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\noindent
\parbox{2.25in}{\bf Math 3510}
\hfill {\Large \bf Homework \#1} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Sept. $5^\mathrm{th}$, 2018}

\vspace{0.2in}

%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
%
%\vspace{0.2in}

\begin{description}
\item[\#1 Jacobians!] Consider $f(x,y) = (3y+x^2y+1,-x+y^2-1,e^{xy^2})$.
\begin{enumerate}[(a)]
\item Find the Jacobian of $f$.
\item Give the linear approximation of $f(x,y)$ based at $(x,y)=(0,-2)$
\item Find something that could be called the ``second derivative'' of $f$ (there are several ways to organize such a beast).
\item Give the quadratic approximation of $f(x,y)$ based at $(x,y)=(0,-2)$.
\end{enumerate}

\vspace{0.1in}

\item[\#2 The Second Derivative Test:] Let $f(x,y,z) = -2x^3-6xy^2+6x+z^4-2z^2$. Find and classify the critical points of $f$ using the second derivative test.

{\it Note:} There are 12 critical points -- sorry. Find them and tell me whether each point is a relative minimum, relative maximum, a saddle point, or if the test does not apply.

{\it Strong Suggestion:} Doing this problem by hand is not advised -- maybe not even possible. You should use some software. Here are possibly relevant Maple commands or stubs of commands\dots

{\color{red}
\verb|with(LinearAlgebra):|\\
\verb|with(VectorCalculus):|

\verb|pts := solve({diff(???,?)=0, ... });|

\verb|H := Hessian(???,[x,y,z]);|

\verb|for pt in pts do|\\
\qquad\verb| Eigenvalues(subs(pt,H));|\\
\verb|end do;|
}

\item[\#3 There's got to be an easier way:] Compute $\det \begin{bmatrix} 1 & 0 & 2 \\ 0 & 3 & 1 \\ 2 & 1 & 2 \end{bmatrix}$ {\it only} using the fact that the determinant is multilinear, alternating/skew, and $\det(I_3)=1$.

\vspace{0.1in}

\item[\#4 Forms Basics:] Let $\omega = xy\,dx + x^2z\,dz$, $\nu = xy^2z\,dx-xyz\,dz$, and $\eta = x^3\,dy\wedge dz+xy\,dz \wedge dx+yz\,dx \wedge dy$. 

In addition, let $\B p = (1,2,-1)$, $\B a = \langle 1,2,3 \rangle$, and $\B b = \langle -1,3,0 \rangle$.

{\it Note:} Simplify answers in term of functions and forms: $dx, dy, dz,  dy\wedge dz, dz \wedge dx, dx \wedge dy,$ and $dx \wedge dy \wedge dz$.

\begin{enumerate}[(a)]
\item Evaluate $\omega_{\B p}(\B a)$ and $\eta_{\B p}(\B a,\B b)$. 
\item Compute $\omega \wedge \nu$ and $\omega \wedge \eta$.
\item Compute $\eta \wedge \omega$ and $\omega \wedge \nu \wedge \eta$ \qquad (this part should be very easy).

\end{enumerate}

\end{description}

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