corr no 2

chapters/chap02/chap02.tex

@@ -12,7 +12,7 @@
 
 This chapter introduces the theoretical foundations for the work presented in
 this thesis. In~\Cref{sec:domain} and~\Cref{sec:differentialEq}, we describe
-differential equations and the underlying theory. In these Sections both the
+differential equations and the underlying theory. In these Sections, both the
 explanations and the approach are based on a book on analysis by
 Rudin~\cite{Rudin2007} and a book about ordinary differential equations by
 Tenenbaum and Pollard~\cite{Tenenbaum1985}. Subsequently, we employ this
@@ -36,7 +36,7 @@ numbers in $\mathbb{R}$.\\
 
 The mapping between variables enables the modeling of a physical process and may
 depict semantics. We use functions in order to facilitate this mapping. Let
-$A, B\subset\mathbb{R}$ be to subsets of the real numbers, then we define a
+$A, B\subset\mathbb{R}$ be two subsets of the real numbers, then we define a
 function as the mapping
 \begin{equation}
   f: A\rightarrow B.
@@ -55,24 +55,24 @@ In this case, time serves as the domain, while the distance is the codomain.
 
 Often, the behavior of a variable or a quantity across a domain is more
 interesting than its current state. Functions are able to give us the latter,
-but do not contain information about the change of a system. The objective
+but do not contain any information about the change of a system. The objective
 is to determine an effective method for calculating the change of a function
 across its domain. Let $f$ be a function and $[a, b]\subset \mathbb{R}$ an
 interval of real numbers. The expression
 \begin{equation}
-  m = \frac{f(b) - f(a)}{a-b}
+  m = \frac{f(b) - f(a)}{b-a}
 \end{equation}
 gives the average rate of change. While the average rate of change is useful in
 many cases, the momentary rate of change is more accurate. To calculate the
-momentary rate of change at $x$, we let the value $t$ approach $x$ thereby
-narrowing down the interval to an infinitesimal. For each $x\in[a, b]$ we
+momentary rate of change at $x$, we let the value $t$ approach $x$ and thereby
+narrow down the interval to an infinitesimal. 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}
 if it exists. As the Tenenbaum and Pollard~\cite{Tenenbaum1985} define,
 $\nicefrac{df}{dx}$ is the \emph{derivative}, which is ``the rate of change of a
-variable with respect to another''. The relation between a variable and its
+variable with respect to another''~\cite{Tenenbaum1985}. The relation between a variable and its
 derivative is modeled in a \emph{differential equation}. The derivative of
 $\nicefrac{df}{dx}$ yields $\nicefrac{d^2f}{dx^2}$, which is the function that
 calculates the rate of change of the rate of change and is called the
@@ -97,17 +97,17 @@ Another example would be a function, that maps its inputs of a location variable
 and a time variable on a height. The term \emph{partial differential equations}
 (PDE) describes differential equations of such functions, which contain
 partial derivatives with respect to each individual domain. In contrast,
-\emph{ordinary differential equations} (ODE) are the single derivatives for a
+\emph{ordinary differential equations} (ODE) contain derivatives for a
 function having only one domain~\cite{Tenenbaum1985}. In this thesis, we
 restrict ourselves to ODE's. Furthermore, a
 \emph{system of differential equations} is the name for a set of differential
-equations. The derivatives in a system of differential equations each have their
+equations. The equations in a system of differential equations each have their
 own codomain, which is part of the problem, while they all share the same
 domain.\\
 
 Tenenbaum and Pollard~\cite{Tenenbaum1985} provide many examples for ODE's,
-including the \emph{Motion of a Particle Along a Straight Line}. Further,
-Newton's second law states that ``the rate of change of the momentum of a body
+including the \emph{Motion of a Particle Along a Straight Line}. This example is based on
+Newton's second law which states that ``the rate of change of the momentum of a body
 ($momentum = mass \cdot velocity$) is proportional to the resultant external
 force $F$ acted upon it''~\cite{Tenenbaum1985}. Let $m$ be the mass of the body
 in kilograms, $v$ its velocity in meters per second and $t$ the time in seconds.