HIGHER MATHEMATICS FOR ENGINEERS AND PHYSICISTS SECOND EDITIONPDF电子书下载

CHAPTER Ⅰ INFINITE SERIES 1

1．Fundamental Concepts 1

2．Series of Constants 6

3．Series of Positive Terms 9

4．Alternating Series 15

5．Series of Positive and Negative Terms 16

6．Algebra of Series 21

7．Continuity of Functions．Uniform Convergence 23

8．Properties of Uniformly Convergent Series 28

9．Power Series 30

10．Properties of Power Series 33

11．Expansion of Functions in Power Series 35

12．Application of Taylor＇s Formula 41

13．Evaluation of Definite Integrals by Means of Power Series 43

14．Rectification of Ellipse．Elliptic Integrals 47

15．Discussion of Elliptic Integrals 48

16．Approximate Formulas in Applied Mathematics 55

CHAPTER Ⅱ FOURIER SERIES 63

17．Preliminary Remarks 63

18．Dirichlet Conditions．Derivation of Fourier Coefficients 65

19．Expansion of Functions in Fourier Series 67

20．Sine and Cosine Series 73

21．Extension of Interval of Expansion 76

22．Complex Form of Fourier Series 78

23．Differentiation and Integration of Fourier Series 80

24．Orthogonal Functions 81

CHAPTER Ⅲ SOLUTION OF EQUATIONS 83

25．Graphical Solutions 83

26．Algebraic Solution of Cubic 86

27．Some Algebraic Theorems 92

28．Horner＇s Method 95

29．Newton＇s Method 97

30．Determinants of the Second and Third Order 102

31．Determinants of the nth Order 106

32．Properties of Determinants 107

33．Minors 110

34．Matrices and Linear Dependence 114

35．Consistent and Inconsistent Systems of Equations 117

CHAPTER Ⅳ PARTIAL DIFFERENTIATION 123

36．Functions of Several Variables 123

37．Partial Derivatives 125

38．Total Differential 127

39．Total Derivatives 130

40．Euler＇s Formula 136

41．Differentiation of Implicit Functions 137

42．Directional Derivatives 143

43．Tangent Plane and Normal Line to a Surface 146

44．Space Curves 149

45．Directional Derivatives in Space 151

46．Higher Partial Derivatives 153

47．Taylor＇s Series for Functions of Two Variables 155

48．Maxima and Minima of Functions of One Variable 158

49．Maxima and Minima of Functions of Several Variables 160

50．Constrained Maxima and Minima 163

51．Differentiation under the Integral Sign 167

CHAPTER Ⅴ MULTIPLE INTEGRALS 173

52．Definition and Evaluation of the Double Integral 173

53．Geometric Interpretation of the Double Integral 177

54．Triple Integrals 179

55．Jacobians．Change of Variable 183

56．Spherical and Cylindrical Coordinates 185

57．Surface Integrals 188

58．Green＇s Theorem in Space 191

59．Symmetrical Form of Green＇s Theorem 194

CHAPTER Ⅵ LINE INTEGRAL 197

60．Definition of Line Integral 197

61．Area of a Closed Curve 199

62．Green＇s Theorem for the Plane 202

63．Properties of Line Integrals 206

64．Multiply Connected Regions 212

65．Line Integrals in Space 215

66．Illustrations of the Application of the Line Integrals 217

CHAPTER Ⅶ ORDINARY DIFFERENTIAL EQUATIONS 225

67．Preliminary Remarks 225

68．Remarks on Solutions 227

69．Newtonian Laws 231

70．Simple Harmonic Motion 233

71．Simple Pendulum 234

72．Further Examples of Derivation of Differential Equations 239

73．Hyperbolic Functions 247

74．First-order Differential Equations 256

75．Equations with Separable Variables 257

76．Homogeneous Differential Equations 259

77．Exact Differential Equations 262

78．Integrating Factors 265

79．Equations of the First Order in Which One of the Variables Does Not Occur Explicitly 267

80．Differential Equations of the Second Order 269

81．Gamma Functions 272

82．Orthogonal Trajectories 277

83．Singular Solutions 279

84．Linear Differential Equations 283

85．Linear Equations of the First Order 284

86．A Non-linear Equation Reducible to Linear Form(Bernoulli＇s Equation) 286

87．Linear Differential Equations of the nth Order 287

88．Some General Theorems 291

89．The Meaning of the Operator 1/Dn+a1Dn-1+．．．+an-1D+anf(x) 295

90．Oscillation of a Spring and Discharge of a Condenser 299

91．Viscous Damping 302

92．Forced Vibrations 308

93．Resonance 310

94．Simultaneous Differential Equations 312

95．Linear Equations with Variable Coefficients 315

96．Variation of Parameters 318

97．The Euler Equation 322

98．Solution in Series 325

99．Existence of Power Series Solutions 329

100．Bessel＇s Equation 332

101．Expansion in Series of Bessel Functions 339

102．Legendre＇s Equation 342

103．Numerical Solution of Differential Equations 346

CHAPTER Ⅷ PARTIAL DIFFERENTIAL EQUATIONS 350

104．Preliminary Remarks 350

105．Elimination of Arbitrary Functions 351

106．Integration of Partial Differential Equations 353

107．Linear Partial Differential Equations with Constant Coefficients 357

108．Transverse Vibration of Elastic String 361

109．Fourier Series Solution 364

110．Heat Conduction 367

111．Steady Heat Flow 369

112．Variable Heat Flow 373

113．Vibration of a Membrane 377

114．Laplace＇s Equation 382

115．Flow of Electricity in a Cable 386

CHAPTER Ⅸ VECTOR ANALYSIS 392

116．Scalars and Vectors 392

117．Addition and Subtraction of Vectors 393

118．Decomposition of Vectors．Base Vectors 396

119．Multiplication of Vectors 399

120．Relations between Scalar and Vector Products 402

121．Applications of Scalar and Vector Products 404

122．Differential Operators 406

123．Vector Fields 409

124．Divergence of a Vector 411

125．Divergence Theorem 415

126．Curl of a Vector 418

127．Stokes＇s Theorem 421

128．Two Important Theorems 422

129．Physical Interpretation of Divergence and Curl 423

130．Equation of Heat Flow 425

131．Equations of Hydrodynamics 428

132．Curvilinear Coordinates 433

CHAPTER Ⅹ COMPLEX VARIABLE 133

133．Complex Numbers 440

134．Elementary Functions of a Complex Variable 444

135．Properties of Functions of a Complex Variable 448

136．Integration of Complex Functions 453

137．Cauchy＇s Integral Theorem 455

138．Extension of Cauchy＇s Theorem 455

139．The Fundamental Theorem of Integral Calculus 457

140．Cauchy＇s Integral Formula 461

141．Taylor＇s Expansion 464

142．Conformal Mapping 465

143．Method of Conjugate Functions 467

144．Problems Solvable by Conjugate Functions 470

145．Examples of Conformal Maps 471

146．Applications of Conformal Representation 479

CHAPTER ⅩⅠ PROBABILITY 492

147．Fundamental Notions 492

148．Independent Events 495

149．Mutually Exclusive Events 497

150．Expectation 500

151．Repeated and Independent Trials 501

152．Distribution Curve 504

153．Stirling＇s Formula 508

154．Probability of the Most Probable Number 511

155．Approximations to Binomial Law 512

156．The Error Function 516

157．Precision Constant．Probable Error 521

CHAPTER ⅩⅡ EMPIRICAL FORMULAS AND CURVE FITTING 525

158．Graphical Method 525

159．Differences 527

160．Equations That Represent Special Types of Data 528

161．Constants Determined by Method of Averages 534

162．Method of Least Squares 536

163．Method of Moments 544

164．Harmonic Analysis 545

165．Interpolation Formulas 550

166．Lagrange＇s Interpolation Formula 552

167．Numerical Integration 554

168．A More General Formula 558

ANSWERS 561

INDEX 575