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\def\red{\color{red}}

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\begin{document}
\noindent{\small
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#1617}{2016-17}:
  \href{http://drorbn.net/?title=1617-257}{Math 257 Analysis II}
  \hfill\url{http://drorbn.net/?title=1617-257}
}

\begin{center}{\bf Term Test 3 --- Problem 4 Marking Key}\end{center}

\noindent{\bf Problem 4. } Consider the forms $\omega = xydx+3dy-yzdz$ and $\eta=xdx-yz^2dy+2xdz$ on $\bbR^3_{xyz}$. Verify by direct computations that $d(d\omega)=0$ and that $d(\omega\wedge\eta)=(d\omega)\wedge\eta - \omega\wedge d\eta$.

\noindent{\bf Solution. }
\begin{multline} d\omega
  = d(xy)\wedge dx + d(3)\wedge dy -d(yz)\wedge dz
  = (ydx+xdy)\wedge dx + 0 - (zdy+ydz)\wedge dz \\
  = x dy\wedge dx - z dy\wedge dz
  = {\red -x dx\wedge dy - z dy\wedge dz}.
\end{multline}
\begin{multline} d(d\omega)
  = d(-x dx\wedge dy - z dy\wedge dz) = 0+0 \\
  {\red = 0}.
\end{multline}
\begin{multline} \omega\wedge\eta = (xydx+3dy-yzdz)\wedge(xdx-yz^2dy+2xdz) \\
  = (-xyyz^2-3x)dx\wedge dy + (3\cdot 2x -yzyz^2)dy\wedge dz + (-yzx-xy2x)dz\wedge dx \\
  = (-xy^2z^2-3x)dx\wedge dy + (6x-y^2z^3)dy\wedge dz + (-xyz-2x^2y)dz\wedge dx
\end{multline}
\vskip -9mm
\null\hfill{\red 6 terms, rough form}
\begin{multline} \label{lhs} d(\omega\wedge\eta)
  =d((-xy^2z^2-3x)dx\wedge dy + (6x-y^2z^3)dy\wedge dz + (-xyz-2x^2y)dz\wedge dx) \\
  = ({\red -2xy^2z}+6-xz-2x^2)dx\wedge dy\wedge dz
\end{multline}
\vskip -9mm
\null\hfill{\red 4 terms, degree 3}
\begin{multline} \label{rhs1} (d\omega)\wedge\eta
  = (-x dx\wedge dy - z dy\wedge dz)\wedge(xdx-yz^2dy+2xdz) \\
  = (-2x^2-xz)dx\wedge dy\wedge dz
\end{multline}
\vskip -9mm
\null\hfill{\red degree 3}
\begin{multline} d\eta
  = d(xdx-yz^2dy+2xdz) \\
  = 2yzdy\wedge dz - 2dz\wedge dx
\end{multline}
\vskip -9mm
\null\hfill{\red 2 terms, degree 2}
\begin{multline} \label{rhs2} \omega\wedge d\eta
  = (xydx+3dy-yzdz)\wedge(2yzdy\wedge dz - 2dz\wedge dx) \\
  = ({\red 2xy^2z}-6)dx\wedge dy\wedge dz
\end{multline}
\vskip -9mm
\null\hfill{\red degree 3}
\begin{equation}
  \eqref{lhs}=\eqref{rhs1}-\eqref{rhs2} \qquad\text{checks!}
\end{equation}

\noindent{\bf Marking Scheme. } Look mostly for the items in {\red red}, about 4 points for each computation.

\noindent{\bf Alternative Marking Scheme. } 10 for $d$, 10 for $\wedge$, 5 for arithemetic

{\red $(-8)$.} Systematic but wrong differentiation of 1-forms.

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