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