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\begin{document}
\begin{center}
Homework 6
\end{center}

\noindent
Due Date: April 9, 2013.\\

\noindent
Page 212 of Rosenlicht: 2, 3, 5, 6, 7, 8, 9.\\

\begin{center}
Additional Problems:
\end{center}

\noindent
1. A function $f:\R^n\times\R^m\to\R^p$ is bilinear if for all $x,x_1,x_2\in \R^n$ and $y,y_1,y_2\in \R^m$ and $a \in\R$ we have
\begin{itemize}
\item[(i)] $f(ax,y)=af(x,y)=f(x,ay)$,
\item[(ii)] $f(x_1+x_2,y)=f(x_1,y)+f(x_2,y)$,
\item[(iii)] $f(x,y_1+y_2)=f(x,y_1)+f(x,y_2)$.
\end{itemize}

\begin{itemize}
\item[(a)] Prove that if $f$ is bilinear then
$$
\lim_{(h,k)\to (0,0)}\frac{\abs{f(h,k)}}{\abs{(h,k)}}=0.
$$
\item[(b)] Prove that $Df(a,b)(h,k)=f(a,k)+f(h,b)$.
\end{itemize}



\noindent
2.  Define $IP:\R^n\times\R^n\to\R$ by $IP(x,y)=\left\langle x,y\right\rangle$ (the inner product between the vectors $x$ and $y$).  
\begin{itemize}
\item[(a)] Compute $D(IP)(x,y)$ and $(IP)'(x,y)$.
\item[(b)] If $f$ and $g$ are differentiable maps from $\R\to\R^n$ and $h(t)=\left\langle f(t), g(t)\right\rangle$ show that
$$
h'(t)=\left\langle f'(t)^{\top}, g(t)\right\rangle+\left\langle f(t), g'(t)^{\top}\right\rangle.
$$
where $y^{\top}$ is the transpose of the matrix/vector $y$.
\end{itemize}

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