\part{Fundamentals}

Mathematical logic
Set theory
Order theory
Real analysis
Ordinary differetial equations
Linear algebra
General topology
Vector analysis 
Fourier analysis



\chapter{Introduction to partial differential equations}

Real analysis allows us to pose ODEs, while vector analysis open the possibility for \emph{partial differential equations} (PDEs); these differential equations are with respect to partial derivatives.

\begin{definition}[Partial differential equation (PDE)]
A \emph{partial differential equation (PDE)} is an equation of the following form, defined by some $f : S \subseteq \mathbb{R}^n \times \mathbb{R}^m$ and $D^k \mathbf{y}$ representing the matrix of possible partial derivatives of order $k$.
\[ f(\mathbf{x},y(\mathbf{x}), D y(\mathbf{x}) , \hdots , D^m y(\mathbf{x})) = 0\]
Solutions for $y : U \to \mathbb{R}$ are (usually, depending on whether all variables can be differentiated $m$ times) in $C^{m}(U)$, for some set $U$ open in (Euclidean topology) $\mathbb{R}^m$.
\end{definition}

The fundamental part of this book will discuss elementary techniques for solving PDEs, generally finding ways to map PDEs to a system of ODEs.

\section{Classification of PDEs}

Linear and honogeneous PDEs
\subsection{Conic sections}

An 

Hyperbolic
Parabolic
Elliptic
\[a f_{xx} + 2b f_{xy} + c f_{yy}\]

The 2nd order linear PDE with constant coefficients is elliptic if $b^2-ac <0$, parabolic if $b^2-ac =0$, and hyperbolic if $b^2-ac >0$.

\subsection{Boundary conditions}

Cauchy
Neumann
Dirichlet
\subsection{Well-posed problems}




\section{Qualitative theorems}
\subsection{Maximum principle}
\subsection{Harnack's theorem}





\section{Laplace equation}
Harmonic function
Importance in complex analysis


\subsection{Maximum principle for harmonic functions}
Extremely important qualitative result on harmonic functions and Laplace equation that have profound implications across mathematics and science.

This allows p
Dirichlet problems on Laplace equation are unique.