Power flow of the guided mode of a waveguide

The integral $\int_\Omega\Phi\Psi\,d\Omega$ taken over an arbitrary plane region $\Omega$ where the scalar functions of the point $\Phi$ and $\Psi$ are the solutions of the Helmholtz two-dimensional equation is presented as a contour, i.e., in the invariant view and in three main orthogonal coordinate systems on a plane, namely, in the Cartesian, polar, and elliptic coordinate systems. An invariant expression in the view of the contour integral for power flow of the guided mode through an arbitrary region of the cross section of a waveguide with constant permittivity has been obtained.