Avoid confusing reuse of 'x' in spatialDistribution description

chapters/operatorsandexpressions.tex

@@ -959,32 +959,35 @@ \subsubsection{spatialDistribution}\label{spatialdistribution}
 \begin{nonnormative}
 Many applications involve the modelling of variable-speed transport of properties.
 One option to model this infinite-dimensional system is to approximate it by an ODE, but this requires a large number of state variables and might introduce either numerical diffusion or numerical oscillations.
-Another option is to use a built-in operator that keeps track of the spatial distribution of $z(x, t)$, by suitable sampling, interpolation, and shifting of the stored distribution.
+Another option is to use a built-in operator that keeps track of the spatial distribution of $z(\xi, t)$, by suitable sampling, interpolation, and shifting of the stored distribution.
 In this case, the internal state of the operator is hidden from the ODE solver.
 \end{nonnormative}
 
 \lstinline!spatialDistribution! allows the infinite-dimensional problem below to be solved efficiently with good accuracy
 \begin{align*}
-\frac{\partial z(x,t)}{\partial t}+v(t)\frac{\partial z(x,t)}{\partial x} &= 0.0\\
-z(0.0, t) &= \mathrm{in}_0(t) \text{ if $v\geq 0$}\\
-z(1.0, t) &= \mathrm{in}_1(t) \text{ if $v<0$}
+\frac{\partial z(\xi, t)}{\partial t} + v(t) \frac{\partial z(\xi, t)}{\partial x} &= 0\\
+z(0, t) &= \mathrm{in}_{0}(t) \quad \text{if $v \geq 0$}\\
+z(1, t) &= \mathrm{in}_{1}(t) \quad \text{if $v < 0$}
 \end{align*}
-where $z(x, t)$ is the transported quantity, $x$ is the normalized spatial coordinate ($0.0 \le x \le 1.0$), $t$ is the time, $v(t)=\mathrm{der}(x)$ is the normalized transport velocity and the boundary conditions are set at either $x=0.0$ or $x=1.0$, depending on the sign of the velocity.
+where $z(\xi, t)$ is the transported quantity, $\xi$ is the normalized spatial coordinate ($0 \le \xi \le 1$), $t$ is the time, $v(t) = \xi'(t)$ is the normalized transport velocity and the boundary conditions are set at either $\xi = 0$ or $\xi = 1$, depending on the sign of the velocity.
+
 The calling syntax is:
 \begin{lstlisting}[language=modelica]
 (out0, out1) = spatialDistribution(in0, in1, x, positiveVelocity,
                                    initialPoints = {0.0, 1.0},
                                    initialValues = {0.0, 0.0});
 \end{lstlisting}
-where \lstinline!in0!, \lstinline!in1!, \lstinline!out0!, \lstinline!out1!, and \lstinline!x! are all subtypes of \lstinline!Real!, \lstinline!positiveVelocity! is a \lstinline!Boolean!, \lstinline!initialPoints! and \lstinline!initialValues! are arrays of subtypes of \lstinline!Real! of equal size, containing the x coordinates and the $z$ values of a finite set of points describing the initial distribution of $z(x, \mathit{t0})$.
-The \lstinline!out0! and \lstinline!out1! are given by the solutions at $z(0.0, t)$ and $z(1.0, t)$; and \lstinline!in0! and \lstinline!in1! are the boundary conditions at $z(0.0, t)$ and $z(1.0, t)$ (at each point in time only one of \lstinline!in0! and \lstinline!in1! is used).
+where \lstinline!in0!, \lstinline!in1!, \lstinline!out0!, \lstinline!out1!, and \lstinline!x! are all subtypes of \lstinline!Real!, \lstinline!positiveVelocity! is a \lstinline!Boolean!, and \lstinline!initialPoints! and \lstinline!initialValues! are arrays of subtypes of \lstinline!Real!.
+The position \lstinline!x! is the integral of the transport velocity $v$, where the constant of integration does not matter.
+The arrays \lstinline!initialPoints! and \lstinline!initialValues! shall be parameter expressions of equal size, containing the $\xi$ coordinates and the $z$ values of a finite set of points describing the initial distribution of $z(\xi, t_{0})$.
+The \lstinline!out0! and \lstinline!out1! are given by the solutions at $z(0, t)$ and $z(1, t)$; and \lstinline!in0! and \lstinline!in1! are the boundary conditions at $z(0, t)$ and $z(1, t)$ (at each point in time only one of \lstinline!in0! and \lstinline!in1! is used).
 Elements in the \lstinline!initialPoints! array must be sorted in non-descending order.
 The operator can not be vectorized according to the vectorization rules described in \cref{scalar-functions-applied-to-array-arguments}.
 The operator can be vectorized only with respect to the arguments \lstinline!in0! and \lstinline!in1! (which must have the same size), returning vectorized outputs \lstinline!out0! and \lstinline!out1! of the same size; the arguments \lstinline!initialPoints! and \lstinline!initialValues! are vectorized accordingly.
 
 The solution, $z$, can be described in terms of characteristics:
 \begin{equation*}
-z(x+\int_{t}^{t+\beta} v(\alpha) \mathrm{d}\alpha, t+\beta) = z(x, t),\quad\text{for all $\beta$ as long as staying inside the domain}
+z(\xi+\int_{t}^{t+\beta}\! v(\alpha) \mathrm{d}\alpha,\, t+\beta) = z(\xi, t), \quad\text{for all $\beta$ as long as staying inside the domain}
 \end{equation*}
 
 This allows the direct computation of the solution based on interpolating the boundary conditions.