HIGHER MATHEMATICS FOR ENGINEERS AND PHYSICISTS SECOND EDITION2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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CHAPTER Ⅰ INFINITE SERIES1

1．Fundamental Concepts1

2．Series of Constants6

3．Series of Positive Terms9

4．Alternating Series15

5．Series of Positive and Negative Terms16

6．Algebra of Series21

7．Continuity of Functions．Uniform Convergence23

8．Properties of Uniformly Convergent Series28

9．Power Series30

10．Properties of Power Series33

11．Expansion of Functions in Power Series35

12．Application of Taylor＇s Formula41

13．Evaluation of Definite Integrals by Means of Power Series43

14．Rectification of Ellipse．Elliptic Integrals47

15．Discussion of Elliptic Integrals48

16．Approximate Formulas in Applied Mathematics55

CHAPTER Ⅱ FOURIER SERIES63

17．Preliminary Remarks63

18．Dirichlet Conditions．Derivation of Fourier Coefficients65

19．Expansion of Functions in Fourier Series67

20．Sine and Cosine Series73

21．Extension of Interval of Expansion76

22．Complex Form of Fourier Series78

23．Differentiation and Integration of Fourier Series80

24．Orthogonal Functions81

CHAPTER Ⅲ SOLUTION OF EQUATIONS83

25．Graphical Solutions83

26．Algebraic Solution of Cubic86

27．Some Algebraic Theorems92

28．Horner＇s Method95

29．Newton＇s Method97

30．Determinants of the Second and Third Order102

31．Determinants of the nth Order106

32．Properties of Determinants107

33．Minors110

34．Matrices and Linear Dependence114

35．Consistent and Inconsistent Systems of Equations117

CHAPTER Ⅳ PARTIAL DIFFERENTIATION123

36．Functions of Several Variables123

37．Partial Derivatives125

38．Total Differential127

39．Total Derivatives130

40．Euler＇s Formula136

41．Differentiation of Implicit Functions137

42．Directional Derivatives143

43．Tangent Plane and Normal Line to a Surface146

44．Space Curves149

45．Directional Derivatives in Space151

46．Higher Partial Derivatives153

47．Taylor＇s Series for Functions of Two Variables155

48．Maxima and Minima of Functions of One Variable158

49．Maxima and Minima of Functions of Several Variables160

50．Constrained Maxima and Minima163

51．Differentiation under the Integral Sign167

CHAPTER Ⅴ MULTIPLE INTEGRALS173

52．Definition and Evaluation of the Double Integral173

53．Geometric Interpretation of the Double Integral177

54．Triple Integrals179

55．Jacobians．Change of Variable183

56．Spherical and Cylindrical Coordinates185

57．Surface Integrals188

58．Green＇s Theorem in Space191

59．Symmetrical Form of Green＇s Theorem194

CHAPTER Ⅵ LINE INTEGRAL197

60．Definition of Line Integral197

61．Area of a Closed Curve199

62．Green＇s Theorem for the Plane202

63．Properties of Line Integrals206

64．Multiply Connected Regions212

65．Line Integrals in Space215

66．Illustrations of the Application of the Line Integrals217

CHAPTER Ⅶ ORDINARY DIFFERENTIAL EQUATIONS225

67．Preliminary Remarks225

68．Remarks on Solutions227

69．Newtonian Laws231

70．Simple Harmonic Motion233

71．Simple Pendulum234

72．Further Examples of Derivation of Differential Equations239

73．Hyperbolic Functions247

74．First-order Differential Equations256

75．Equations with Separable Variables257

76．Homogeneous Differential Equations259

77．Exact Differential Equations262

78．Integrating Factors265

79．Equations of the First Order in Which One of the Variables Does Not Occur Explicitly267

80．Differential Equations of the Second Order269

81．Gamma Functions272

82．Orthogonal Trajectories277

83．Singular Solutions279

84．Linear Differential Equations283

85．Linear Equations of the First Order284

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

87．Linear Differential Equations of the nth Order287

88．Some General Theorems291

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

90．Oscillation of a Spring and Discharge of a Condenser299

91．Viscous Damping302

92．Forced Vibrations308

93．Resonance310

94．Simultaneous Differential Equations312

95．Linear Equations with Variable Coefficients315

96．Variation of Parameters318

97．The Euler Equation322

98．Solution in Series325

99．Existence of Power Series Solutions329

100．Bessel＇s Equation332

101．Expansion in Series of Bessel Functions339

102．Legendre＇s Equation342

103．Numerical Solution of Differential Equations346

CHAPTER Ⅷ PARTIAL DIFFERENTIAL EQUATIONS350

104．Preliminary Remarks350

105．Elimination of Arbitrary Functions351

106．Integration of Partial Differential Equations353

107．Linear Partial Differential Equations with Constant Coefficients357

108．Transverse Vibration of Elastic String361

109．Fourier Series Solution364

110．Heat Conduction367

111．Steady Heat Flow369

112．Variable Heat Flow373

113．Vibration of a Membrane377

114．Laplace＇s Equation382

115．Flow of Electricity in a Cable386

CHAPTER Ⅸ VECTOR ANALYSIS392

116．Scalars and Vectors392

117．Addition and Subtraction of Vectors393

118．Decomposition of Vectors．Base Vectors396

119．Multiplication of Vectors399

120．Relations between Scalar and Vector Products402

121．Applications of Scalar and Vector Products404

122．Differential Operators406

123．Vector Fields409

124．Divergence of a Vector411

125．Divergence Theorem415

126．Curl of a Vector418

127．Stokes＇s Theorem421

128．Two Important Theorems422

129．Physical Interpretation of Divergence and Curl423

130．Equation of Heat Flow425

131．Equations of Hydrodynamics428

132．Curvilinear Coordinates433

CHAPTER Ⅹ COMPLEX VARIABLE133

133．Complex Numbers440

134．Elementary Functions of a Complex Variable444

135．Properties of Functions of a Complex Variable448

136．Integration of Complex Functions453

137．Cauchy＇s Integral Theorem455

138．Extension of Cauchy＇s Theorem455

139．The Fundamental Theorem of Integral Calculus457

140．Cauchy＇s Integral Formula461

141．Taylor＇s Expansion464

142．Conformal Mapping465

143．Method of Conjugate Functions467

144．Problems Solvable by Conjugate Functions470

145．Examples of Conformal Maps471

146．Applications of Conformal Representation479

CHAPTER ⅩⅠ PROBABILITY492

147．Fundamental Notions492

148．Independent Events495

149．Mutually Exclusive Events497

150．Expectation500

151．Repeated and Independent Trials501

152．Distribution Curve504

153．Stirling＇s Formula508

154．Probability of the Most Probable Number511

155．Approximations to Binomial Law512

156．The Error Function516

157．Precision Constant．Probable Error521

CHAPTER ⅩⅡ EMPIRICAL FORMULAS AND CURVE FITTING525

158．Graphical Method525

159．Differences527

160．Equations That Represent Special Types of Data528

161．Constants Determined by Method of Averages534

162．Method of Least Squares536

163．Method of Moments544

164．Harmonic Analysis545

165．Interpolation Formulas550

166．Lagrange＇s Interpolation Formula552

167．Numerical Integration554

168．A More General Formula558

ANSWERS561

INDEX575