d2y/dx2 + b dy/dx + cy = 0......Page 499 51.2 Procedure to solve differentialequations of the form a d2y/dx2 +b dy/dx +cy = f(x) ......Page 502 51.3 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 503 51.4 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 505 51.5 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 507 51.6 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 509 52.2 Higher order differential coefficients as series......Page 512 52.3 Leibniz’s theorem......Page 514 52.4 Power series solution by the Leibniz–Maclaurin method......Page 516 52.5 Power series solution by the Frobenius method......Page 519 52.6 Bessel’s equation and Bessel’s functions......Page 525 52.7 Legendre’s equation and Legendre polynomials......Page 530 53.2 Partial integration......Page 534 53.3 Solution of partial differential equations by direct partial integration......Page 535 53.5 Separating the variables......Page 537 53.6 The wave equation......Page 538 53.7 The heat conduction equation......Page 542 53.8 Laplace’s equation......Page 544 Revision Test 15......Page 547 54.1 Some statistical terminology......Page 548 54.2 Presentation of ungrouped data......Page 549 54.3 Presentation of grouped data......Page 553 55.2 Mean, median andmode for discrete data......Page 560 55.3 Mean, median andmode for grouped data......Page 561 55.4 Standard deviation......Page 563 55.5 Quartiles, deciles and percentiles......Page 565 56.1 Introduction to probability......Page 567 56.3 Worked problemson probability......Page 568 56.4 Further worked problems on probability......Page 570 Revision Test 16......Page 573 57.1 The binomial distribution......Page 575 57.2 The Poisson distribution......Page 578 58.1 Introduction to the normal distribution......Page 581 58.2 Testing for a normal distribution......Page 585 59.2 The product-moment formula for determining the linear correlation coefficient......Page 589 59.4 Worked problems on linear correlation......Page 590 60.2 The least-squares regression lines......Page 594 60.3 Worked problems on linear regression......Page 595 Revision Test 17......Page 600 61.4 Laplace transforms of elementary functions......Page 601 61.5 Worked problems on standard Laplace transforms......Page 602 62.2 Laplace transforms of the form eat f(t) ......Page 606 62.3 The Laplace transforms of derivatives......Page 608 62.4 The initial and final value theorems......Page 610 63.2 Inverse Laplace transforms of simple functions......Page 612 63.3 Inverse Laplace transforms using partial fractions......Page 615 63.4 Poles and zeros......Page 617 64.3 Worked problems on solving differential equations using Laplace transforms......Page 619 65.3 Worked problems on solving simultaneous differential equations by using Laplace transforms......Page 624 Revision Test 18......Page 629 66.3 Fourier series......Page 630 66.4 Worked problems on Fourier series of periodic functions of period 2π......Page 631 67.2 Worked problems on Fourier series of non-periodic functions over a range of 2π......Page 636 68.2 Fourier cosine and Fourier sine series......Page 642 68.3 Half-range Fourier series......Page 645 69.1 Expansion of a periodic function of period L......Page 649 69.2 Half-range Fourier series for functions defined over range L......Page 653 70.2 Harmonic analysis on data given in tabular or graphical form......Page 656 70.3 Complex waveform considerations......Page 660 71.2 Exponential or complexnotation......Page 663 71.3 The complex coefficients......Page 664 71.4 Symmetry relationships......Page 668 71.5 The frequency spectrum......Page 671 71.6 Phasors......Page 672 Revision Test 19......Page 677 Essential formulae......Page 678 Index......Page 694

Includes:
• 1,000 Worked Examples
• 1,750 Further Problems
• 19 Assignments

John Bird’s approach, based on numerous worked examples and interactive problems, is ideal for students from a wide range of academic backgrounds, and can be worked through at the student’s own pace. Basic mathematical theories are explained in the simplest of terms, supported by practical engineering examples and applications from a wide variety of engineering disciplines, to ensure the reader can relate the theory to actual engineering practice. This extensive and thorough topic coverage makes this an ideal text for a range of university degree modules, Foundation Degrees, and HNC/D units.

An established text which has helped many thousands of students to gain exam success, now in its fifth edition Higher Engineering Mathematics has been further extended with new topics to maximise the book’s applicability for first year engineering degree students, and those following Foundation Degrees. New material includes: inequalities; differentiation of parametric equations; differentiation of hyperbolic functions; t = tan è/2 substitution; and homogeneous first order differential equations.

This book also caters specifically for the engineering mathematics units of the Higher National Engineering schemes from Edexcel, including the core unit Analytical Methods for Engineers, and the two specialist units Further Analytical Methods for Engineers and Engineering Mathematics in their entirety, common to both the electrical/electronic engineering and mechanical engineering pathways. A mapping grid is included showing precisely which topics are required for the learning outcomes of each unit, for ease of reference.

The book is supported by a suite of free web downloads:
• Introductory-level algebra: To enable students to revise basic algebra needed for engineering courses - available at http://books.elsevier.com/companions/0750681527
• Instructor's Manual: Featuring full worked solutions and mark scheme for all 19 assignments in the book and the remedial algebra assignment - available on http://www.textbooks.elsevier.com for lecturers only
• Extensive Solutions Manual: 640 pages featuring worked solutions for 1,000 of the further problems and exercises in the book - available on http://www.textbooks.elsevier.com for lecturers only

Booknews

An engineering math textbook catering to students in BTEC Higher National Certificate and Higher National Diploma courses, as well as first year undergraduates struggling with the key mathematical concepts and skills essential to engineering studies. Bird (former Head of Applied Electronics, Highbury College) introduces 38 major theories providing a concise summary of each, followed by numerous examples demonstrating how the problems are solved. This new edition contains material on Maclaurin's and Taylor's series, as well as statistics, determinants, and complex numbers. Annotation c. Book News, Inc., Portland, OR (booknews.com)

John Bird's approach, based on numerous worked examples and interactive problems, is ideal for students from a wide range of academic backgrounds, and can be worked through at the student's own pace. Basic mathematical theories are explained in a straightforward manner, being supported by practical engineering examples and applications in order to ensure that readers can relate theory to practice. The extensive and thorough topic coverage makes this an ideal text for a range of university degree modules, foundation degrees, and HNC/D units. Now in its sixth edition, Higher Engineering Mathematics is an established textbook that has helped many thousands of students to gain exam success. It has been updated to maximise the book's suitability for first year engineering degree students and those following foundation degrees. This book also caters specifically for the engineering mathematics units of the Higher National Engineering schemes from Edexcel. As such it includes the core unit, Analytical Methods for Engineers, and two specialist units, Further Analytical Methods for Engineers and Engineering Mathematics, both of which are common to the electrical/electronic engineering and mechanical engineering pathways. For ease of reference a mapping grid is included that shows precisely which topics are required for the learning outcomes of each unit. The book is supported by a suite of free web downloads: . Introductory-level algebra: To enable students to revise the basic algebra needed for engineering courses - available at http://books.elsevier.com/companions/XXXXXXXXX . Instructor's Manual: Featuring full worked solutions and mark schemes for all of the assignments in the book and the remedial algebra assignment - available at http://www.textbooks.elsevier.com (for lecturers only) . Extensive Solutions Manual: 640 pages featuring worked solutions for 1,000 of the further problems and exercises in the book - available on http://www.textbooks.elsevier.com (for lecturers only) . Unique in introducing higher mathematical concepts from an engineering perspective, ensuring that readers understand what they need to do in order to turn theory into practice . Fully mapped to BTEC Higher National Engineering and Foundation Degree unit specifications . Free instructor's manual availablenbsp, onlinenbsp, - contains worked solutions and a suggested mark scheme