Vector Analysis by M.L. Krasnov; A.I. Kiselev; G.I. Makarenko.

The present collection of problems in vector analysis contains the required minimum of problems and exercises for the course of vector analysis of engineering colleges.

Each section starts with a brief review of theory and detailed solutions of a sufficient number of typical problems. The text contains 100 worked problems and there are 314 problems left to the student. There are also a certain number of problems of an applied nature that have been chosen so that their analysis does not require supplementary information in specialized fields. The material of the sixth chapter is devoted to curvilinear coordinates and the basic operations of vector analysis in curvilinear coordinates. Its purpose is to give the reader at least a few problems to develop the necessary skills.

The exposition in this text follows closely the lines currently employed at the chair of higher mathematics of the Moscow Power Institute.

The present text may be regarded as a short course in vector analysis in which the basic facts are given without proof but with illustrative examples of a practical nature. Hence this problem book may be used in a recapitulation of the essentials of vector analysis or as a text for readers who wish merely to master the techniques of vector analysis, while dispensing with the proofs of propositions and theorems.

This collection of problems is designed for students of day and evening departments at engineering colleges and also for correspondence students with a background of vector algebra and calculus as given in the first two years of college study.

The book was translated from the Russian by George Yankovsky and was first published by Mir in 1983.

All credits to the original uploader.

Contents

Preface 7

CHAPTER I. THE VECTOR FUNCTION OF A SCALAR ARGUMENT
Sec. 1. The hodograph of a vector function 9
Sec. 2. The limit and continuity of a vector function of a scalar argument 11
Sec. 3. The derivative of a vector function with respect to a scalar argument 14
Sec. 4, Integrating a vector function of a scalar argument 18
Sec. 5. The first and second derivatives of a vector with
respect to the arc length of a curve. The curvature of a curve. The principal normal. 27
Sec. 6. Osculating plane. Binormal. Torsion. The Frenet formulas. 31

CHAPTER II. SCALAR FIELDS
Sec. 7. Examples of scalar fields. Level surfaces and level linea 35
Sec. 8. Directional derivative 39
Sec. 9. The gradient of a scalar field 44

CHAPTER III. VECTOR FIELDS
Sec. 10. Vector linea. Differential equations of vector linea 52
Sec. 11. The flux of a vector field. Methods of calculating flux 58
Sec. 12. The flux of a vector through a closed surface. The Gauss-Ostrogradsky Theorem. 89
Sec. 13. The divergence of a vector field. Solenoidal fields. 89
See. 14. A line integral in a vector field. The circulation of a vector field 96
Sec. 15. The curl (rotation) of a vector field 108
Sec. 16. Stokes' theorem 111
Sec. 17. The independence of a line integral of the path
of integration. Green's formula 115

CHAPTER IV. POTENTIAL FIELDS
See. 18. The criterion for the potentiality of a vector field t2t
See. 19. Computing a line integral in a potential field 124

CHAPTER V. THE HAMILTONIAN OPERATOR. SECOND-ORDER DIFFERENTIAL OPER~
ATIONS. THE LAPLACE OPERATOR
See. 20. The Hamiltonian operator del 130
See. 21. Second-order differential operations. The Laplace operator 135
See. 22. Vector potential 146

CHAPTER VI. CURVILINEAR COORDINATES. BASIC OPERATIONS OF VECTOR ANALYSIS IN CURVILINEAR COORDINATES
See. 23. Curvilinear coordinates 152
See. 24. Basic operations of vector analysis in curvilinear coordinates 156
See. 25. The Laplace operator in orthogonal coordinates 174

ANSWERS 177
APPENDIX I 184
APPENDIX II 186
BIBLIOGRAPHY 187
INDEX 188