Principles of Electromagnetic Waves and Materials, 2nd Edition

Appendix 1A: Vector Formulas and Coordinate Systems

1A.1Vector Transformations

This appendix closely follows the development of the materials given in [1], which is a standard development used in many textbooks. The three coordinate systems, (a) rectangular, (b) cylindrical, and (c) spherical, are shown in Figure 1A.1.

FIGURE 1A.1(a) Rectangular, (b) cylindrical, and (c) spherical coordinate systems.

1A.1.1Rectangular to Cylindrical (and Cylindrical to Rectangular) Transformation

Referring to Figure 1A.1b, the coordinate transformation from rectangular (x, y, z) to cylindrical (ρ, ϕ, z) coordinates is given by

(1A.1)$$\begin{array}{c}x=\mathrm{\rho }cos\mathrm{\varphi },\hfill \\ y=\mathrm{\rho }sin\mathrm{\varphi },\hfill \end{array}$$