《ADVANCED CALCULUS NEW EDITION》

CHAPTER Ⅰ.PRELIMINARY1

1.Functions1

2.Continuity2

3.The derivative5

4.Composite functions7

5.Rolle’s theorem7

6.Theorem of the mean8

7.Taylor’s series with a remainder10

8.The form 0/015

9.The form ∞/∞16

10.Other indeterminate forms18

11.Infinitesimals19

12.Fundamental theorems on infinitesimals22

13.Some geometric theorems involving infinitesimals23

14.The first differential28

15.Higher differentials29

16.Change of variable32

CHAPTER Ⅱ.POWER SERIES38

17.Definitions38

18.Comparison test for convergence40

19.The ratio test for convergence41

20.Region of convergence42

21.Uniform convergence45

22.Function defined by a power series45

23.Integral and derivative of a power series46

24.Taylor’s series48

25.Operations with two power series51

26.The exponential and trigonometric functions53

27.Hyperbolic functions55

28.Dominant functions57

29.Conditionally convergent series58

CHAPTER Ⅲ.PARTIAL DIFFERENTIATION65

30.Functions of two or more variables65

31.Partial derivatives66

32.Order of differentiation68

33.Differentiation of composite functions69

34.Euler’s theorem on homogeneous functions73

35.Directional derivative74

36.The first differential78

37.Higher differentials84

38.Taylor’s series85

CHAPTER Ⅳ.IMPLICIT FUNCTIONS91

39.One equation，two variables91

40.One equation，more than two variables93

41.Two equations，four variables95

42.Three equations，six variables97

43.The general case98

44.Jacobians99

CHAPTER Ⅴ.APPLICATIONS TO GEOMETRY106

45.Element of arc106

46.Straight line108

47.Surfaces109

48.Planes110

49.Behavior of a surface near a point112

50.Maxima and minima116

51.Curves118

52.Curvature and torsion121

53.Curvilinear coordinates124

CHAPTER Ⅵ.THE DEFINITE INTEGRAL134

54.Definition134

55.Existence proof135

56.Properties of definite integrals137

57.Evaluation of a definite integral138

58.Simpson’s rule139

59.Change of variables140

60.Differentiation of a definite integral141

61.Integration under the integral sign145

62.Infinite limit146

63.Differentiation and integratron of an integral with an infinite limit148

64.Infinite integrand151

65.Certain definite integrals153

66.Multiple integrals156

CHAPTER Ⅶ.THE GAMMA AND BETA FUNCTIONS164

67.The Gamma function164

68.The Beta function166

69.Dirichlet’s integrals167

70.Special relations169

CHAPTER Ⅷ.LINE，SURFACE，AND SPACE INTEGRALS174

71.Line integrals174

72.Plane area as a line integral177

73.Green’s theorem in the plane181

74.Dependence upon the path of integration183

75.Exact differentials185

76.Area of a curved surface187

77.Surface integrals190

78.Green’s theorem in space192

79.Other forms of Green’s theorem195

80.Stokes’s theorem197

CHAPTER Ⅸ.VECTOR NOTATION203

81.Vectors203

82.The scalar product204

83.The vector product206

84.Curves208

85.Areas209

86.The gradient210

87.The divergence211

88.The curl212

CHAPTER Ⅹ.DIFFERENTIAL EQUATIONS OF THE FIRST ORDER216

89.Introduction216

90.Existence proof217

91.Equations of the first degree219

92.Equations not of the first degree225

93.Envelope of a family of plane curves229

94.Envelope as locus of limit points231

95.Singular solutions232

96.Evolute and involute233

97.Orthogonal trajectories of plane curves236

98.Differential equation of the first order in three variables237

99.Simultaneous equations in three variables243

100.Existence proof246

CHAPTER Ⅺ.DIFFERENTIAL EQUATIONS OF HIGHER ORDER252

101.Existence proof252

102.The linear differential equation253

103.Method of variation of constants255

104.The linear differential equation with constant coefficients257

105.The complementary function259

106.The particular integral260

107.Equations reducible to linear equations with constant coefficients262

108.Simultaneous linear differential equations with constant coefficients263

109.Equations of the second order264

110.Legendre’s equation268

CHAPTER Ⅻ.BESSEL FUNCTIONS275

111.Bessel’s differential equation275

112.Bessel functions of integral order276

113.Roots of Bessel functions of integral order279

114.Bessel functions of integral order as definite integrals280

115.The function In（x）283

116.The Bessel function of fractional order284

117.Bessel functions of the second kind285

CHAPTER ⅩⅢ.PARTIAL ？？？？？？？？？ EQUATIONS291

118.Introduction291

119.Special forms of partial differential eq？ations291

120.The linear partial differential equation of the ？？？292

121.The Fourier series295

122.The Fourier series with sines or cosines only299

123.Laplace’sequation in two variables301

124.Application to flow of heat303

125.The Laplace equation in three variables306

126.Application to potential308

127.Harmonic functions310

CHAPTER ⅩⅣ.CALCULUS OF VARIATIONS317

128.The simplest case317

129.Solution by differentials320

130.Variable limits323

131.Constrained variation324

132.Any number of variables326

133.Hamilton’s principle； Lagrange’s328

CHAPTER ⅩⅤ.FUNCTI？？？？？ A COMPLEX VARIABLE332

134.Complex numbers332

135.Graphical representation ？？？？？？？？orm333

136.Powers and roots335

137.The square root337

138.Exponential and trig339

139.The hyperbolic functions341

140.The logarithmic function342

141.The inverse hyperbolic and trigonometric functions343

142.Functions of a complex variable in general344

143.Conjugate functions347

144.Conformal representation348

145.Integral of a complex function351

146.Cauchy’s theorem352

147.Taylor’s series353

148.Poles and residues355

149.Application to real integrals356

150.Application to Bessel functions360

CHAPTER ⅩⅥ.ELLIPTIC INTEGRALS365

151.Introduction365

152.The functions sn u，cn u，dn u367

153.Application to the pendulum369

154.Formulas of differentiation and series expansion371

155.Addition formulas372

156.The periods373

157.Limiting cases375

158.Elliptic integrals in the complex plane376

159.Elliptic integrals of the second kind and of the third kind379

160.The function p（u）381

161.Applications382

ANSWERS387

INDEX395