Mathematical Physics Vol 1

Field theory

II

77

Chapter 4 Field theory 79 4.1 Scalarfield .....................................7.9 4.1.1 Directional derivative. Gradient . . . . . . . . . . . . . . . . . . . . . . 8. 0 4.1.2 Partial gradient of a scalar function . . . . . . . . . . . . . . . . . . . . 8. 6 4.1.3 Propertiesofgradient............................8.6 4.1.4 Nabla operator or Hamilton operator . . . . . . . . . . . . . . . . . . . . 8. 7 4.1.5 Laplaceordeltaoperator..........................8.8 4.2 Vectorfield .....................................8.9 4.2.1 Vectorfunction.Vectorfield . . . . . . . . . . . . . . . . . . . . . . . . 8.9 4.2.2 Divergenceandrotor.............................91 4.2.3 Classificationofvectorfields . . . . . . . . . . . . . . . . . . . . . . . . 9.2 4.2.4 Potential ..................................9.3 4.2.5 Examplesofpotential ...........................9.4 4.2.6 A brief overview of introduced concepts . . . . . . . . . . . . . . . . . . 9. 8 4.2.7 Spatialderivation..............................9.9 4.2.8 Integraltheorems..............................10.2 4.3 Examplesofsomefields ..............................10.3 4.4 Generalizedcoordinates...............................10.8 4.4.1 Arcandvolumeelements. . . . . . . . . . . . . . . . . . . . . . . . . .1.11 4.4.2 Gradient, divergence, rotor and Laplacian - expressed by generalized coordinates .................................11.3 4.5 Specialcoordinatesystems .............................11.4 4.6 Examples ......................................11.9 4.6.1 Gradient...................................11.9 4.6.2 Divergence .................................12.6 4.6.3 Rotor ....................................13.3 4.6.4 Mixedproblems ..............................1.41 4.6.5 Invariant ..................................14.9 4.6.6 Integrals, integral theorems . . . . . . . . . . . . . . . . . . . . . . . . .15. 4 4.6.7 Variousexamples..............................18.7 4.6.8 Generalised orthogonal systems . . . . . . . . . . . . . . . . . . . . . .1.91 4.6.9 Gradient, divergence and rotor in generalized orthogonal coordinates . .20. 3 4.6.10 Surfaces in terms of orthogonal generalized coordinates . . . . . . . . .20. 9 4.6.11 Generalizedsystems ............................21.0 4.6.12 Variousproblems..............................21.5 221 Chapter 5 Series Solutions of Differential Equations. Special functions 223 5.1 Functionalseries.Powerseries . . . . . . . . . . . . . . . . . . . . . . . . . . .22.3 5.2 Series Solutions of Differential Equations . . . . . . . . . . . . . . . . . . . . .22. 7 5.2.1 Solutions of Differential Equations using Power Series . . . . . . . . . .22. 7 5.3 Legendre: equation, function, polynomial . . . . . . . . . . . . . . . . . . . . .22. 8 5.4 Bessel equation. Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . .2.31 III Solving differential equations