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\begin{enumerate}
\item Show that the composition 
\[
T^{\mathrm{phys}}_pM\to T^{\mathrm{geom}}_pM\to 
T^{\mathrm{alg}}_pM \to T^{\mathrm{phys}}_pM
\]
is the identity.

{\sc solution:}

\item  Consider the projective plane $\proj^2$ with homogeneous
  coordinates $(x,y,z)$, and let $p=(1,1,1)\in\proj^2$.  Define
\[
f(x,y,z)=\frac{x}{y}.
\]
\begin{enumerate}
\item Show that $f$ is a well-defined function in a neighborhood of
  the point~$p$.

{\sc solution:}

\item Consider the curve in $\alpha(t)=(1+t,1+t^2,1+t^3)\in\proj^2$ for $t$ in a
  small open interval about $0$.  The curve $\alpha$ determines a derivation, $v_{\alpha}$, of
  germs at $p$.  What is $v_{\alpha}(f)$?

{\sc solution:}

\item\label{Ux} Consider the standard chart $(U_x,\phi_x)$ at $p$,
  i.e., $U_x=\{(x,y,z)\in\proj^2\mid x\neq 0\}$ with $\phi_x(x,y,z)=(y/x,z/x)$. Let $(u,v)$ denote the coordinates on $\R^2$ here.  Fixing this chart gives a basis for
  $T_p\proj^2$ of the form 
\[
\left(\frac{\partial}{\partial u}\right)_p, \left(\frac{\partial}{\partial v}\right)_p. 
\]
What is the tangent vector determined by $\alpha$ in terms of these
coordinates?

{\sc solution:}

\item\label{U_y} Repeat the previous exercise, (\ref{Ux}), with respect to the chart
  $(U_y,\phi_y)$.

{\sc solution:}

\item Show that your solution to (\ref{Ux}) is sent to your solution
  to (\ref{U_y}) by the derivative of the change of basis mapping
  $\phi_y\circ\phi_x^{-1}$. 

{\sc solution:}

\end{enumerate}
\item {}[See exercise 2.3 in our text.]  Consider the manifold
  $M=(0,\infty)\subset\R$.  Let $f\colon M\to\R$ be a differentiable
  mapping, and let $p\in M$.  Let $\mathcal{D}_p(M)$ be the
  collection of charts at~$p$.  The gradient relative to each chart
  gives a mapping
  \begin{align*}
\mathcal{D}_p(M)&\to\R\\
(U,h)&\mapsto\nabla (f\circ h^{-1})(h(p))=\frac{d}{dx}(f\circ h^{-1})(h(p)).
\end{align*}
Show that this mapping does not, in general, give a tangent vector in
  $T^{\mathrm{phys}}_pM$. [Explicitly choose a point $p$, two charts
  at $p$, and a function $f$ to illustrate your point.]

{\sc solution:}

\item 
Is the cross-product mapping
\begin{align*}
\omega\colon\R^3\times\R^3&\to\R^3\\
(u,v)&\mapsto u\times v
\end{align*}
multilinear and alternating?  Explain.

{\sc solution:}

\end{enumerate}
\end{document}
