A Course of Higher Mathematics
Adiwes International Series in Mathematics, Volume 2
- 1st Edition - January 1, 1964
- Imprint: Pergamon
- Author: V. I. Smirnov
- Editor: A. J. Lohwater
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 1 - 6 9 2 7 - 9
- eBook ISBN:9 7 8 - 1 - 4 8 3 1 - 8 5 0 8 - 8
A Course of Higher Mathematics, Volume II: Advanced Calculus covers the theory of functions of real variable in advanced calculus. This volume is divided into seven chapters and… Read more
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Request a sales quoteA Course of Higher Mathematics, Volume II: Advanced Calculus covers the theory of functions of real variable in advanced calculus. This volume is divided into seven chapters and begins with a full discussion of the solution of ordinary differential equations with many applications to the treatment of physical problems. This topic is followed by an account of the properties of multiple integrals and of line integrals, with a valuable section on the theory of measurable sets and of multiple integrals. The subsequent chapters deal with the mathematics necessary to the examination of problems in classical field theories in vector algebra and vector analysis and the elements of differential geometry in three-dimensional space. The final chapters explore the Fourier series and the solution of the partial differential equations of classical mathematical physics. This book will prove useful to advanced mathematics students, engineers, and physicists.
Introduction
Preface to the Sixth Edition
Preface to the Fourteenth Edition
Chapter I Ordinary Differential Equations
§ 1. Equations of the First Order
1. General Principles
2. Equations with Separable Variables
3. Homogeneous Equations
4. Linear Equations; Bernoulli's Equation
5. Finding the Solution of a Differential Equation with a given Initial Condition
6. The Euler—Cauchy Method
7. The General Solution
8. Clairaut's Equation
9. Lagrangian Equations
10. The Envelope of a Family of Curves, and Singular Solutions
11. Equations Quadratic in y'
12. Isogonal Trajectories
§ 2. Differential Equations of Higher Orders; Systems of Equations
13. General Principles
14. Graphical Methods of Integrating Second Order Differential Equations
15. The Equation y(n) = ƒ (X)
16. Bending of a Beam
17. Lowering the Order of a Differential Equation
18. Systems of Ordinary Differential Equations
19. Examples
20. Systems of Equations and Equations of Higher Orders
21. Linear Partial Differential Equations
22. Geometrical Interpretation
23. Examples
Chapter II Linear Differential Equations. Supplementary Remarks on the Theory of Differential Equations
§ 3. General Theory; Equations with Constant Coefficients
24. Linear Homogeneous Equations of the Second Order
25. Non-Homogeneous Linear Equations of the Second Order
26. Linear Equations of Higher Orders
27. Homogeneous Equations of the Second Order with Constant Coefficients
28. Non-Homogeneous Linear Equations of the Second Order with Constant Coefficients
29. Particular Cases
30. Linear Equations of Higher Orders with Constant Coefficients
31. Linear Equations and Oscillatory Phenomena
32. Free and Forced Oscillations
33. Sinusoidal External Forces and Resonance
34. Impulsive External Forces
35. Statical External Forces
36. The Strength of a Thin Elastic Rod, Compressed by Longitudinal Forces (Euler's Problem)
37. Rotating Shaft
38. Symbolic Method
39. Linear Homogeneous Equations of Higher Orders with Constant Coefficients
40. Linear Non-Homogeneous Equations with Constant Coefficients
41. Example
42. Euler's Equation
43. Systems of Linear Equations with Constant Coefficients
44. Examples
§ 4. Integration with the Aid of Power Series
45. Integration of a Linear Equation, Using a Power Series
46. Examples
47. Expansion of Solutions into Generalized Power Series
48. BessePs Equation
49. Equations Reducible to Bessel's Equation
§ 5. Supplementary Notes on the Theory of Differential Equations
50. The Method of Successive Approximations for Linear Equations
51. The Case of a Non-Linear Equation
52. Singular Points of First Order Differential Equations
53. The Streamlines of Collinear Plane Fluid Motion
Chapter III Multiple and Line Integrals. Improper Integrals that Depend on a Parameter
§ 6. Multiple Integrals
54. Volumes
55. Double Integrals
56. Evaluation of Double Integrals
57. Curvilinear Coordinates
58. Triple Integrals
59. Cylindrical and Spherical Coordinates
60. Curvilinear Coordinates in Space
61. Basic Properties of Multiple Integrals
62. Surface Areas
63. Integrals Over a Surface and Ostrogradskii's Formula
64. Integrals Over a given Side of a Surface
65. Moments
§ 7. Line Integrals
66. Definition of a Line Integral
67. Work Done by a Field of Force. Examples
68. Areas and Line Integrals
69. Green's Formula
70. Stokes' Formula
71. Independence of a Line Integral on the Path in a Plane
72. Multiply Connected Domains
73. Independence of a Line Integral on the Path in Space
74. Steady-State Flow of Fluids
75. Integrating Factors
76. Exact Differential Equations in the Case of Three Variables
77. Change of Variables in Double Integrals
§ 8. Improper Integrals and Integrals that Depend on a Parameter
78. Integration Under the Integral Sign
79. Dirichlet's Formula
80. Differentiation Under the Integral Sign
81. Examples
82. Improper Integrals
83. Conditionally Convergent Integrals
84. Uniformly Convergent Integrals
85. Examples
86. Improper Multiple Integrals
87. Examples
§ 9. Supplementary Remarks on the Theory of Multiple Integrals
88. Preliminary Concepts
89. Basic Theorems Regarding Sets
90. Interior and Exterior Areas
91. Measurable Sets
92. Independence on the Choice of Axes
93. The Case of any Number of Dimensions
94. Darboux's Theorem
95. Integrable Functions
96. Properties of Integrable Functions
97. Evaluation of Double Integrals
98. n-Tuple Integrals
99. Examples
Chapter IV Vector Analysis and Field Theory
§ 10. Basic Vector Algebra
100. Addition and Subtraction of Vectors
101. Multiplication of a Vector by a Scalar. Coplanar Vectors
102. Resolution of a Vector into Three Non-Coplanar Components
103. Scalar Product
104. Vector Products
105. The Relationship between Scalar and Vector Products
106. The Velocities at Points of a Rotating Rigid Body; the Moment of a Vector
§ 11. Field Theory
107. Differentiation of Vectors
108. Scalar Field and Gradient
109. Vector Fields. Curl and Divergence
110. Lamellar and Solenoidal Fields
111. Directed Elementary Areas
112. Some Formulae of Vector Analysis
113. Motion of a Rigid Body and Small Deformations
114. Equation of Continuity
115. Hydrodynamical Equations for an Ideal Fluid
116. Equations of Sound Propagation
117. Equation of Thermal Conduction
118. Maxwell's Equations
119. Laplace's Operator in Orthogonal Coordinates
120. Differentiation in the Case of a Variable Field
Chapter V Foundations of Differential Geometry
§ 12. Curves on a Plane and in Space
121. The Curvature of a Plane Curve; the Evolute
122. Involutes
123. The Natural Equation of a Curve
124. The Fundamental Elements of Curves in Space
125. Frenet's Formula
126. The Osculating Plane
127. The Helix
128. Field of Unit Vectors
§ 13. Elementary Theory of Surfaces
129. The Parametric Equations of a Surface
130. Gauss First Differential Form
131. Gauss Second Differential Form
132. The Curvature of Lines Ruled on Surfaces
133. Dupin's Indicatrix and Euler's Formula
134. Finding the Principal Radii of Curvature and Principal Directions
135. Line of Curvature
136. Dupin's Theorem
137. Examples
138. Gaussian Curvature
139. The Variation of an Elementary Area and the mean Curvature
140. Envelopes of Surfaces and Curves
141. Developable Surfaces
Chapter VI Fourier Series
§ 14. Harmonic Analysis
142. Orthogonality of the Trigonometric Functions
143. Diriehlet's Theorem
144. Examples
145. Expansion in the Interval (0, π)
146. Periodic Functions of Period 2l
147. Average Error
148. General Orthogonal Systems of Functions
149. Practical Harmonic Analysis
§ 15. Supplementary Remarks on the Theory of Fourier Series
150. Expansion in Fourier Series
151. Second mean Value Theorem
152. Dirichlet Integrals
153. Diriehlet's Theorem
154. Polynomial Approximations to Continuous Functions
155. The Closure Equation
156. Properties of Closed Systems of Functions
157. The Character of the Convergence of Fourier Series
158. Improving the Convergence of Fourier Series
159. Example
§ 16. Fourier Integrals and Multiple Fourier Series
160. Fourier's Formula
161. Fourier Series in Complex Form
162. Multiple Fourier Series
Chapter VII The Partial Differential Equations of Mathematical Physics
§ 17. The Wave Equation
163. The Equation of the Vibration of a String
164. D'Alembert's Solution
165. Particular Cases
166. Finite String
167. Fourier's Method
168. Harmonics and Standing Waves
169. Forced Vibrations
170. Concentrated Force
171. Poisson's Formula
172. Cylindrical Waves
173. The Case of n-Dimensional Space
174. Non-Homogeneous Wave Equation
175. Point Sources
176. The Transverse Vibrations of a Membrane
177. Rectangular Membrane
178. Circular Membranes
179. The Uniqueness Theorem
180. Applications of Fourier Integrals
§ 18. The Equation of Telegraphy
181. Fundamental Equations
182. Steady-State Processes
183. Transient Processes
184. Examples
185. Generalized Equation of Vibration of a String
186. The General Case of an Infinite Circuit
187. Fourier's Method for a Finite Circuit
188. The Generalized Wave Equation
§ 19. The Vibrations of Rods
189. Fundamental Equations
190. Particular Solutions
191. The Expansion of an Arbitrary Function
§ 20. Laplace's Equation
192. Harmonic Functions
193. Green's Formula
194. The Fundamental Properties of Harmonic Functions
195. The Solution of Dirichlet's Problem for a Circle
196. Poisson Integrals
197. Dirichlet's Problem for a Sphere
198. Green's Function
199. The Case of a Halfspace
200. Potential of a Distributed Mass
201. Poisson's Equation
202. Kirchhoff's Formula
§ 21. The Equation of Thermal Conduction
203. Fundamental Equations
204. Infinite Rod
205. Semi-Infinite Rods
206. Rods Bounded at both Ends
207. Supplementary Remarks
208. The Case of a Sphere
209. The Uniqueness Theorem
Index
- Edition: 1
- Published: January 1, 1964
- No. of pages (eBook): 644
- Imprint: Pergamon
- Language: English
- Paperback ISBN: 9781483169279
- eBook ISBN: 9781483185088
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