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\chapter{Weitere Formeln zur trilinearen Interpolation}
\label{CTrilinearAbleitungen}

Die partiellen Ableitungen erster Ordnung von \eqref{ETrilinearInter}
lauten (man beachte den konstanten Faktor):
\begin{multline}\label{ETrilinearInterX}
F_{G_0}^{G_1}(x,y,z)\frac{d}{dx} = \frac{1}{x_{step}} \cdot \biggl(\\
\begin{split}&- \frac{y_1-y}{y_{step}}\cdot \frac{z_1-z}{z_{step}}\cdot V_G\begin{pmatrix}x_0\\y_0\\z_0\end{pmatrix}
 - \frac{y_1-y}{y_{step}}\cdot \frac{z-z_0}{z_{step}}\cdot V_G\begin{pmatrix}x_0\\y_0\\z_1\end{pmatrix}\\
&- \frac{y-y_0}{y_{step}}\cdot \frac{z_1-z}{z_{step}}\cdot V_G\begin{pmatrix}x_0\\y_1\\z_0\end{pmatrix}
 - \frac{y-y_0}{y_{step}}\cdot \frac{z-z_0}{z_{step}}\cdot V_G\begin{pmatrix}x_0\\y_1\\z_1\end{pmatrix}\\
&+ \frac{y_1-y}{y_{step}}\cdot \frac{z_1-z}{z_{step}}\cdot V_G\begin{pmatrix}x_1\\y_0\\z_0\end{pmatrix}
 + \frac{y_1-y}{y_{step}}\cdot \frac{z-z_0}{z_{step}}\cdot V_G\begin{pmatrix}x_1\\y_0\\z_1\end{pmatrix}\\
&+ \frac{y-y_0}{y_{step}}\cdot \frac{z_1-z}{z_{step}}\cdot V_G\begin{pmatrix}x_1\\y_1\\z_0\end{pmatrix}
 + \frac{y-y_0}{y_{step}}\cdot \frac{z-z_0}{z_{step}}\cdot V_G\begin{pmatrix}x_1\\y_1\\z_1\end{pmatrix} \biggr)
\end{split}\end{multline}

%Let both formulas get on this side
\enlargethispage{3\baselineskip}

\begin{multline}\label{ETrilinearInterY}
F_{G_0}^{G_1}(x,y,z)\frac{d}{dy} = \frac{1}{y_{step}} \cdot \biggl(\\
\begin{split}&-\frac{x_1-x}{x_{step}}\cdot \frac{z_1-z}{z_{step}}\cdot V_G\begin{pmatrix}x_0\\y_0\\z_0\end{pmatrix}
 - \frac{x_1-x}{x_{step}}\cdot \frac{z-z_0}{z_{step}}\cdot V_G\begin{pmatrix}x_0\\y_0\\z_1\end{pmatrix}\\
&+ \frac{x_1-x}{x_{step}}\cdot \frac{z_1-z}{z_{step}}\cdot V_G\begin{pmatrix}x_0\\y_1\\z_0\end{pmatrix}
 + \frac{x_1-x}{x_{step}}\cdot \frac{z-z_0}{z_{step}}\cdot V_G\begin{pmatrix}x_0\\y_1\\z_1\end{pmatrix}\\
&- \frac{x-x_0}{x_{step}}\cdot \frac{z_1-z}{z_{step}}\cdot V_G\begin{pmatrix}x_1\\y_0\\z_0\end{pmatrix}
 - \frac{x-x_0}{x_{step}}\cdot \frac{z-z_0}{z_{step}}\cdot V_G\begin{pmatrix}x_1\\y_0\\z_1\end{pmatrix}\\
&+ \frac{x-x_0}{x_{step}}\cdot \frac{z_1-z}{z_{step}}\cdot V_G\begin{pmatrix}x_1\\y_1\\z_0\end{pmatrix}
 + \frac{x-x_0}{x_{step}}\cdot \frac{z-z_0}{z_{step}}\cdot V_G\begin{pmatrix}x_1\\y_1\\z_1\end{pmatrix}
\end{split}\end{multline}


\begin{multline}\label{ETrilinearInterZ}
F_{G_0}^{G_1}(x,y,z)\frac{d}{dz} = \frac{1}{z_{step}} \cdot \biggl(\\
\begin{split}&-\frac{x_1-x}{x_{step}}\cdot \frac{y_1-y}{y_{step}}\cdot V_G\begin{pmatrix}x_0\\y_0\\z_0\end{pmatrix}
 + \frac{x_1-x}{x_{step}}\cdot \frac{y_1-y}{y_{step}}\cdot V_G\begin{pmatrix}x_0\\y_0\\z_1\end{pmatrix}\\
&- \frac{x_1-x}{x_{step}}\cdot \frac{y-y_0}{y_{step}}\cdot V_G\begin{pmatrix}x_0\\y_1\\z_0\end{pmatrix}
 + \frac{x_1-x}{x_{step}}\cdot \frac{y-y_0}{y_{step}}\cdot V_G\begin{pmatrix}x_0\\y_1\\z_1\end{pmatrix}\\
&- \frac{x-x_0}{x_{step}}\cdot \frac{y_1-y}{y_{step}}\cdot V_G\begin{pmatrix}x_1\\y_0\\z_0\end{pmatrix}
 + \frac{x-x_0}{x_{step}}\cdot \frac{y_1-y}{y_{step}}\cdot V_G\begin{pmatrix}x_1\\y_0\\z_1\end{pmatrix}\\
&- \frac{x-x_0}{x_{step}}\cdot \frac{y-y_0}{y_{step}}\cdot V_G\begin{pmatrix}x_1\\y_1\\z_0\end{pmatrix}
 + \frac{x-x_0}{x_{step}}\cdot \frac{y-y_0}{y_{step}}\cdot V_G\begin{pmatrix}x_1\\y_1\\z_1\end{pmatrix}
\end{split}\end{multline}


Die partiellen Ableitungen zweiter Ordnung von \eqref{ETrilinearInter}
lauten (man beachte die konstanten Faktoren):

\begin{equation}\label{ETrilinearInterXXYYZZ}
F_{G_0}^{G_1}(x,y,z)\frac{d}{d^2x} \; = \; F_{G_0}^{G_1}(x,y,z)\frac{d}{d^2y} \; = \; F_{G_0}^{G_1}(x,y,z)\frac{d}{d^2z} \; = \; \vec 0
\end{equation}


\begin{multline}\label{ETrilinearInterXY}
F_{G_0}^{G_1}(x,y,z)\frac{d}{dx\,dy} = \frac{1}{x_{step}} \cdot \frac{1}{y_{step}} \cdot \biggl(\\
\begin{split}&\frac{z_1-z}{z_{step}}\cdot V_G\begin{pmatrix}x_0\\y_0\\z_0\end{pmatrix}
 + \frac{z-z_0}{z_{step}}\cdot V_G\begin{pmatrix}x_0\\y_0\\z_1\end{pmatrix}\\
 -&\frac{z_1-z}{z_{step}}\cdot V_G\begin{pmatrix}x_0\\y_1\\z_0\end{pmatrix}
 - \frac{z-z_0}{z_{step}}\cdot V_G\begin{pmatrix}x_0\\y_1\\z_1\end{pmatrix}\\
 -&\frac{z_1-z}{z_{step}}\cdot V_G\begin{pmatrix}x_1\\y_0\\z_0\end{pmatrix}
 - \frac{z-z_0}{z_{step}}\cdot V_G\begin{pmatrix}x_1\\y_0\\z_1\end{pmatrix}\\
 +&\frac{z_1-z}{z_{step}}\cdot V_G\begin{pmatrix}x_1\\y_1\\z_0\end{pmatrix}
 + \frac{z-z_0}{z_{step}}\cdot V_G\begin{pmatrix}x_1\\y_1\\z_1\end{pmatrix} \biggr)
\end{split}\end{multline}


\begin{multline}\label{ETrilinearInterXZ}
F_{G_0}^{G_1}(x,y,z)\frac{d}{dx\,dz} = \frac{1}{x_{step}} \cdot \frac{1}{z_{step}} \cdot \biggl(\\
\begin{split}&\frac{y_1-y}{y_{step}}\cdot V_G\begin{pmatrix}x_0\\y_0\\z_0\end{pmatrix}
 - \frac{y_1-y}{y_{step}}\cdot V_G\begin{pmatrix}x_0\\y_0\\z_1\end{pmatrix}\\
 +&\frac{y-y_0}{y_{step}}\cdot V_G\begin{pmatrix}x_0\\y_1\\z_0\end{pmatrix}
 - \frac{y-y_0}{y_{step}}\cdot V_G\begin{pmatrix}x_0\\y_1\\z_1\end{pmatrix}\\
 -&\frac{y_1-y}{y_{step}}\cdot V_G\begin{pmatrix}x_1\\y_0\\z_0\end{pmatrix}
 + \frac{y_1-y}{y_{step}}\cdot V_G\begin{pmatrix}x_1\\y_0\\z_1\end{pmatrix}\\
 -&\frac{y-y_0}{y_{step}}\cdot V_G\begin{pmatrix}x_1\\y_1\\z_0\end{pmatrix}
 + \frac{y-y_0}{y_{step}}\cdot V_G\begin{pmatrix}x_1\\y_1\\z_1\end{pmatrix} \biggr)
\end{split}\end{multline}


\begin{multline}\label{ETrilinearInterYZ}
F_{G_0}^{G_1}(x,y,z)\frac{d}{dy\,dz} = \frac{1}{y_{step}} \cdot \frac{1}{z_{step}} \cdot \biggl(\\
\begin{split}&\frac{x_1-x}{x_{step}}\cdot V_G\begin{pmatrix}x_0\\y_0\\z_0\end{pmatrix}
 - \frac{x_1-x}{x_{step}}\cdot V_G\begin{pmatrix}x_0\\y_0\\z_1\end{pmatrix}\\
 -&\frac{x_1-x}{x_{step}}\cdot V_G\begin{pmatrix}x_0\\y_1\\z_0\end{pmatrix}
 + \frac{x_1-x}{x_{step}}\cdot V_G\begin{pmatrix}x_0\\y_1\\z_1\end{pmatrix}\\
 +&\frac{x-x_0}{x_{step}}\cdot V_G\begin{pmatrix}x_1\\y_0\\z_0\end{pmatrix}
 - \frac{x-x_0}{x_{step}}\cdot V_G\begin{pmatrix}x_1\\y_0\\z_1\end{pmatrix}\\
 -&\frac{x-x_0}{x_{step}}\cdot V_G\begin{pmatrix}x_1\\y_1\\z_0\end{pmatrix}
 + \frac{x-x_0}{x_{step}}\cdot V_G\begin{pmatrix}x_1\\y_1\\z_1\end{pmatrix}
\end{split}\end{multline}


Aufgrund von \eqref{ETrilinearInterXXYYZZ} vereinfacht sich
die Formel \eqref{EAbleitungTangentenkurveAllgemein3} über
den 3. Ableitungsvektor einer Tangentenkurve für stückweise trilinear
interpolierte Vektorfelder wie folgt:\footnote{Der Übersichtlichkeit halber
sei $F_{G_0}^{G_1}(x,y,z) = F = (f^u,f^v,f^w)^T$.}

\begin{equation}\begin{split}
\dddot{L} & = f^u\cdot (f^u_x\cdot F_x + f^v_x\cdot F_y + f^v\cdot F_{xy} + f^w_x\cdot F_z + f^w\cdot F_{xz})\\
& + f^v\cdot (f^u_y\cdot F_x + f^u\cdot F_{xy} + f^v_y\cdot F_y + f^w_y\cdot F_z + f^w\cdot F_{yz})\\
& + f^w\cdot (f^u_z\cdot F_x + f^u\cdot F_{xz} + f^v_z\cdot F_y + f^v\cdot F_{yz} + f^w_z\cdot F_z)
\end{split}\end{equation}

