%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% Scientific Word Wrap/Unwrap Version 2.5 %
% Scientific Word Wrap/Unwrap Version 3.0 %
% %
% If you are separating the files in this message by hand, you will %
% need to identify the file type and place it in the appropriate %
% directory. The possible types are: Document, DocAssoc, Other, %
% Macro, Style, Graphic, PastedPict, and PlotPict. Extract files %
% tagged as Document, DocAssoc, or Other into your TeX source file %
% directory. Macro files go into your TeX macros directory. Style %
% files are used by Scientific Word and do not need to be extracted. %
% Graphic, PastedPict, and PlotPict files should be placed in a %
% graphics directory. %
% %
% Graphic files need to be converted from the text format (this is %
% done for e-mail compatability) to the original 8-bit binary format. %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% Files included: %
% %
% "/document/hwk06sol.tex", Document, 4483, 8/7/2001, 3:24:04, "" %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%% Start /document/hwk06sol.tex %%%%%%%%%%%%%%%%%%%%
\documentclass{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%TCIDATA{OutputFilter=LATEX.DLL}
%TCIDATA{Created=Wednesday, August 23, 2000 15:54:58}
%TCIDATA{LastRevised=Monday, August 06, 2001 23:24:03}
%TCIDATA{}
%TCIDATA{}
%TCIDATA{CSTFile=LaTeX article (bright).cst}
\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\input{tcilatex}
\begin{document}
\begin{center}
{\LARGE Exercise 06\medskip }
\end{center}
Please attempt all of the following problems before the due date. Your grade
on this assignment will be calculated from the best three answers.\vspace{%
0.5in}
{\LARGE Problem 06.1\medskip }
Show that the tensor $\omega ^{r}\otimes \omega ^{s}$ evaluated on the
vector pair $u,v$ is just the product of their components:
\[
\omega ^{r}\otimes \omega ^{s}\left( u,v\right) =u^{r}v^{s}
\]%
\bigskip
{\large Answer 06.1}{\LARGE \medskip }
Use the definition of the tensor cross product%
\[
\alpha \otimes \beta \left( u,v\right) =\alpha \left( u\right) \beta \left(
v\right)
\]%
and the expression for the components of a form%
\[
u^{r}=u\left( \omega ^{r}\right)
\]%
$\omega ^{r}\otimes \omega ^{s}\left( u,v\right) =\omega ^{r}\left( u\right)
\omega ^{s}\left( v\right) =u\left( \omega ^{r}\right) v\left( \omega
^{s}\right) =u^{r}v^{s}.$\vspace{0.5in}
{\LARGE Problem 06.2\medskip }
Use the summation convention to repeat the demonstration in the text that
the tensor contraction defined by%
\[
K\left( \alpha ,u\right) =\sum\limits_{i=1}^{n}K\left( \alpha ,\omega
^{i},u,e_{i}\right)
\]%
is independent of the choice of basis vectors.
{\large Answer 06.2}{\LARGE \medskip }
Copy the demonstration in the text and delete all the summations signs.
$e_{i}^{\prime }=U_{i}{}^{j}e_{j}$
$\omega ^{\prime r}=\omega ^{s}\left( U^{-1}\right) _{s}{}^{r}$
$K^{\prime }\left( \alpha ,u\right) =K\left( \alpha ,\omega ^{\prime
i},u,e_{i}^{\prime }\right) =K\left( \alpha ,\left( U^{-1}\right)
_{s}{}^{i}\omega ^{s},u,U_{i}{}^{j}e_{j}\right) $
$=\left( U^{-1}\right) _{s}{}^{i}U_{i}{}^{j}K\left( \alpha ,\omega
^{s},u,e_{j}\right) =\delta _{s}^{j}K\left( \alpha ,\omega
^{s},u,e_{j}\right) =K\left( \alpha ,\omega ^{j},u,e_{j}\right) =K\left(
\alpha ,u\right) .$\vspace{0.5in}
{\LARGE Problem 06.3\medskip }
Show that the contraction of the tensor product $\alpha \otimes v$ of a form
$\alpha $ and a vector $v$ is equal to $\alpha \left( v\right) $.\bigskip
{\large Answer 06.3}{\LARGE \medskip }
$\alpha \otimes v\left( e_{r},\omega ^{r}\right) =\alpha \left( e_{r}\right)
v\left( \omega ^{r}\right) =\alpha _{r}v^{r}=\alpha \left( v\right) .$%
\vspace{0.5in}
{\LARGE Problem 06.4\medskip }
Consider a vector $k$ and a tensor that assigns the number $K\left(
u,v,w\right) $ to the vectors $u,v,w$. Show that the contraction of the
tensor product $K\otimes k$ on its last two arguments is the same as the
tensor that assigns the number $K\left( u,v,k\right) $ to each pair of
vectors $u,v$.\bigskip
{\large Answer 06.4}{\LARGE \medskip }
$K\otimes k\left( u,v,w,\alpha \right) =$ $K\left( u,v,w\right) k\left(
\alpha \right) $
$K\otimes k\left( u,v,e_{r},\omega ^{r}\right) =K\left( u,v,e_{r}\right)
k\left( \omega ^{r}\right) =K\left( u,v,e_{r}\right) k^{r}=K\left(
u,v,k^{r}e_{r}\right) =K\left( u,v,k\right) .$\vspace{0.5in}
\end{document}
%%%%%%%%%%%%%%%%%%%%%% End /document/hwk06sol.tex %%%%%%%%%%%%%%%%%%%%%