\relax \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Functions of Several Variables.}}{1}} \newlabel{ex:LinearProjection}{{2.2}{2}} \@writefile{lof}{\contentsline {figure}{\numberline {2.1}{\ignorespaces Projections in $\ensuremath {\mathbb R}^3$ onto the $xy$-plane (at left), and the unit sphere $S^2$ (at right).\relax }}{2}} \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}} \newlabel{fig:Projections}{{2.1}{2}} \newlabel{def:Graph1}{{2.4}{3}} \newlabel{def:Graphn}{{2.5}{3}} \@writefile{lof}{\contentsline {figure}{\numberline {2.2}{\ignorespaces For $f:X\subset \ensuremath {\mathbb R}^n\to \ensuremath {\mathbb R}$, $\textbf {graph}(f)\subset \ensuremath {\mathbb R}^{n+1}$.\relax }}{3}} \newlabel{fig:Graph}{{2.2}{3}} \@writefile{lof}{\contentsline {figure}{\numberline {2.3}{\ignorespaces Projections in $\ensuremath {\mathbb R}^3$ onto the $xy$-plane (at left), and the unit sphere $S^2$ (at right).\relax }}{4}} \newlabel{fig:TrajPhaseSpace}{{2.3}{4}} \@writefile{lof}{\contentsline {figure}{\numberline {2.4}{\ignorespaces Parameterization of $S^1\subset \ensuremath {\mathbb R}^2$ via $h:\ensuremath {\mathbb R}\to \ensuremath {\mathbb R}^2$, $h(t) = (\qopname \relax o{cos}t,\qopname \relax o{sin}t)$.\relax }}{4}} \newlabel{fig:CircleParam}{{2.4}{4}} \newlabel{ex:Sphere}{{2.6}{5}} \@writefile{lof}{\contentsline {figure}{\numberline {2.5}{\ignorespaces Parameterization of $S^2\subset \ensuremath {\mathbb R}^3$ via $\Phi :D\to \ensuremath {\mathbb R}^3$.\relax }}{5}} \newlabel{fig:Sphere}{{2.5}{5}} \@writefile{lof}{\contentsline {figure}{\numberline {2.6}{\ignorespaces $\textbf {image}(F)\subset \ensuremath {\mathbb R}^3$ in Example\nonbreakingspace 2.7\hbox {} is not the graph of a function defined on the $xy$-plane in $\ensuremath {\mathbb R}^3$.\relax }}{5}} \newlabel{fig:Carpet}{{2.6}{5}} \newlabel{ex:Carpet}{{2.7}{5}} \newlabel{ex:cLevelSet}{{2.9}{6}} \@writefile{lof}{\contentsline {figure}{\numberline {2.7}{\ignorespaces The $z=8$-section and $c=8$-level set of $z = f(x,y)=x^2+y^2$.\relax }}{6}} \newlabel{fig:cLevelSet}{{2.7}{6}} \@writefile{lof}{\contentsline {figure}{\numberline {2.8}{\ignorespaces The 2 coordinate slices through $\textbf {graph}(f)$, for $f:X\subset \ensuremath {\mathbb R}^2\to \ensuremath {\mathbb R}$.\relax }}{6}} \newlabel{fig:x&ySlices}{{2.8}{6}} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{17.25pt} \newlabel{tocindent1}{0pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} \@writefile{lof}{\contentsline {figure}{\numberline {2.9}{\ignorespaces The $y$-slice at $y=2$ of the function $z = f(x,y) = x^2 + y^2$.\relax }}{7}} \newlabel{fig:y2Slice}{{2.9}{7}}