Engineering Mathematics

最新数学手册，内容可拷贝。 Contents 1 Engineering Conversions, Constants and Symbols 1 1.1 General conversions 1 1.2 Greek alphabet 2 1.3 Basic SI units, derived units and common prefixes 3 1.4 Some physical and mathematical constants 5 1.5 Recommended mathematical symbols 7 1.6 Symbols for physical quantities 10 2 Some Algebra Topics 20 2.1 Polynomial division 20 2.2 The factor theorem 21 2.3 The remainder theorem 23 2.4 Continued fractions 24 2.5 Solution of quadratic equations by formula 25 2.6 Logarithms 28 2.7 Exponential functions 31 2.8 Napierian logarithms 32 2.9 Hyperbolic functions 36 2.10 Partial fractions 41 3 Some Number Topics 46 3.1 Arithmetic progressions 46 3.2 Geometric progressions 47 3.3 The binomial series 49 3.4 Maclaurin’s theorem 54 3.5 Limiting values 57 3.6 Solving equations by iterative methods 58 3.7 Computer numbering systems 65 4 Areas and Volumes 73 4.1 Area of plane figures 73 4.2 Circles 77 4.3 Volumes and surface areas of regular solids 82 4.4 Volumes and surface areas of frusta of pyramids and cones 88 vi Contents 4.5 The frustum and zone of a sphere 92 4.6 Areas and volumes of irregular figures and solids 95 4.7 The mean or average value of a waveform 101 5 Geometry and Trigonometry 105 5.1 Types and properties of angles 105 5.2 Properties of triangles 106 5.3 Introduction to trigonometry 108 5.4 Trigonometric ratios of acute angles 109 5.5 Evaluating trigonometric ratios 110 5.6 Fractional and surd forms of trigonometric ratios 112 5.7 Solution of right-angled triangles 113 5.8 Cartesian and polar co-ordinates 116 5.9 Sine and cosine rules and areas of any triangle 119 5.10 Graphs of trigonometric functions 124 5.11 Angles of any magnitude 125 5.12 Sine and cosine waveforms 127 5.13 Trigonometric identities and equations 134 5.14 The relationship between trigonometric and hyperbolic functions 139 5.15 Compound angles 141 6 Graphs 149 6.1 The straight line graph 149 6.2 Determination of law 152 6.3 Logarithmic scales 158 6.4 Graphical solution of simultaneous equations 163 6.5 Quadratic graphs 164 6.6 Graphical solution of cubic equations 170 6.7 Polar curves 171 6.8 The ellipse and hyperbola 178 6.9 Graphical functions 180 7 Vectors 188 7.1 Scalars and vectors 188 7.2 Vector addition 189 7.3 Resolution of vectors 191 7.4 Vector subtraction 192 7.5 Relative velocity 195 7.6 Combination of two periodic functions 197 7.7 The scalar product of two vectors 200 7.8 Vector products 203 8 Complex Numbers 206 8.1 General formulae 206 8.2 Cartesian form 206 Contents vii 8.3 Polar form 209 8.4 Applications of complex numbers 211 8.5 De Moivre’s theorem 213 8.6 Exponential form 215 9 Matrices and Determinants 217 9.1 Addition, subtraction and multiplication of matrices 217 9.2 The determinant and inverse of a 2 by 2 matrix 218 9.3 The determinant of a 3 by 3 matrix 220 9.4 The inverse of a 3 by 3 matrix 221 9.5 Solution of simultaneous equations by matrices 223 9.6 Solution of simultaneous equations by determinants 226 9.7 Solution of simultaneous equations using Cramer’s rule 230 9.8 Solution of simultaneous equations using Gaussian elimination 232 10 Boolean Algebra and Logic Circuits 234 10.1 Boolean algebra and switching circuits 234 10.2 Simplifying Boolean expressions 238 10.3 Laws and rules of Boolean algebra 239 10.4 De Morgan’s laws 241 10.5 Karnaugh maps 242 10.6 Logic circuits and gates 248 10.7 Universal logic gates 253 11 Differential Calculus and its Applications 258 11.1 Common standard derivatives 258 11.2 Products and quotients 259 11.3 Function of a function 261 11.4 Successive differentiation 262 11.5 Differentiation of hyperbolic functions 263 11.6 Rates of change using differentiation 264 11.7 Velocity and acceleration 265 11.8 Turning points 267 11.9 Tangents and normals 270 11.10 Small changes using differentiation 272 11.11 Parametric equations 273 11.12 Differentiating implicit functions 276 11.13 Differentiation of logarithmic functions 279 11.14 Differentiation of inverse trigonometric functions 281 11.15 Differentiation of inverse hyperbolic functions 284 11.16 Partial differentiation 289 11.17 Total differential 292 11.18 Rates of change using partial differentiation 293 11.19 Small changes using partial differentiation 294 11.20 Maxima, minima and saddle points of functions of two variables 295 viii Contents 12 Integral Calculus and its Applications 303 12.1 Standard integrals 303 12.2 Non-standard integrals 307 12.3 Integration using algebraic substitutions 307 12.4 Integration using trigonometric and hyperbolic substitutions 310 12.5 Integration using partial fractions 317 12.6 The t tan θ 2 substitution 319 12.7 Integration by parts 323 12.8 Reduction formulae 326 12.9 Numerical integration 331 12.10 Area under and between curves 336 12.11 Mean or average values 343 12.12 Root mean square values 345 12.13 Volumes of solids of revolution 347 12.14 Centroids 350 12.15 Theorem of Pappus 354 12.16 Second moments of area 359 13 Differential Equations 366 13.1 The solution of equations of the form dy dx f(x) 366 13.2 The solution of equations of the form dydxf(y) 367 13.3 The solution of equations of the form dy dxf(x).f(y) 368 13.4 Homogeneous first order differential equations 371 13.5 Linear first order differential equations 373 13.6 Second order differential equations 2 13.7 Second order differential equations cy f(x) 13.8 Numerical methods for first order differential equations 385 13.9 Power series methods of solving ordinary differential equations 394 13.10 Solution of partial differential equations 405 14 Statistics and Probability 416 14.1 Presentation of ungrouped data 416 14.2 Presentation of grouped data 420 Contents ix 14.3 Measures of central tendency 424 14.4 Quartiles, deciles and percentiles 429 14.5 Probability 431 14.6 The binomial distribution 434 14.7 The Poisson distribution 435 14.8 The normal distribution 437 14.9 Linear correlation 443 14.10 Linear regression 445 14.11 Sampling and estimation theories 447 14.12 Chi-square values 454 14.13 The sign test 457 14.14 Wilcoxon signed-rank test 460 14.15 The Mann-Whitney test 464 15 Laplace Transforms 472 15.1 Standard Laplace transforms 472 15.2 Initial and final value theorems 477 15.3 Inverse Laplace transforms 480 15.4 Solving differential equations using Laplace transforms 483 15.5 Solving simultaneous differential equations using Laplace transforms 487 16 Fourier Series 492 16.1 Fourier series for periodic functions of period 2 π 492 16.2 Fourier series for a non-periodic function over range 2 π 496 16.3 Even and odd functions 498 16.4 Half range Fourier series 501 16.5 Expansion of a periodic function of period L 504 16.6 Half-range Fourier series for functions defined over range L 508 16.7 The complex or exponential form of a Fourier series 511 16.8 A numerical method of harmonic analysis 518 16.9 Complex waveform considerations 522 Index 525 《工程数学》是一本详尽的数学参考手册，由John Bird撰写，专为工程领域的实践者和学习者设计。本书涵盖了广泛的主题，旨在提供在工程领域工作中常用的数学工具和技术。 第1章“工程转换、常量和符号”介绍了基本的单位转换、希腊字母、国际单位制（SI）的基本单位及其前缀、物理和数学常量，以及推荐的数学符号，这些都是进行计算和表达的基础。 第2章“代数主题”涉及多项式除法、因式定理、余数定理、连分数、二次方程公式求解、对数、指数函数、奈培对数、双曲函数和部分分数。这些内容是解决复杂代数问题的关键。 第3章“数字主题”包括等差序列、等比序列、二项级数、麦克劳林公式、极限值、迭代法解方程、计算机数字系统。这些知识在处理序列和近似计算时非常有用。 第4章“面积与体积”讲解了平面图形的面积、圆的性质、规则立体的体积和表面积、锥台和圆锥的体积与表面积、不规则形状的面积和体积，以及波形的平均值。这些内容在工程设计和物理问题中经常出现。 第5章“几何与三角学”涵盖了角的类型和性质、三角形的属性、三角学的简介、锐角的三角比、三角比的评估、有理化和无理形式的三角比、解决直角三角形、笛卡尔坐标和极坐标、正弦和余弦规则、任意角度的三角比、正弦和余弦波形以及三角恒等式和方程。这些知识是解决几何问题和导航计算的基础。 第6章“图形”讲解了直线图、确定规律、对数比例尺、同时方程的图形解法、二次曲线、立方方程的图形解、极坐标曲线、椭圆和双曲线，以及图形函数。这是理解和绘制数学关系的重要部分。 第7章“向量”涉及标量和向量、向量加法、分解向量、向量减法、相对速度、两个周期函数的组合、向量点积和向量积。向量理论在物理学和工程学中至关重要，特别是在动力学和电磁学中。 第8章“复数”涵盖了复数的一般公式、直角坐标形式、极坐标形式、复数的应用、棣莫弗定理、指数形式。复数是解决某些实数问题的有力工具。 第9章“矩阵与行列式”涉及矩阵的加减乘法、2x2矩阵的行列式和逆矩阵、3x3矩阵的行列式和逆矩阵、用矩阵解线性方程组、用行列式解线性方程组、克莱姆法则和高斯消元法。这些是线性代数的基础，对控制系统和网络分析特别重要。 第10章“布尔代数与逻辑电路”介绍了布尔代数、开关电路、简化布尔表达式、布尔定律、德摩根定律、卡诺图、逻辑门和通用逻辑门。这部分内容与数字电子学密切相关。 第11章“微积分及其应用”涵盖了常见导数、乘积与商、复合函数、连续导数、超双曲函数的导数、利用导数求速率变化、速度与加速度、拐点、切线与法线、微小变化、参数方程、隐函数的导数、对数函数的导数、反三角函数的导数、反双曲函数的导数、偏导数、全微分、偏导数求速率变化和微小变化、以及多元函数的极值点。这些是理解变化率和优化问题的核心概念。 第12章“积分及其应用”包括标准积分、非标准积分、代数和三角代换法求积分、部分分数法、t-tanθ替换法、分部积分法、递推公式、数值积分、曲线下的面积、均值或平均值、平方根均值、旋转体的体积、质心、帕普斯定理、面积的第二矩。积分在解决物理问题和求解面积、体积等问题时扮演关键角色。 第13章“微分方程”涉及一阶微分方程的解法、齐次微分方程、线性一阶微分方程、二阶微分方程、一阶微分方程的数值方法、常微分方程的幂级数解法和偏微分方程的解法。微分方程在描述动态系统的行为时极其重要。 第14章“统计与概率”涵盖未分组数据和分组数据的呈现、集中趋势度量、四分位数、百分位数、概率、二项分布、泊松分布、正态分布、线性相关性、线性回归、采样和估计理论、卡方值、符号检验、威尔科逊符号秩检验、曼-惠特尼检验。这些统计概念在数据分析和假设检验中必不可少。 第15章“拉普拉斯变换”讲解标准拉普拉斯变换、初值和终值定理、逆拉普拉斯变换、使用拉普拉斯变换解微分方程以及使用拉普拉斯变换解耦合微分方程。拉普拉斯变换是信号处理和控制理论中的关键工具。 第16章“傅里叶级数”讨论周期函数和非周期函数的傅里叶级数、偶函数和奇函数、半范围傅里叶级数、周期函数的傅里叶级数、半范围傅里叶级数、傅里叶级数的复数或指数形式、谐波分析的数值方法以及复波形的考虑。傅里叶级数在信号分析和频谱分析中具有重要应用。 《工程数学》是一部全面的资源，提供了工程师和学生所需的所有数学工具，帮助他们解决实际工程问题。从基础的代数和几何到高级的微积分和统计学，这本书为理解和应用数学原理提供了扎实的基础。