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Use 'aligned' instead of 'eqnarray*' to match style in other places
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henrikt-ma committed Mar 3, 2021
1 parent 4bb4a3b commit 7339307
Showing 1 changed file with 18 additions and 12 deletions.
30 changes: 18 additions & 12 deletions chapters/functions.tex
Expand Up @@ -1440,12 +1440,15 @@ \subsection{Using the Derivative Annotation}\label{using-the-derivative-annotati
end fGeneralDer;
\end{lstlisting}
In the example above \lstinline!zeroDerivative=y! and \lstinline!zeroDerivative=offset! imply that
\begin{eqnarray*}
\frac{d}{dt}f(x(t),y(t),o(t))&=&\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}+\frac{\partial f}{\partial o}\frac{do}{dt}\\
&=&\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\cdot 0+\frac{\partial f}{\partial o}\cdot 0\\
&=&\frac{\partial f}{\partial x}\frac{dx}{dt}\\
&=&fDer\cdot\frac{dx}{dt}
\end{eqnarray*}
\begin{equation*}
\begin{aligned}
\frac{d}{dt}f(x(t),y(t),o(t))
&= \frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}+\frac{\partial f}{\partial o}\frac{do}{dt}\\
&= \frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\cdot 0+\frac{\partial f}{\partial o}\cdot 0\\
&= \frac{\partial f}{\partial x}\frac{dx}{dt}\\
&= fDer\cdot\frac{dx}{dt}
\end{aligned}
\end{equation*}
\end{nonnormative}

\begin{itemize}
Expand Down Expand Up @@ -1493,12 +1496,15 @@ \subsection{Using the Derivative Annotation}\label{using-the-derivative-annotati
effort of \lstinline!fg!.

Therefore \lstinline!h! indirectly includes the derivative with respect to \lstinline!y! as follows:
\begin{eqnarray*}
\frac{d}{dt}fg(x(t))&=&\frac{d}{dt}f(x(t),g(x(t)))\\
&=&\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}\frac{dx}{dt}\\
&=&\left(\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}\right)\frac{dx}{dt}\\
&=&h(x,y)\frac{dx}{dt}
\end{eqnarray*}
\begin{equation*}
\begin{aligned}
\frac{d}{dt}fg(x(t))
&= \frac{d}{dt}f(x(t),g(x(t)))\\
&= \frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}\frac{dx}{dt}\\
&= \left(\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}\frac{\partial g}{\partial x}\right)\frac{dx}{dt}\\
&= h(x,y)\frac{dx}{dt}
\end{aligned}
\end{equation*}
\end{nonnormative}

\subsection{Partial Derivatives of Functions}\label{partial-derivatives-of-functions}
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