
%%latex file
%\documentstyle[12pt,amssymb]{amsart}
\documentclass[12pt]{amsart} \usepackage{amssymb} \usepackage{mathabx} \usepackage{bm} \usepackage{wasysym} \usepackage{graphicx} \usepackage{stmaryrd}
\usepackage{imakeidx}
\usepackage{color,hyperref}
\hypersetup{colorlinks,breaklinks,
             linkcolor=blue,urlcolor=blue,
           anchorcolor=blue,citecolor=blue}
\newenvironment{pf}{\proof[\proofname]}{\endproof}

\makeindex[name=terms,title=Index of terms, columns=2]

\renewcommand\baselinestretch{1.1}
\renewcommand\theequation{\arabic{section}.\arabic{equation}}

\newcommand{\wrlab}[1]{\label{#1}}
\renewcommand{\arraystretch}{1}

\newtheorem{thm}{THEOREM}[section]
\numberwithin{thm}{subsection}
\newtheorem{axiom}[thm]{AXIOM}
\newtheorem{defn}[thm]{DEFINITION}

\newcommand{\secref}[1]{\S\ref{#1}}

\newcommand{\tref}[1]{Theorem~\ref{#1}}
\newcommand{\aref}[1]{Axiom~\ref{#1}}
\newcommand{\dref}[1]{Definition~\ref{#1}}

% BLACK BOARD BOLD

\def\Bbb{\mathbb}  %needed in Nancy

\def\A{{\mathbb A}} \def\B{{\mathbb B}} \def\C{{\mathbb C}} \def\D{{\mathbb D}}
\def\N{{\mathbb N}} \def\I{{\mathbb I}} \def\Q{{\mathbb Q}} \def\R{{\mathbb R}}
\def\Z{{\mathbb Z}} 

\def\T{{\mathbb T}}

\def\P{{\mathbb P}} 

% BOLD
\def\bfzero{{\bf 0}}
\def\bfP{{\bf P}}
\def\bfQ{{\bf Q}}
\def\bfS{{\bf S}}
\def\bfT{{\bf T}}

% CAL 
\def\Cal{\mathcal} \def\scra{{\mathcal A}} \def\scrb{{\mathcal B}}
\def\scrd{{\mathcal D}} \def\scrf{{\mathcal F}} \def\scrg{{\mathcal G}}
\def\scri{{\mathcal I}} \def\scrk{{\mathcal K}} \def\scrl{{\mathcal L}}
\def\scrm{{\mathcal M}} \def\scrp{{\mathcal P}} \def\scrq{{\mathcal Q}}
\def\scrr{{\mathcal R}} \def\scrs{{\mathcal S}} \def\scrt{{\mathcal T}}
\def\scru{{\mathcal U}} \def\scrv{{\mathcal V}} \def\scrw{{\mathcal W}}
\def\scre{{\mathcal E}} \def\scrx{{\mathcal X}} \def\scry{{\mathcal Y}}
\def\scrz{{\mathcal Z}} \def\scrh{{\mathcal H}} \def\scrc{{\mathcal C}}
\def\scrn{{\mathcal N}} \def\scro{{\mathcal O}}
\def\scrpf{{\scrp\scrf}}
\def\scral{{\scra\scrl}}
\def\scrsb{{\scrs\scrb}}

% HATTED ROMAN
\def\hatd{{\widehat{d}}}
\def\hatX{{\widehat{X}}}

% HATTEDCAL
\def\hatscrp{{\widehat{\scrp}}}
\def\hatscrp{{\widehat{\scrp}}}
\def\hatpi{{\widehat{\pi}}}

% HATTED BB BOLD
\def\hatbbR{{\widehat{\R}}}

\newcommand\scrlito{
  \mathchoice
    {{\scriptstyle\mathcal{O}}}% \displaystyle
    {{\scriptstyle\mathcal{O}}}% \textstyle
    {{\scriptscriptstyle\mathcal{O}}}% \scriptstyle
    {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}}%\scriptscriptstyle
  }

\def\scrbigo{{\widehat{\scro}}}

% LIE ALGEBRAS

\font\tenolde=eufm10 at 10pt
\font\sevenolde=eufm7
\font\fiveolde=eufm5
\newfam\oldefam
\textfont\oldefam=\tenolde
\scriptfont\oldefam=\sevenolde
\scriptscriptfont\oldefam=\fiveolde
\def\fraktwo{\fam\oldefam\tenolde}

\newfam\cyrfam
\font\tencyr=wncyr10
\def\cyr{\fam\cyrfam\tencyr}

\newcommand\sbullet[1][.75]{\mathbin{\vcenter{\hbox{\scalebox{#1}{$\bullet$}}}}}

\def\aff{{\mathfrak{aff}(\R)}} \def\La{{\mathfrak a}}
\def\Lb{{\mathfrak b}} \def\Lc{{\mathfrak c}} \def\Lg{{\mathfrak g}}
\def\Lh{{\mathfrak h}} \def\Lk{{\mathfrak k}} \def\Ll{{\mathfrak l}}
\def\Lm{{\mathfrak m}} \def\Ln{{\mathfrak n}} \def\Lr{{\mathfrak r}}
\def\Ls{{\mathfrak s}} \def\Lt{{\mathfrak t}} \def\Lw{{\mathfrak w}}
\def\Lq{{\mathfrak q}} \def\Lu{{\mathfrak u}} \def\Lv{{\mathfrak v}}
\def\Lz{{\mathfrak z}} \def\Lsl{\mathfrak{sl}} \def\Lgl{\mathfrak{gl}}
\def\Lp{{\mathfrak p}} \def\Lf{{\mathfrak f}} \def\Lx{{\mathfrak x}}
\def\Lpsl{\mathfrak{psl}}
\def\Lso{{\mathfrak{so}}} 
\def\Lsu{{\mathfrak{su}}} 
\def\Lsp{{\mathfrak{sp}}} 
\def\orthog{{\mathfrak{o}}} 
\def\tilso{{\mathfrak{so}}}
\def\tilco{{\mathfrak{co}}} 
\def\co{\mathfrak{co}}

% OVERLINE
\def\barscru{{\overline{\scru}}}
\def\barPsi{{\overline{\Psi}}}
\def\barA{{\overline{A}}}
\def\barB{{\overline{B}}}
\def\barC{{\overline{C}}}
\def\barD{{\overline{D}}}
\def\barE{{\overline{E}}}
\def\barF{{\overline{F}}}
\def\barbarF{{\overline{\overline{F}}}}
\def\barG{{\overline{G}}}
\def\barH{{\overline{H}}}
\def\barI{{\overline{I}}}
\def\barJ{{\overline{J}}}
\def\barK{{\overline{K}}}
\def\barL{{\overline{L}}}
\def\barM{{\overline{M}}}
\def\barN{{\overline{N}}}
\def\barP{{\overline{P}}}
\def\barbarP{{\overline{\overline{P}}}}
\def\barQ{{\overline{Q}}}
\def\barR{{\overline{R}}}
\def\barS{{\overline{S}}}
\def\barT{{\overline{T}}}
\def\barU{{\overline{U}}}
\def\barV{{\overline{V}}}
\def\barW{{\overline{W}}}
\def\barbarW{{\overline{\overline{W}}}}
\def\barX{{\overline{X}}}
\def\barY{{\overline{Y}}}
\def\barZ{{\overline{Z}}}
\def\barLr{{\overline{\Lr}}}
\def\barLg{{\overline{\Lg}}}
\def\barLh{{\overline{\Lh}}}
\def\barLp{{\overline{\Lp}}}
\def\barLn{{\overline{\Ln}}}
\def\barLm{{\overline{\Lm}}}
\def\barLl{{\overline{\Ll}}}
\def\bara{{\overline{a}}}
\def\bare{{\overline{e}}}
\def\barf{{\overline{f}}}
\def\barg{{\overline{g}}}
\def\bark{{\overline{k}}}
\def\baru{{\overline{U}}}
\def\barv{{\overline{v}}}
\def\barw{{\overline{w}}}
\def\barz{{\overline{z}}}
\def\barbeta{{\overline{\beta}}}
\def\bargamma{{\overline{\gamma}}}
\def\barI{{\overline{I}}}

% TILDE
\def\tilvec{\widetilde{\vec}}

% HAT
\def\hatA{{\widehat{A}}}
\def\hatC{{\widehat{C}}}
\def\hatH{{\widehat{H}}}
\def\hatI{{\widehat{I}}}
\def\hatN{{\widehat{N}}}
\def\hatQ{{\widehat{Q}}}
\def\hatR{{\widehat{R}}}
\def\hatS{{\widehat{S}}}
\def\hatV{{\widehat{V}}}
\def\hatt{{\widehat{t}}}
\def\hatu{{\widehat{u}}}
\def\hatv{{\widehat{v}}}
\def\hatw{{\widehat{w}}}
\def\hatx{{\widehat{x}}}
\def\haty{{\widehat{y}}}
\def\hatz{{\widehat{z}}}
\def\hatalpha{{\widehat{\alpha}}}
\def\hatrho{{\widehat{\rho}}}
\def\hatkappa{{\widehat{\kappa}}}
\def\hatlambda{{\widehat{\lambda}}}
\def\hatsigma{{\widehat{\sigma}}}
\def\hatmu{{\widehat{\mu}}}
\def\hatnu{{\widehat{\nu}}}
\def\hattau{{\widehat{\tau}}}

% CHECK
\def\ckt{{\widecheck{t}}}
\def\ckw{{\widecheck{w}}}
\def\ckx{{\widecheck{x}}}
\def\cky{{\widecheck{y}}}
\def\ckz{{\widecheck{z}}}
\def\cksigma{{\widecheck{\sigma}}}
\def\cktau{{\widecheck{\tau}}}
\def\ckH{{\widecheck{H}}}
\def\ckN{{\widecheck{N}}}
\def\ckS{{\widecheck{S}}}

\def\ds{\displaystyle}
\def\bfindex[#1]{{\bf#1}\index[terms]{#1}}

\def\timestimes{{\times\!\!\times}}

%operators
\def\Qd{{\rm Qd}}
\def\IC{{\rm IC}}
\def\RC{{\rm RC}}
\def\ES{{\rm ES}}
\def\PF{{\rm PF}}
\def\set{{\rm set}}
\def\bi{{\rm bi}}
\def\fn{{\rm fn}}
\def\ss{{\rm setset}}
\def\sf{{\rm setfn}}
\def\fs{{\rm fnset}}
\def\seq{{\rm seq}}
\def\EUCL{{\rm EUCL}}
\def\TNSR{{\rm TNSR}}
\def\starstar{{**}}
\def\fincap{{\hbox{{\rm fin}}\,\cap}}
\def\fincup{{\hbox{{\rm fin}}\,\cup}}
\def\ctblcap{{\hbox{{\rm ctbl}}\,\cap}}
\def\ctblcup{{\hbox{{\rm ctbl}}\,\cup}}
\def\finamalg{{\hbox{{\rm fin}}\,\amalg}}
\def\two{{\rm two}}
\def\three{{\rm three}}
\def\ten{{\rm ten}}
\def\Lin{{\rm Lin}}
\def\Bilin{{\rm Bilin}}
\def\Trilin{{\rm Trilin}}
\def\PS{{\rm PS}}
\def\UE{{\rm UE}}
\def\FV{{\rm FV}}
\def\CH{{\rm CH}}
\def\DQ{{\rm DQ}}
\def\SS{{\rm SS}}
\def\SUMS{{\rm SUMS}}
\def\res{{\rm res}}
\def\add{{\rm add\,}}
\def\FS{{\rm FS}}
\def\LIMS{{\rm LIMS}}
\def\NOT{{\rm NOT}}
\def\ORB{{\rm ORB}}
\def\dlin{{\rm dlin}}
\def\LINS{{\rm LINS}}
\def\opLIMS{\mathop{\LIMS}}
\def\IDS{{\rm IDS}}
\def\UL{{\rm UL}}
\def\UB{{\rm UB}}
\def\LB{{\rm LB}}
\def\LP{{\rm LP}}
\def\LPD{{\rm LPD}}
\def\Isol{{\rm Isol}}
\def\Cl{{\rm Cl}}
\def\sCl{{\rm sCl}}
\def\id{{\rm id}}
\def\dim{{\rm dim\,}}
\def\DIMS{{\rm DIMS\,}}
\def\Bas{{\rm Bas\,}}
\def\Int{{\rm Int}}
\def\IntD{{\rm IntD}}
\def\det{{\rm det\,}}
\def\exp{{\rm exp}}
\def\End{{\rm End}}
\def\SO{{\rm SO}}
\def\CO{{\rm CO}}
\def\O{{\rm O}}
\def\Stab{{\rm Stab}}
\def\Fix{{\rm Fix}}
\def\GL{{\rm GL}}
\def\Ad{{\rm Ad}}
\def\Aut{{\rm Aut}}
\def\der{{\rm der}}
\def\ad{{\rm ad}}
\def\diag{{\rm diag}}
\def\Mink{{\rm Mink}}
\def\Tay{{\rm Tay}}
\def\lin{{\rm lin}}
\def\switch{{\rm switch}}
\def\SOtwortwo{{\SO(2)\ltimes\R^2}}
\def\O{{\rm O}}
\def\OB{{\rm OB}}
\def\SO{{\rm SO}}
\def\SL{{\rm SL}}
\def\PSL{{\rm PSL}}
\def\SBF{{\rm SBF}}
\def\STF{{\rm STF}}
\def\SM{{\rm SM}}
\def\QF{{\rm QF}}
\def\Id{{\rm Id}}
\def\dom{{\rm dom}}
\def\im{{\rm im}}
\def\dd{{\rm dd}}
\def\dct{{\rm dct}}
\def\sym{{\rm sym}}
\def\Sym{{\rm Sym}}
\def\as{{\rm as}}
\def\mm{{\rm mm}}
\def\cof{{\rm cof}}
\def\adj{{\rm adj}}
\def\det{{\rm det}}
\def\DET{{\rm DET}}
\def\ADJ{{\rm ADJ}}
\def\DNZ{{\hbox{\rm DNZ}}}
\def\CVZ{{\hbox{\rm CVZ}}}
\def\BNZ{{\hbox{\rm BNZ}}}
\def\ev{{\rm ev}}
\def\sl{{\rm s\ell}}

%misc
\def\vep{{\varepsilon}}
\def\bksl{{\backslash}}
\def\sigmaalg{{\rm s.alg}}
\def\drta{\dashrightarrow}
\def\ltlt{{<\!\!<}}
\def\phx{\phantom{x}}

\def\Xint#1{\mathchoice
{\XXint\displaystyle\textstyle{#1}}%
{\XXint\textstyle\scriptstyle{#1}}%
{\XXint\scriptstyle\scriptscriptstyle{#1}}%
{\XXint\scriptscriptstyle\scriptscriptstyle{#1}}%
\!\int}
\def\XXint#1#2#3{{\setbox0=\hbox{$#1{#2#3}{\int}$ }
\vcenter{\hbox{$#2#3$ }}\kern-.6\wd0}}
\def\ddashint{\Xint=}
\def\dashint{\Xint-}

\begin{document}
\centerline{\bf Homework for MATH 4604 (Advanced Calculus II)}
\centerline{\bf Spring 2020}
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 8: Due on Tuesday 31 March}
\vskip.1in\noindent
8-1. Let \quad $V,W\in\ES$, \quad $f:V\drta W$, \quad $q,u\in V$.\\
\phx\quad $\,$ Assume: \quad $f=0_W$ near $q$ in $V$.\\
\phx\quad $\,$ Show: \quad $\nabla_uf=0_W$ near $q$ in $V$.
\vskip.1in\noindent
8-2. Let \quad $V,W\in\ES$, \quad $f:V\drta W$, \quad $q\in V$.\\
\phx\quad $\,$ Assume: \quad $f=0_W$ near $q$ in $V$.\\
\phx\quad $\,$ Show: \quad $\forall k\in\N$, \quad $\forall u_1,\ldots,u_k\in V$,\\
\phx\qquad\qquad\qquad\qquad $\nabla_{u_k}\cdots\nabla_{u_1}f=0_W$ near $q$ in $V$.
\vskip.1in\noindent
8-3. Let \quad $W\in\ES$, \quad $\phi:\R\drta W$, \quad $t\in\R$.\\
\phx\quad $\,$ Assume: \quad $\phi=0$ near $t$ in $\R$.\\
\phx\quad $\,$ Show: \quad $\phi'_t=0$.
\phx\quad $\,$ 
\phx\quad $\,$ 
\vskip.1in\noindent
8-4. Let \quad $V:=\R^2$, \quad $f:V\drta\R$.\\
\phx\quad $\,$ Let \quad $S:=\{(x,y)\in\R^2\,|\,x^2\le y\le3x^2\}$.\\
\phx\quad $\,$ Assume: \quad $f=0$ on $V\backslash S$.\\
\phx\quad $\,$ Show: \quad $\forall k\in\N$, \quad $\forall u_1,\ldots,u_k\in V$,\\
\phx\qquad\qquad\qquad\qquad $\nabla_{u_k}\cdots\nabla_{u_1}f=0$ on $V\backslash S$.
\vskip.1in\noindent
8-5. Let \quad $V:=\R^2$, \quad $f:V\drta\R$, \quad $u\in V$.\\
\phx\quad $\,$ Let \quad $S:=\{(x,y)\in\R^2\,|\,x^2\le y\le3x^2\}$.\\
\phx\quad $\,$ Assume: \quad $f=0$ on $V\backslash S$. \qquad Assume: \quad $f_{0_V}=0$.\\
\phx\quad $\,$ Show: \quad $(\nabla_uf)_{0_V}=0$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 7: Due on Wednesday 25 March}
\vskip.1in\noindent
7-1. Let \quad $V\in\ES$, \quad $x,u\in V$, \quad $s\in\R$.\\
\phx\quad $\,$ Show: \quad $(\alpha_x^u)_s^\T\,\,=\,\,\alpha_{0_V}^u$.
\vskip.1in\noindent
7-2. Let \quad $V\in\ES$, \quad $x,u\in V$, \quad $s\in\R$.\\
\phx\quad $\,$ Show: \quad $(\alpha_x^u)'_s\,\,=\,\,u$.
\vskip.1in\noindent
7-3. Let \quad $V\in\ES$, \quad $f:V\drta W$, \quad $q,u\in V$.\\
\phx\quad $\,$ Define $S:V\to V$ by: \quad $\forall x\in V$, $\,$ $S_x=x+q$.\\
\phx\quad $\,$ Show: \quad $\nabla_u(f\circ S)\,\,=\,\,(\nabla_uf)\circ S$.
\vskip.1in\noindent
7-4. Let \quad $V,W\in\ES$, \quad $C\in\scrc^V_W$, \quad $u\in V$.\\
\phx\quad $\,$ Show: \quad $\nabla_uC\,\,=\,\,\bfzero^V_W$.
\vskip.1in\noindent
7-5. Let \quad $V,W\in\ES$, \quad $f,g:V\drta W$, \quad $x,u\in V$.\\
\phx\quad $\,$ Assume: \quad $f=g$ $\,$ near $x$ in $V$.\\
\phx\quad $\,$ Show: \quad $f\circ\alpha_x^u\,\,=\,\,g\circ\alpha_x^u$ $\,$ near $0$ in $\R$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 6: Due on Wednesday 18 March}
\vskip.1in\noindent
6-1. Let $V,W\in\ES$, let $*\in\scrb^{VV}_W$ and let $\#:=\Sym^*$.\\
\phx\quad $\,$ Show: \qquad $\#\in\scrsb^V_W$ \qquad and \qquad $\Qd^\#=\Qd^*$.
\vskip.1in\noindent
6-2. Let $V,W\in\ES$, let $*\in\scrsb^V_W$ and let $Q:=\Qd^*$.\\
\phx\qquad\qquad Show: \quad $\forall u,v\in V$, \quad $\ds{u*v=\frac{Q_{u+v}-Q_u-Q_v}{2}}$.\\
\vskip.1in\noindent
6-3. Let $V:=\R^2$ and $W:=\R$. \qquad Define $Q:V\to W$ by:\\
\phx\qquad\qquad\qquad\qquad\qquad\qquad  $\forall x,y\in\R$, \qquad $Q(x,y)=3x^2-4xy+5y^2$.\\
\phx\quad $\,$ Let $B\in\scrsb^V_W$. \quad Assume: $\Qd^B=Q$. \quad Show: $\ds{[B]=\left[\begin{array}{cc}3&-2\\-2&5\end{array}\right]}$.
\vskip.1in\noindent
6-4. Let $V:=\R^2$. \qquad Define $Q:V\to\R$ by:\\
\phx\qquad\qquad\qquad\qquad\qquad\qquad $\forall x,y\in\R$, \qquad $Q(x,y)=2x^2+6xy+5y^2$.\\
\phx\quad $\,$ Show: $Q>0$ on $V_{0_V}^\times$.
\vskip.1in\noindent
6-5. Let $V:=\R^2$. \qquad Define $Q:V\to\R$ by:\\
\phx\qquad\qquad\qquad\qquad\qquad\qquad $\forall x,y\in\R$, \qquad $Q(x,y)=2x^2+6xy+4y^2$.\\
\phx\quad $\,$ Find $x,y\in\R$ s.t.~$Q(x,y)<0$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 5: Due on Tuesday 3 March}
\vskip.1in\noindent
5-1. Let \quad $\alpha,\beta\in\N$, \quad $V:=\R^\alpha$, \quad $W:=\R^\beta$, \quad $L\in\scra\scrl^V_W$.\\
\phx\quad $\,$ Show: \quad $\Lin^{[L]}\,\,=\,\,L$.
\vskip.1in\noindent
5-2. Let \quad $\alpha,\beta\in\N$, \quad $V:=\R^\alpha$, \quad $W:=\R^\beta$, \quad $S\in W\otimes V$.\\
\phx\quad $\,$ Show: \quad $[\,\Lin^S\,]\,\,=\,\,S$.
\vskip.1in\noindent
5-3. Let \quad $\alpha,\beta,\gamma\in\N$, \quad $V:=\R^\alpha$, \quad $W:=\R^\beta$, \quad $X:=\R^\gamma$.\\
\phx\quad $\,$ Let $L\in\scrl^V_W$ and let $M\in\scrl^W_X$. \qquad Show: \quad $[M\circ L]\,\,=\,\,[M]\cdot[L]$.
\vskip.1in\noindent
5-4. Let \quad $\alpha,\beta,\gamma\in\N$, \quad $V:=\R^\alpha$, \quad $W:=\R^\beta$, \quad $X:=\R^\gamma$.\\
\phx\quad $\,$ Let $B\in\scrb^{VW}_X$. \qquad Show: \quad $\Bilin^{[B]}=B$.
\vskip.1in\noindent
5-5. Let \quad $\alpha,\gamma\in\N$, \quad $V:=\R^\alpha$, \quad $X:=\R^\gamma$, \quad $*\in\scrb^{VV}_X$.\\
\phx\quad $\,$ Define $Q:V\to X$ by: $\forall v\in V$, $Q_v=v*v$. \quad Show: $Q\in\scrq^V_X$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 4: Due on Tuesday 25 February}
\vskip.1in\noindent
4-1. Define $a\in\R^{\N_0}$ by: $\forall j\in\N_0$, $a_j=1/(j!)$. \qquad Show: $\RC_a=\infty$.
\vskip.1in\noindent
4-2. Let $\,$ $a\in\R^{\N_0}$, $\,$ $x,r\in\R$, $\,$ $i,j\in\N_0$, $\,$ $b:=P_\bullet SS_x^a$, $\,$ $c:=P_\bullet SS_r^{|a|}$.\\
\phx\quad $\,$ Assume: $|x|\le r$. \qquad Show: $|b_j-b_i|\le|c_j-c_i|$.
\vskip.1in\noindent
4-3. Let $\,$ $\alpha\in[0;\infty)^{\N_0}$, $\,$ $r\in[0;\RC_\alpha)$, $\,$ $c:=P_\bullet SS_r^\alpha$.\\
\phx\quad $\,$ Show: $c$ is convergent in $\R$.
\vskip.1in\noindent
4-4. Let $a\in\R^{\N_0}$ and let $\alpha:=|a|$. \qquad Show: $\RC_a=\RC_\alpha$.
\vskip.1in\noindent
4-5. Let $a\in\R^{\N_0}$. \qquad Show: $\RC_a=\RC_{a^*}$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 3: Due on Tuesday 18 February}
\vskip.1in\noindent
3-1. Let $X$ be a metric space, \quad $s\in X^\N$, \quad $q\in X$, \quad $m\in\N$.\\
\phx\quad $\,$ Define $t\in X^\N$ by: \quad $\forall j\in\N$, $\,$ $t_j=s_{j+m}$.\\
\phx\quad $\,$ Assume: $t\to q$ in $X$. \qquad Show: $s\to q$ in $X$.
\vskip.1in\noindent
3-2. Let $X$ and $Y$ be metric spaces, \quad $s\in X^\N$, \quad $t\in Y^\N$.\\
\phx\quad $\,$ Show: \qquad\quad $\,[\,\,(s,t)\hbox{ is Cauchy in }X\times Y\,\,]$\\
\phx\qquad\qquad\qquad $\Leftrightarrow\,\,[\,\,(\,\,s\hbox{ is Cauchy in }X\,\,)\,\,\&\,\,(\,\,t\hbox{ is Cauchy in }Y\,\,)\,\,]$.
\vskip.1in\noindent
3-3. Let $X$ and $Y$ be complete metric spaces.\\
\phx\quad $\,$ Show $X\times Y$ is complete.
\vskip.1in\noindent
3-4. Let $p,q,r\in\N$, $A\in\R^{p\times q}$, $B\in\R^{q\times r}$. \qquad
Show: $|AB|\,\,\le\,\,|A|\cdot|B|$.
\vskip.1in\noindent
3-5. Let $V\in\ES$. \qquad Show:\\
\phx\qquad\qquad\qquad (a) $|\bullet|_V$ is Lipschitz-1 from $V$ to $\R$\\
\phx\qquad and \qquad (b) $\|\bullet\|_V$ is Lipschitz-$\sqrt{\#\scri_V}$ from $V$ to $\R$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 2: Due on Tuesday 11 February}
\vskip.1in\noindent
2-1. Show: \quad $\forall\ell\in\N$, $\,$ $((\bullet)^\ell)'=\ell\cdot(\bullet)^{\ell-1}$.
\vskip.1in\noindent
2-2. Let $m\in\R$, $L:\R\to\R$. \qquad Assume: $\forall x\in\R,\,\,L_x=m\cdot x$.\\
Show: $(\,\,\forall x\in\R, \,\, L'_x=m\,\,)\,\,\&\,\,(\,\,\forall x\in\R, \,\, L''_x=0\,\,)$.
\vskip.1in\noindent
2-3. Let $c\in\R$, $Q:\R\to\R$. \qquad Assume: $\forall x\in\R,\,\,Q_x=c\cdot x^2$.\\
Show: $(\,\,\forall x\in\R, \,\, Q'_x=2\cdot c\cdot x\,\,)\,\,\&\,\,(\,\,\forall x\in\R, \,\, Q''_x=2\cdot c\,\,)$.
\vskip.1in\noindent
2-4. Let $\alpha,\beta:\R\drta\R$. \qquad Assume: $\forall x\in\R,\,\,\,\,\beta_x\,\,=^*\,\,\alpha_x$.\\
\phx\qquad\qquad\qquad\qquad\qquad\qquad $\,\,\,\,\,\,$ Show: $\forall x\in\R,\,\,\,\,\beta'_x\,\,=^*\,\,\alpha'_x$.
\vskip.1in\noindent
2-5. Let $a\in\R$. \qquad Define $T:\R\to\R$ by: $\,\,\,\forall h\in\R$, $\,\,T_h=h+a$.\\
\phx\qquad\qquad\qquad\qquad\qquad\quad Show: $\,\,\,T'\,=\,C_\R^1$.
\vskip.1in\noindent
\hrule
\vskip.1in\noindent
\centerline{Homework 1: Due on Tuesday 4 February}
\vskip.1in\noindent
1-1. Let $f:\R\drta\R$ and let $p\in\D_f$.\\
\phx\quad $\,$ Show: \qquad\qquad $(\,\,f\hbox{ has a global strict-maximum at }p\,\,)$\\
\phx\quad\qquad\qquad\quad $\,$ $\Leftrightarrow\,\,\,\,(\,\,f_p^\T\hbox{ has a global strict-maximum at }0\,\,)$.
\vskip.1in\noindent
1-2. Let $f:\R\drta\R$, let $c\in\R$, let $L\in\LINS_cf$ and let $\vep>0$.\\
\phx\quad $\,$ Show: \quad $L-\vep\cdot|\bullet|\,\,\le\,\,f_c^\T\,\,\le\,\,L+\vep\cdot|\bullet|$\quad near $0$ in $\R$.
\vskip.1in\noindent
1-3. Let $f:\R\drta\R$, let $c\in\R$ and let $L\in\LINS_cf$.\\
\phx\quad $\,$ Assume: \quad $f$ has a local semi-maximum at $c$ in $\R$.\\
\phx\quad $\,$ Show: \qquad $L=\bfzero$.
\vskip.1in\noindent
1-4. Let $\phi:\R\drta\R$, let $a>0$ and let $Q:=a\cdot(\bullet)^2$.\\
\phx\quad $\,$ Assume: \quad $\phi-Q\,\,\in\,\,\scrlito_2$.\\
\phx\quad $\,$ Show: \qquad $\phi$ has a local strict-minimum at $0$ in $\R$.
\vskip.1in\noindent
1-5. Let $f,g:\R\drta\R$ and let $c\in\R$.\\
\phx\quad $\,$ Assume: \quad $f=g$ near $c$ in $\R$.\\
\phx\quad $\,$ Show: \qquad $\LINS_cf\,\,\subseteq\,\,\LINS_cg$.\\
\vskip.1in\noindent
\hrule
\end{document}

