Vector analysis

Preface 7
CHAPTER I. THE VECTOR FUNCTION OF A SC A LA R ARGUMENT 9
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 35
Sec. 7. Examples of scalar fields. Level surfaces and level lines 35
Sec. 8. Directional derivative 39
Sec. 9. The gradient of a scalar field 44
CHAPTER III. VECTOR FIELDS 52
Sec. 10. Vector lines. Differential equations of vector lines 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 Gauas-Ostrogradaky theorem 85
Sec. 13. The divergence of a vector field. Solenoidal fields 89
Sec. 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 121
Sec. 18. The criterion for the potentiality of a vector field 121
Sec. 19. Computing a line integral in a potential field 124
CHAPTER V. THE HAMILTONIAN OPERATOR. SECOND-ORDER DIFFERENTIAL OPERATIONS. THE LAPLACE OPERATOR 130
Sec. 20. The Hamiltonian operator del 130
Sec. 21. Second-order differential operations. The Laplace operator 135
Sec. 22. Vector potential 146
CHAPTER VI. CURVILINEAR COORDINATES. BASIC OPERATIONS OF VECTOR ANALYSIS IN CURVILINEAR COORDINATES 152
Sec. 23. Curvilinear coordinates 152
Sec. 24. Basic operations of vector analysis in curvilinear coordinates 156
Sec. 25. The Laplace operator in orthogonal coordinates 174
ANSWERS 177
APPENDIX I 184
APPENDIX II 186
BIBLIOGRAPHY 187
INDEX 188