A Course of Higher Mathematics
Adiwes International Series in Mathematics, Volume 2
Published on 9. May 2014
644 pages
978-1-4831-8508-8 (ISBN)
Description
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 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.
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Content
IntroductionPreface to the Sixth EditionPreface to the Fourteenth EditionChapter 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) = f (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. ExamplesChapter 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 MotionChapter 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.
