\chapter{Notations}
\label{chap:dg-notations}

Differential geometry is the study of things invariant under change of notation. 

\begin{tabular}{lll}
    \textbf{Notation} & \textbf{Description} & \textbf{Source} \\
    \hline
    $\mathcal{H}(E;F)$, $\mathcal{R}(E;F)$ & Space of derivatives; space of remainders in an $\mathcal{HR}$-system. & \autoref{definition:derivative-system} \\
    $D_{\mathcal{HR}} f(x_0)$ & $\mathcal{HR}$-derivative of $f$ at $x_0$. & \autoref{definition:space-differentiability} \\
    $\mathcal{R}_\sigma^n(E; F)$, $\mathcal{R}_\sigma(E;F)$ & $\sigma$-small functions of order $n$; order 1. & \autoref{definition:differentiation-small} \\
    $D_\sigma f(x_0)$ & $\sigma$-derivative of $f$ at $x_0$. & \autoref{definition:derivative-sets} \\
    $D_\sigma^n f$ & $n$-fold $\sigma$-derivative. & \autoref{definition:n-differentiable-sets} \\
    $D_\sigma^n(U; F)$ & $n$-fold $\sigma$-differentiable functions. & \autoref{definition:differentiable-space} \\
    $L^{(n)}_\sigma(E; F)$ & Codomain of derivatives. $L^{(0)}_\sigma(E; F) = F$, $L^{(n)}_\sigma(E; F) = L(E; L_\sigma^{(n-1)}(E; F))$, equipped with the $\sigma$-uniform topology. & \autoref{definition:higher-derivatives-codomain} \\
    $x^{(k)}$ & Tuple of $x$ repeated $k$ times. & \autoref{theorem:taylor-peano} \\
    $D^+f(x)$ & Right derivative of $f$ at $x$. & \autoref{definition:right-differentiable-mvt}
\end{tabular}
