Chapter 10 - Several important theorems and formulas

Abstract

This chapter reviews several important theorems and formulas which form the bases for the derivation of various equations. Gauss's theorem, Green's theorem, and Stokes' theorem are foundations of electromagnetic theory. The integral theorem of Helmholtz and Kirchhoff and the Fresnel-Kirchhoff diffraction formula are basic theories for solving diffraction problems. The representation of the surface integral on the upper and lower surface is called Gauss's theorem. It states that the summation of the divergence of a vector in a volume space is equal to the sum of the outward normal components on the surface enclosing the space. Using the Helmholtz and Kirchhoff theorem of integration, the amplitude of light at an arbitrary observation point can be obtained by knowing the field distribution of light on the surface enclosing the observation point. The integration theorem of Helmholtz and Kirchhoff is used to find the diffraction pattern of an aperture when illuminated by a point source and projected onto a screen. An important modification of the Fresnel-Kirchhoff diffraction formula implies that if the amplitude distribution of the light across the aperture is known, then the field at the point of observation can be obtained. The chapter also summarizes the formulas in cylindrical and spherical coordinates.