Advanced Diagnostics for the Study of Linearly Polarized Emission. I:
  Derivation
  Linearly polarized emission is described, in general, in terms of the Stokes
parameters $Q$ and $U$, from which the polarization intensity and polarization
angle can be determined. Although the polarization intensity and polarization
angle provide an intuitive description of the polarization, they are affected
by the limitations of interferometric data, such as missing single-dish data in
the u-v plane, from which radio frequency interferometric data is visualized.
To negate the effects of these artefacts, it is desirable for polarization
diagnostics to be rotationally and translationally invariant in the $Q$-$U$
plane. One rotationally and translationally invariant quantity, the
polarization gradient, has been shown to provide a unique view of spatial
variations in the turbulent interstellar medium when applied to diffuse radio
frequency synchrotron emission. In this paper we develop a formalism to derive
additional rotationally and translationally invariant quantities. We present
new diagnostics that can be applied to diffuse or point-like polarized emission
in any waveband, including a generalization of the polarization gradient, the
polarization directional curvature, polarization wavelength derivative, and
polarization wavelength curvature. In Paper II we will apply these diagnostics
to observed and simulated images of diffuse radio frequency synchrotron
emission.
