update domain and diff

chapters/chap02/chap02.tex

@@ -7,7 +7,7 @@
 %         summary of the content in this chapter
 % Version:  26.07.2024
 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\chapter{Theoretical background}
+\chapter{Theoretical Background}
 \label{chap:background}
 
 This chapter is set to introduce the theoretical knowledge on which the work in this thesis is founded on. First we talk about domain mathematics and differential equations.
@@ -19,41 +19,45 @@ by M. Tenenbaum and H. Pollard. %\cite{Tenenbaum1985}. %TODO introduce other sec
 \section{Domain Mathematics and Functions}
 \label{sec:domain}
 
-% describe the meaning of a domain and codomain space to its problem and then shivvy to the function that translates from one to another.
-When looking at problems from a mathematical it is possible to describe it by putting it into a system. Then the base   A function $f$ assigns each element $x$ of the set $A$ to an
-element of the set $B$.
-\begin{equation}
-  f: A \rightarrow B
-\end{equation}
-In this case $A$ is called the domain, while $B$ is called the codomain. For further explanations we refer to $f$ as:
+In order to model a mathematical problem, it is necessary to define a set of fundamental numbers or quantities upon which the subsequent calculations will be based. These sets may
+represent a specific time interval or a distance, for instance. The term "domain" is used to describe these fundamental sets of numbers or quantities. For the purpose this thesis
+we will restrict ourselves to domains of real numbers in $\mathbb{R}$.\\
+
+The mapping of a value from one domain to a value from another domain enables the modelling and depiction of semantics. In order to facilitate this mapping, the use of functions is
+employed. The function $f$ is defined as
 \begin{equation}
-  f: \mathbb{R} \rightarrow \mathbb{R}
+  f: A\rightarrow B
 \end{equation}
-Functions are able to describe the condition of a system given certain parameters.
+In other words, $f$ assigns each value $x$ of an interval $A$ to a value $y$ of $B$. $A$ is referred to as the domain and $B$ as the codomain of $f$. In our case, both $A$ and $B$
+are subsets of $\mathbb{R}$. Functions are capable of representing the state of a system as a value based on an input value from their domain. One illustrative example is a function
+that maps a time point to the distance covered since the last time point. In this case, time serves as the domain, while the distance is the codomain.
 
 % -------------------------------------------------------------------
 
 \section{Basics of Differential Equations}
 \label{sec:differentialEq}
 
-In the real world states of system are under constant change. While functions are able to show the state of a system for a certain set of parameters that
-are living in the domain space, they can only indirectly give information about the change of the system under different sets of input. This shows the
-need of a way to retrieve the information of change from a function. One way would be taking the rate of change across a certain interval $[a, b]\subseteq\mathbb{R}$ of
-the domain of a function $f$, by calculating
+Often, the change of a system is more interesting than its current state. Functions are able to give us the latter, but only passively give information about the change of a system.
+Now the problem at hand is a way of calculating the change of a function across its domain. Given a function $f$ and an interval $[a, b]\subset \mathbb{R}$,
 \begin{equation}
   m = \frac{f(b) - f(a)}{a-b}
 \end{equation}
-$m$ is the average rate of change across the interval $[a, b]$. Since in most cases we want to find the rate of change in a specific spot and the average will
-not be sufficient. For this reason instead of looking across the whole interval we single out every $x\in[a, b]$. We narrow the interval down to be infinitesimal small
-and then calculate the average rate of change
+is the average rate of change across $[a, b]$. While the average change of rate is helpful in many cases, the momentary rate of change is more accurate. To find this the interval is
+narrowed down, to be infinitesimal small. For each $x\in[a, b]$ we calculate
 \begin{equation} \label{eqn:differential}
   \frac{df}{dx} = \lim_{t\to x} \frac{f(t) - f(x)}{t-x}
 \end{equation}
-where this value exists. $\frac{df}{dx}$ is the rate of the change or derivative of the function $f$ undergoes in respect of its parameter $x$. This now is able to give information about
-the rate of change for a specific set of parameters. By re-iterating this process it is possible for us to calculate the rate of the rate of change as well, which is called the derivative of the second
-order. Equation \ref{eqn:differential} shows how to theoretically come from the function to its assigned derivative, but in many cases (applications) differential equations are built from
-semantics and logics. For this thesis we would like to concentrate on ordinary differential equations, which have only one input parameter.
-In our case this is $t$ the point of time. \\
+if it exists. $\frac{df}{dx}$ is called the derivative or differential equation, it returns the momentary rate of change of $f$ for each value $x$ of $f$'s
+domain. When repeating this process on $\frac{df}{dx}$ we get $\frac{d^2f}{dx^2}$ which is the function that calculates the rate of change of the rate of change and is called the second
+derivative. Iterating this $n$ times results in $\frac{d^nf}{dx^n}$, the derivative of the $n$'th order. Another way to get a differential equation is to create it from the semantics of
+a problem. This way is often chosen if no basic function exists for a system.\\
+
+For functions, it is possible to have multiple domains, meaning that function has more than one parameter. For example a function working with 2D parameters or a function that, works with
+time and locations as its inputs. These functions need differential equations for each of their domains, which are called "partial differential equations" (PDE's). The ones for functions with one
+single domain are called "ordinary differential equations" (ODE's). For this thesis only ODE's will be of interest.\\
+
+If a system has several codomains, it will have a differential equation for each codomain all sharing the same domain. The collective of these equations is called a differential system.\\
+
 
 For illustrating the functionality of a derivative we will look upon the specific problem. For this we