Appendix a Orthogonal Curvilinear Coordinates
Appendix A Orthogonal Curvilinear Coordinates Given a symmetry for a system under study, the calculations can be simplified by choosing, instead of a Cartesian coordinate system, another set of coordinates which takes advantage of that symmetry. For example calculations in spherical coordinates result easier for systems with spherical symmetry. In this chapter we will write the general form of the differential operators used in electrodynamics and then give their expressions in spherical and cylindrical coordi- nates.1 A.1 Orthogonal curvilinear coordinates A system of coordinates u1, u2, u3, can be defined so that the Cartesian coordinates x, y and z are known functions of the new coordinates: x = x(u1, u2, u3) y = y(u1, u2, u3) z = z(u1, u2, u3) (A.1) Systems of orthogonal curvilinear coordinates are defined as systems for which locally, nearby each point P(u1, u2, u3), the surfaces u1 = const, u2 = const, u3 = const are mutually orthogonal. An elementary cube, bounded by the surfaces u1 = const, u2 = const, u3 = const, as shown in Fig. A.1, will have its edges with lengths h1du1, h2du2, h3du3 where h1, h2, h3 are in general functions of u1, u2, u3. The length ds of the line- element OG, one diagonal of the cube, in Cartesian coordinates is: ds = (dx)2 + (dy)2 + (dz)2 1The orthogonal coordinates are presented with more details in J.A. Stratton, Electromagnetic The- ory, McGraw-Hill, 1941, where many other useful coordinate systems (elliptic, parabolic, bipolar, spheroidal, paraboloidal, ellipsoidal) are given. © Springer International Publishing Switzerland 2016 189 F. Lacava, Classical Electrodynamics, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-3-319-39474-9 190 Appendix A: Orthogonal Curvilinear Coordinates Fig.
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