  The Spatial Parallel Diffusion Coefficient (SPDC) is one of the important
quantities describing energetic charged particle transport. There are three
different definitions for the SPDC, i.e., the Displacement Variance definition
$\kappa_{zz}^{DV}=\lim_{t\rightarrow t_{\infty}}d\sigma^2/(2dt)$, the Fick's
Law definition $\kappa_{zz}^{FL}=J/X$ with $X=\partial{F}/\partial{z}$, and the
TGK formula definition $\kappa_{zz}^{TGK}=\int_0^{\infty}dt \langle
v_z(t)v_z(0) \rangle$. For constant mean magnetic field, the three different
definitions of the SPDC give the same result. However, for focusing field it is
demonstrated that the results of the different definitions are not the same. In
this paper, from the Fokker-Planck equation we find that different methods,
e.g., the general Fourier expansion and perturbation theory, can give the
different Equations of the Isotropic Distribution Function (EIDFs). But it is
shown that one EIDF can be transformed into another by some Derivative
Iterative Operations (DIOs). If one definition of the SPDC is invariant for the
DIOs, it is clear that the definition is also an invariance for different
EIDFs, therewith it is an invariant quantity for the different Derivation
Methods of EIDF (DMEs). For the focusing field we suggest that the TGK
definition $\kappa_{zz}^{TGK}$ is only the approximate formula, and the Fick's
Law definition $\kappa_{zz}^{FL}$ is not invariant to some DIOs. However, at
least for the special condition, in this paper we show that the definition
$\kappa_{zz}^{DV}$ is the invariant quantity to the kinds of the DIOs.
Therefore, for spatially varying field the displacement variance definition
$\kappa_{zz}^{DV}$, rather than the Fick's law definition $\kappa_{zz}^{FL}$
and TGK formula definition $\kappa_{zz}^{TGK}$, is the most appropriate
definition of the SPDCs.
