  This study extends the functional perturbation theory, originally developed
for orbits and periodic orbits in dynamical systems, \textit{e.g.} a map
$\mathcal{P}$, to derive formulae for the deformation of an $d$-dimensional
invariant torus, on which one can define $\mathcal{P}^k$ for $k\in \mathbb{R}$
and even $\mathbf{k}\in\mathbb{R}^d$ to continuize and generalize the
exponential function $\mathcal{P}^{\mathbf{k}}$ in $k$. The invariant tori,
which manifest as closed flux surfaces in magnetic confinement fusion (MCF)
machines, are the dominant structure influencing the radial transport and,
consequently, the overall confinement performance.
