\part{Fundamentals}
\chapter{Differential rings}

\begin{definition}[Derivation]
A \emph{derivation} is a function $\partial : R \to R$ acting on the ring with the following properties.
\[ \partial (ar+bs) = a\partial(r)+b\partial(s) \]
\[ \partial (rs) = \partial(r)s+r\partial(s) \]
\end{definition}


\begin{definition}[Differential ring]
- Differential ring
- Ordinary differential rings
- Partial differential rings
\end{definition}

\section{Algebra of dual numbers}

\[ \varepsilon^2 = 0 \]

\section{Differential field}
\section{Differential ideal}
\section{Liouville's theorem (differential algebra)}

There is an intuitive sense that integration is 'harder' than integration in that the integral of some elementary functions can turn out to be nonelementary; differential algebra gives a way to formalize how this is so. Liouville proved that differentiation is closed under Liouvillian functions (his definition of an elementary function), while integration is not.
- Liouvillian function
- Liouville's theorem 
\section{Locally nilpotent derivation}

\section{Differential Galois theory}