# LaTeX2HTML 2008 (1.71)
# Associate images original text with physical files.


$key = q/displaystyleL_1^*q={sum_{r,s}}L_2^*[{delta({svectorx}-{{svectorx}_r})}({D_{rs}(t)}-L_2P)w].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img74.png"
 ALT="$\displaystyle L_1^* q = {\sum_{r,s}}L_2^* [ {\delta ({\svector x}- {{\svector x}_r})}({D_{rs}(t)}- L_2 P) w] .$">|; 

$key = q/displaystyleleft{{1over{{c({svectorx})}^2{rho({svectorx})}}}{{partial^2}over{paro({svectorx})}}}{svectornabla}right]right}{q{({svectorx},t)}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img112.png"
 ALT="$\displaystyle \left \{ {1 \over {{c({\svector x})}^2 {\rho({\svector x})}}}{{\p...
...rho({\svector x})}}}{\svector \nabla}\right ] \right \} {q {({\svector x}, t)}}$">|; 

$key = q/{displaymath}{{f_s}(t)}={{svectornabla}cdot}[{svector{f}(t)}slashrho({{svectorx}_s})],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="\begin{displaymath}
{{f_s}(t)}= {{\svector \nabla}\cdot}[ {\svector{f}(t)}/ \rho({{\svector x}_s})],
\end{displaymath}">|; 

$key = q/s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="$s$">|; 

$key = q/deltaP;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img72.png"
 ALT="$\delta P$">|; 

$key = q/displaystyle{{int_0^T}!{int_{Omega}}}{q{({svectorx},t)}}delta({svectorx})delta{{f_s}(t)}{~d{svectorx},dt}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img97.png"
 ALT="$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}{q {({\svector x}, t)}}\delta ({\svector x}) \delta {{f_s}(t)}{&nbsp;d {\svector x}  d t}.$">|; 

$key = q/displaystyle{sum_{r,s}}{int_{Omega}}delta{svectorm}cdot{int_0^T}[{{svectornabla}orx}-{{svectorx}_r})}({D_{rs}(t)}-L_2P_s)~w]dt{~d{svectorx}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="501" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="$\displaystyle {\sum_{r,s}}{\int_{\Omega}}\delta {\svector m}\cdot {\int_0^T}[ {...
...svector x}- {{\svector x}_r})}( {D_{rs}(t)}- L_2 P_s)&nbsp;w] dt {&nbsp;d {\svector x}}.$">|; 

$key = q/displaystyle{int_{Omega}}delta{svectorm}{int_0^T}[{{svectornabla}_{mbox{{scriptsize{{boldmath{m}}}(L_1P)]^*q~dt~d{svectorx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img76.png"
 ALT="$\displaystyle {\int_{\Omega}}\delta {\svector m}{\int_0^T}[ {{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}(L_1 P)]^* q&nbsp; dt&nbsp; d {\svector x}$">|; 

$key = q/{D_{rs}(t)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="${D_{rs}(t)}$">|; 

$key = q/{{svectornabla}_{mbox{{scriptsize{{boldmath{m}}}(L_1P);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="${{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}(L_1 P)$">|; 

$key = q/{deltaP_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="${\delta P_s}$">|; 

$key = q/displaystyle{deltaJoverdelta{c({svectorx})}}={int_0^T}{1over{{c({svectorx})}^3{r{P_s{({svectorx},t)}}}over{partialt^2}}{q{({svectorx},t)}}dt.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img118.png"
 ALT="$\displaystyle {\delta J \over \delta {c({\svector x})}} =
{\int_0^T}{1 \over {{...
...ial^2 {P_s {({\svector x}, t)}}}\over{\partial t^2}}{q {({\svector x}, t)}}dt .$">|; 

$key = q/{displaymath}{{int_0^T}!{int_{Omega}}}{q{({svectorx},t)}}L_1[{{svectorm}{({svect}{~d{svectorx},dt}=0,~forall{q{({svectorx},t)}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="\begin{displaymath}
{{\int_0^T}\! {\int_{\Omega}}}{q {({\svector x}, t)}}L_1 [ {...
...{&nbsp;d {\svector x}  d t}= 0,&nbsp; \forall {q {({\svector x}, t)}}.
\end{displaymath}">|; 

$key = q/displaystyle{sum_{r,s}}{int_{Omega}}delta{svectorm}{int_0^T}[{{svectornabla}_{mbvectorx}-{{svectorx}_r})}({D_{rs}(t)}-L_2P)w]~dt~d{svectorx}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="478" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img78.png"
 ALT="$\displaystyle {\sum_{r,s}}{\int_{\Omega}}\delta {\svector m}{\int_0^T}[ {{\svec...
...\svector x}- {{\svector x}_r})}( {D_{rs}(t)}- L_2 P) w ]&nbsp; dt&nbsp; d {\svector x}.$">|; 

$key = q/delta{svectorm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="$\delta {\svector m}$">|; 

$key = q/displaystyleJ[rho,c,{f_s}];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="$\displaystyle J [ \rho, c, {f_s}]$">|; 

$key = q/displaystyle{sum_{r,s}}{int_0^T}[{{svectornabla}_{mbox{{scriptsize{{boldmath{m}}^*[{delta({svectorx}-{{svectorx}_r})}({D_{rs}(t)}-L_2P)w]~dt.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="405" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img82.png"
 ALT="$\displaystyle {\sum_{r,s}}{\int_0^T}[ {{\svector \nabla}_{\mbox{\scriptsize\bol...
... ]^* [ {\delta ({\svector x}- {{\svector x}_r})}( {D_{rs}(t)}- L_2 P) w ]&nbsp;dt .$">|; 

$key = q/{displaymath}mbox{where}{{svectorm}{({svectorx},t)}}=[{c({svectorx})},{rho({svectorx})},{f_s}].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="\begin{displaymath}
\mbox{where } {{\svector m}{({\svector x}, t)}}= [ {c({\svector x})}, {\rho({\svector x})}, {f_s}].
\end{displaymath}">|; 

$key = q/textstyle+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="$\textstyle +$">|; 

$key = q/a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="$a$">|; 

$key = q/displaystyle{int_{Omega}}{{svectornabla}cdot}(a{svectorb}){~d{svectorx}}-{int_{Omega}}{svectorb}cdot{svectornabla}a{~d{svectorx}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img99.png"
 ALT="$\displaystyle {\int_{\Omega}}{{\svector \nabla}\cdot}(a {\svector b}) {&nbsp;d {\svector x}}- {\int_{\Omega}}{\svector b}\cdot {\svector \nabla}a {&nbsp;d {\svector x}}$">|; 

$key = q/z=x_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$z = x_1$">|; 

$key = q/displaystyle{svectornabla}_sJ(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img123.png"
 ALT="$\displaystyle {\svector \nabla}_s J(t)$">|; 

$key = q/L_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="$L_1$">|; 

$key = q/displaystyle{{int_0^T}!{int_{Omega}}}left[{1over{{c({svectorx})}^2{rho({svectorxla}right)right]~{delta{P_s{({svectorx},t)}}}{~d{svectorx},dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="438" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img94.png"
 ALT="$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}\left [ {1 \over {{c({\svector x})}...
...a}\right ) \right ]&nbsp; {\delta {P_s {({\svector x}, t)}}}{&nbsp;d {\svector x}  d t}$">|; 

$key = q/displaystyle{sum_{r,s}}{{int_0^T}!{int_{Omega}}}deltaP_sL_2^*[{delta({svectorx}-{{svectorx}_r})}({D_{rs}(t)}-L_2P_s)~w]{~d{svectorx},dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="401" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="$\displaystyle {\sum_{r,s}}{{\int_0^T}\! {\int_{\Omega}}}\delta P_s L_2^* [ {\de...
...ector x}- {{\svector x}_r})}( {D_{rs}(t)}- L_2 P_s)&nbsp;w] {&nbsp;d {\svector x}  d t}$">|; 

$key = q/displaystyle{{int_0^T}!{int_{Omega}}}{q{({svectorx},t)}}left[{1over{{c({svectorx}over{partialt^2}}right]delta{c({svectorx})}{~d{svectorx},dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="376" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img95.png"
 ALT="$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}{q {({\svector x}, t)}}\left [ {1 \...
...}}}\over{\partial t^2}}\right ] \delta {c({\svector x})}{&nbsp;d {\svector x}  d t}$">|; 

$key = q/{V_{max}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="${V_{max}}$">|; 

$key = q/displaystyle+{int_0^T}!{int_{partialOmega}}delta{rho({svectorx})}left[{1over{{rh_s}(t)}delta({svectorx}){q{({svectorx},t)}}{~d{svectorx},dt}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="602" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img109.png"
 ALT="$\displaystyle + {\int_0^T}\! {\int_{\partial \Omega}}\delta {\rho({\svector x})...
...{{f_s}(t)}\delta ({\svector x}) {q {({\svector x}, t)}}{&nbsp;d {\svector x}  d t}.$">|; 

$key = q/{displaymath}J[{{svectorm}{({svectorx},t)}}]={1over2}{sum_{r,s}}{{int_0^T}!{int_ctorx}-{{svectorx}_r})}{~w(t)}{~d{svectorx},dt}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="625" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="\begin{displaymath}
J [ {{\svector m}{({\svector x}, t)}}] = {1 \over 2} {\sum_{...
...\svector x}- {{\svector x}_r})}{&nbsp;w(t)}{&nbsp;d {\svector x}  d t}.
\end{displaymath}">|; 

$key = q/{displaymath}left[{1over{{c({svectorx})}^2{rho({svectorx})}}}{{partial^2}over{pa_s}(t)}delta({svectorx}-{{svectorx}_s})delta(t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="494" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="\begin{displaymath}
\left [ {1 \over {{c({\svector x})}^2 {\rho({\svector x})}}}...
...= {{f_s}(t)}\delta ({\svector x}-{{\svector x}_s}) \delta(t) ,
\end{displaymath}">|; 

$key = q/textstyle=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$\textstyle =$">|; 

$key = q/{c({svectorx})};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="${c({\svector x})}$">|; 

$key = q/displaystyle=-{sum_{r,s}}{{partial}over{partialz}}left{{delta({svectorx}-{{svect}}}{{partial{P_s{({svectorx},t)}}}over{partialz}}right]right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="448" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img113.png"
 ALT="$\displaystyle = - {\sum_{r,s}}{{\partial}\over{\partial z}}\left \{ {\delta ({\...
...x})}}}{{\partial {P_s {({\svector x}, t)}}}\over{\partial z}}\right ] \right \}$">|; 

$key = q/displaystyle+{{int_0^T}!{int_{Omega}}}{delta{P_s{({svectorx},t)}}}{1over{{c({sve}}{partialoverpartialn}{delta{P_s{({svectorx},t)}}},dsigma,dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="631" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img105.png"
 ALT="$\displaystyle +{{\int_0^T}\! {\int_{\Omega}}}{\delta {P_s {({\svector x}, t)}}}...
...{\partial \over \partial n} {\delta {P_s {({\svector x}, t)}}}  d \sigma   dt$">|; 

$key = q/textstylehfil;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="5" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img93.png"
 ALT="$\textstyle \hfil$">|; 

$key = q/{svectorm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="${\svector m}$">|; 

$key = q/displaystyle{{int_0^T}!{int_{Omega}}}P_1{({svectorx},t)}P_2{({svectorx},t)}{~d{svectorx},dt},mbox{and};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}P_1 {({\svector x}, t)}P_2 {({\svector x}, t)}{&nbsp;d {\svector x}  d t}, \mbox{ and}$">|; 

$key = q/displaystyle{{svectorx}~|~z=0~~mbox{or}~parallel{svectorx}parallel=T{V_{max}}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$\displaystyle \{ {\svector x}&nbsp;\vert&nbsp; z = 0&nbsp;&nbsp; \mbox{or}&nbsp; \parallel {\svector x}\parallel
= T {V_{max}}\},$">|; 

$key = q/<a|Lb>_a=<L^*a|b>_b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="$&lt;a \vert Lb&gt;_a = &lt;L^* a\vert b&gt;_b$">|; 

$key = q/displaystyle-{sum_{r,s}}{int_0^T}{{delta({svectorx}-{{svectorx}_r})}over{rho({svright]{{partial{P_s{({svectorx},t)}}}over{partialz}}{~w(t)}dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="496" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img122.png"
 ALT="$\displaystyle - {\sum_{r,s}}{\int_0^T}{{\delta ({\svector x}- {{\svector x}_r})...
...al z}}\right ]
{{\partial {P_s {({\svector x}, t)}}}\over{\partial z}}{&nbsp;w(t)}dt$">|; 

$key = q/displaystyle{svectornabla}_{rho}J({svectorx});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img119.png"
 ALT="$\displaystyle {\svector \nabla}_{\rho} J({\svector x})$">|; 

$key = q/displaystyle{int_0^T}[{{svectornabla}_{mbox{{scriptsize{{boldmath{m}}}(L_1P)]^*q~dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img81.png"
 ALT="$\displaystyle {\int_0^T}[ {{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}(L_1 P)]^* q&nbsp;dt$">|; 

$key = q/{svectorm}({svectorx});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img79.png"
 ALT="${\svector m}({\svector x})$">|; 

$key = q/deltaJ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img116.png"
 ALT="$\delta J$">|; 

$key = q/frac{1}{c^3rho}frac{partial^2P}{partialt^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img125.png"
 ALT="$\frac{1}{c^3 \rho}\frac{\partial^2P}{\partial t^2}$">|; 

$key = q/{q{({svectorx},t)}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="${q {({\svector x}, t)}}$">|; 

$key = q/displaystyle{{int_0^T}!{int_{Omega}}}q_1{({svectorx},t)}q_2{({svectorx},t)}{~d{svectorx},dt},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}q_1 {({\svector x}, t)}q_2 {({\svector x}, t)}{&nbsp;d {\svector x}  d t},$">|; 

$key = q/displaystyle{sum_{r,s}}{{int_0^T}!{int_{Omega}}}{delta({svectorx}-{{svectorx}_r})}({D_{rs}(t)}-L_2P_s)L_2deltaP_s~w~{~d{svectorx},dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="$\displaystyle {\sum_{r,s}}{{\int_0^T}\! {\int_{\Omega}}}{\delta ({\svector x}- ...
...ector x}_r})}( {D_{rs}(t)}- L_2 P_s) L_2 \delta P_s&nbsp;w&nbsp; {&nbsp;d {\svector x}  d t}$">|; 

$key = q/{{{svectorx}_r}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img83.png"
 ALT="$\{{{\svector x}_r}\}$">|; 

$key = q/{displaymath}left.{P_s{({svectorx},t)}}right|_{zleq0}=0,~left.{P_s{({svectorx},trtialt^2}}{P_s{({svectorx},t)}}right|_{tleq0}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="472" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="\begin{displaymath}
\left. {P_s {({\svector x}, t)}}\right\vert _{z \leq 0} = 0 ...
...al t^2}}{P_s {({\svector x}, t)}}\right\vert _{t \leq 0} = 0 ,
\end{displaymath}">|; 

$key = q/displaystyle{int_{partialOmega}}{svector{hatn}}cdot{svectorb},a,dsigma-{int_{Omega}}{svectorb}cdot{svectornabla}a{~d{svectorx}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img100.png"
 ALT="$\displaystyle {\int_{\partial \Omega}}{\svector {\hat n} }\cdot {\svector b}  ...
...\sigma - {\int_{\Omega}}{\svector b}\cdot {\svector \nabla}a {&nbsp;d {\svector x}}.$">|; 

$key = q/{svectorx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="${\svector x}$">|; 

$key = q/displaystyle{{int_0^T}!{int_{Omega}}}left{left[{1over{{c({svectorx})}^2{rho({sveorx}_s})delta(t)right}{q{({svectorx},t)}}{~d{svectorx},dt}=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="712" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}\left \{ \left [ {1 \over {{c({\sve...
...r x}_s}) \delta(t) \right \}
{q {({\svector x}, t)}}{&nbsp;d {\svector x}  d t}= 0,$">|; 

$key = q/displaystyle{sum_{r,s}}{{int_0^T}!{int_{Omega}}}{delta({svectorx}-{{svectorx}_r}{boldmath{m}}}(L_2P_s)cdotdelta{svectorm}~w~{~d{svectorx},dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="474" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="$\displaystyle {\sum_{r,s}}{{\int_0^T}\! {\int_{\Omega}}}{\delta ({\svector x}- ...
...ize\boldmath m}}}(L_2 P_s) \cdot \delta {\svector m}&nbsp;w&nbsp; {&nbsp;d {\svector x}  d t}$">|; 

$key = q/textstyle,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="8" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img121.png"
 ALT="$\textstyle  $">|; 

$key = q/{{svectorx}_r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="${{\svector x}_r}$">|; 

$key = q/{svectorb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img101.png"
 ALT="${\svector b}$">|; 

$key = q/displaystyle{{int_0^T}!{int_{Omega}}}{q{({svectorx},t)}}left{{1over{{c({svectorxtorx},t)}}right]right}delta{rho({svectorx})}{~d{svectorx},dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="584" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img96.png"
 ALT="$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}{q {({\svector x}, t)}}\left \{ {1 ...
...r x}, t)}}\right ]
\right \} \delta {\rho({\svector x})}{&nbsp;d {\svector x}  d t}$">|; 

$key = q/displaystyle{1over2}{sum_{r,s}}{int_0^T}left[{D_{rs}(t)}-{1overrho({{svectorx}_r})}{partialP({{svectorx}_r},t)overpartialz}right]^2{~w(t)}dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="$\displaystyle {1 \over 2} {\sum_{r,s}}{\int_0^T}
\left [ {D_{rs}(t)}- {1 \over ...
...}_r})}
{\partial P ({{\svector x}_r}, t ) \over \partial z}\right ]^2 {&nbsp;w(t)}dt$">|; 

$key = q/displaystyle<P_1|P_2>_P;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="$\displaystyle &lt; P_1 \vert P_2 &gt;_P$">|; 

$key = q/displaystyle{int_{Omega}}m_1({svectorx})m_2({svectorx})~d{svectorx}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$\displaystyle {\int_{\Omega}}m_1 ({\svector x}) m_2 ({\svector x})&nbsp; d {\svector x}.$">|; 

$key = q/L^*;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="$L^*$">|; 

$key = q/displaystyle{{int_0^T}!{int_{Omega}}}{deltaP_s}L_1^*q{~d{svectorx},dt}=-{int_{Omcriptsize{{boldmath{m}}}(L_1P)]^*q~dt{~d{svectorx}},~forallq.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="461" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}{\delta P_s}L_1^* q {&nbsp;d {\svector x...
...box{\scriptsize\boldmath m}}}(L_1 P)]^* q&nbsp; dt {&nbsp;d {\svector x}},&nbsp; \forall q .$">|; 

$key = q/displaystylembox{thesurface}partialOmega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="$\displaystyle \mbox{ the surface }
\partial \Omega$">|; 

$key = q/displaystyle-deltaJ[rho,c,{f_s}]={sum_{r,s}}{{int_0^T}!{int_{Omega}}}{delta{P_s{ctorx},t)}}}over{partialz}}right]right}{~d{svectorx},w(t),dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="727" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img90.png"
 ALT="$\displaystyle - \delta J [ \rho, c, {f_s}] = {\sum_{r,s}}{{\int_0^T}\! {\int_{\...
...r x}, t)}}}\over{\partial z}}\right ] \right \} {&nbsp;d {\svector x}  w(t)   d t}$">|; 

$key = q/partialOmega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$\partial \Omega$">|; 

$key = q/{svectorm}_{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img85.png"
 ALT="${\svector m}_{n+1}$">|; 

$key = q/{{svectorm}{({svectorx},t)}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="${{\svector m}{({\svector x}, t)}}$">|; 

$key = q/displaystyle<m_1|m_2>_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$\displaystyle &lt; m_1 \vert m_2 &gt;_m$">|; 

$key = q/displaystyle{{svectornabla}_{mbox{{scriptsize{{boldmath{m}}}J[{svectorm}({svectoft.{deltaJoverdelta{svectorm}}right|_{{svectorm}({svectorx})};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img80.png"
 ALT="$\displaystyle {{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}J [{\svector m...
...{\delta J \over \delta {\svector m}} \right \vert _{{\svector m}({\svector x})}$">|; 

$key = q/displaystyle{sum_{r,s}}{{int_0^T}!{int_{Omega}}}{delta({svectorx}-{{svectorx}_r}o({svectorx})}over{rho({svectorx})}^2}{~d{svectorx},w(t),dt}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="571" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img88.png"
 ALT="$\displaystyle {\sum_{r,s}}{{\int_0^T}\! {\int_{\Omega}}}{\delta ({\svector x}- ...
...ho({\svector x})}\over {\rho({\svector x})}^2} {&nbsp;d {\svector x}  w(t)   d t}.$">|; 

$key = q/displaystyle{{int_0^T}!{int_{Omega}}}deltaPL_1^*q{~d{svectorx},dt}={sum_{r,s}}{{torx}-{{svectorx}_r})}({D_{rs}(t)}-L_2P)~w]{~d{svectorx},dt}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="562" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img73.png"
 ALT="$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}\delta P L_1^* q {&nbsp;d {\svector x} ...
...svector x}- {{\svector x}_r})}({D_{rs}(t)}- L_2P)&nbsp;w ] {&nbsp;d {\svector x}  d t}.$">|; 

$key = q/displaystyle{{int_0^T}!{int_{Omega}}}delta{c({svectorx})}left[{1over{{c({svector}}over{partialt^2}}right]{q{({svectorx},t)}}{~d{svectorx},dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="376" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img107.png"
 ALT="$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}\delta {c({\svector x})}\left [ {1 ...
...)}}}\over{\partial t^2}}\right ] {q {({\svector x}, t)}}{&nbsp;d {\svector x}  d t}$">|; 

$key = q/{~w(t)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="${&nbsp;w(t)}$">|; 

$key = q/textstyle-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img77.png"
 ALT="$\textstyle -$">|; 

$key = q/H^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$H^2$">|; 

$key = q/0leqtleqT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$0 \leq t \leq T$">|; 

$key = q/Omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img103.png"
 ALT="$\Omega$">|; 

$key = q/displaystyle{svectornabla}_cJ({svectorx});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img117.png"
 ALT="$\displaystyle {\svector \nabla}_c J({\svector x})$">|; 

$key = q/rho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="$\rho$">|; 

$key = q/displaystyle+{sum_{r,s}}{{int_0^T}!{int_{Omega}}}delta{rho({svectorx})}left{{del({svectorx},t)}}}over{partialz}}right}{~d{svectorx},w(t),dt}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="661" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img91.png"
 ALT="$\displaystyle + {\sum_{r,s}}{{\int_0^T}\! {\int_{\Omega}}}\delta {\rho({\svecto...
...\svector x}, t)}}}\over{\partial z}}
\right \} {&nbsp;d {\svector x}  w(t)   d t}.$">|; 

$key = q/q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$q$">|; 

$key = q/P;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$P$">|; 

$key = q/{displaymath}{{svectornabla}_{mbox{{scriptsize{{boldmath{m}}}G({svectorm})cdotdeight|_{epsilon=0},mbox{where}G({svectorm})=L_1P,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="\begin{displaymath}
{{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}G({\svect...
...vert _{\epsilon = 0}, \mbox{ where } G({\svector m}) = L_1 P ,
\end{displaymath}">|; 

$key = q/displaystylemin_alphaJ[{svectorm}_n+alpha{{svectornabla}_{mbox{{scriptsize{{boldvectornabla}_{mbox{{scriptsize{{boldmath{m}}}J({svectorm}_n).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="479" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img84.png"
 ALT="$\displaystyle \min_\alpha J [ {\svector m}_n + \alpha {{\svector \nabla}_{\mbox...
...+ \alpha {{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}J({\svector m}_n) .$">|; 

$key = q/displaystyledeltaJ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img75.png"
 ALT="$\displaystyle \delta J$">|; 

$key = q/c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="$c$">|; 

$key = q/{{svectorx}_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="${{\svector x}_s}$">|; 

$key = q/b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="$b$">|; 

$key = q/{svectornabla}_{rho}J_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img126.png"
 ALT="${\svector \nabla}_{\rho} J_1$">|; 

$key = q/displaystyle{1over2}{sum_{r,s}}{{int_0^T}!{int_{Omega}}}left[{D_{rs}(t)}-{1over{t]^2{delta({svectorx}-{{svectorx}_r})}{~d{svectorx},w(t),dt}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="472" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="$\displaystyle {1 \over 2} {\sum_{r,s}}{{\int_0^T}\! {\int_{\Omega}}}
\left [ {D...
...t ]^2
{\delta ({\svector x}- {{\svector x}_r})}{&nbsp;d {\svector x}  w(t)   d t}.$">|; 

$key = q/delta{P_s{({svectorx},t)}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img92.png"
 ALT="$\delta {P_s {({\svector x}, t)}}$">|; 

$key = q/{svector{hatn}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img102.png"
 ALT="${\svector {\hat n} }$">|; 

$key = q/displaystyle-{{int_0^T}!{int_{Omega}}}{delta{P_s{({svectorx},t)}}}{{svectornabla)}}}{svectornabla}{q{({svectorx},t)}}right]{~d{svectorx},dt}=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img106.png"
 ALT="$\displaystyle - {{\int_0^T}\! {\int_{\Omega}}}{\delta {P_s {({\svector x}, t)}}...
...})}}}{\svector \nabla}{q {({\svector x}, t)}}\right ] {&nbsp;d {\svector x}  d t}
=$">|; 

$key = q/z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img89.png"
 ALT="$z$">|; 

$key = q/{{f_s}(t)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="${{f_s}(t)}$">|; 

$key = q/displaystyleleft.{int_{Omega}}{q{({svectorx},t)}}{1over{{c({svectorx})}^2{rho({srx})}}}{delta{P_s{({svectorx},t)}}}{~d{svectorx}}right|_{t=T};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="622" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img104.png"
 ALT="$\displaystyle \left. {\int_{\Omega}}{q {({\svector x}, t)}}{1 \over {{c({\svect...
...r x})}}}
{\delta {P_s {({\svector x}, t)}}}{&nbsp;d {\svector x}}\right \vert _{t=T}$">|; 

$key = q/partialPslashpartialn={svector{hatn}}cdot{svectornabla}P;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img111.png"
 ALT="$\partial P / \partial n = {\svector {\hat n} }\cdot {\svector \nabla}P$">|; 

$key = q/displaystyleleft.{q{({svectorx},t)}}right|_{t=T}=left.{{partial}over{partialt}}{right|_{t=T}=left.{q{({svectorx},t)}}right|_{partialOmega}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img115.png"
 ALT="$\displaystyle \left. {q {({\svector x}, t)}}\right \vert _{t=T} =
\left. {{\par...
...rt _{t=T} =
\left. {q {({\svector x}, t)}}\right \vert _{\partial \Omega} = 0 .$">|; 

$key = q/{displaymath}{{int_0^T}!{int_{Omega}}}qL_1{deltaP_s}{~d{svectorx},dt}=-{{int_0^TP)cdotdelta{svectorm}{~d{svectorx},dt},~forallq.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="443" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="\begin{displaymath}
{{\int_0^T}\! {\int_{\Omega}}}q L_1 {\delta P_s}{&nbsp;d {\svecto...
...cdot \delta {\svector m}{&nbsp;d {\svector x}  d t},
&nbsp; \forall q.
\end{displaymath}">|; 

$key = q/L_1^*;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="$L_1^*$">|; 

$key = q/{{f_s}(t)}=delta(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img127.png"
 ALT="${{f_s}(t)}= \delta(t)$">|; 

$key = q/1slashrho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="$1/\rho$">|; 

$key = q/{svector{f}(t)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="${\svector{f}(t)}$">|; 

$key = q/{displaymath}left.{P_s{({svectorx},t)}}right|_{mathbf{x}inpartialOmega}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="\begin{displaymath}
\left. {P_s {({\svector x}, t)}}\right\vert _{\mathbf{x} \in \partial \Omega} = 0.
\end{displaymath}">|; 

$key = q/displaystylembox{fortheregion}Omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$\displaystyle \mbox{ for the region }
\Omega$">|; 

$key = q/J;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$J$">|; 

$key = q/w(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="$w(t)$">|; 

$key = q/t=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img110.png"
 ALT="$t=0$">|; 

$key = q/displaystyle-deltaJ[rho,c,{f_s}];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img86.png"
 ALT="$\displaystyle - \delta J [ \rho, c, {f_s}]$">|; 

$key = q/{rho({svectorx})};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="${\rho({\svector x})}$">|; 

$key = q/{P_s{({svectorx},t)}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="${P_s {({\svector x}, t)}}$">|; 

$key = q/displaystyle-deltaJ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="$\displaystyle - \delta J$">|; 

$key = q/L;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="$L$">|; 

$key = q/t=T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img114.png"
 ALT="$t=T$">|; 

$key = q/{f_s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="${f_s}$">|; 

$key = q/displaystyle{{svectorx}~|~zgeq0~~mbox{and}~parallel{svectorx}parallelleqT{V_{max}}}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$\displaystyle \{ {\svector x}&nbsp;\vert&nbsp; z \geq 0&nbsp;&nbsp; \mbox{and}&nbsp;
\parallel {\svector x}\parallel \leq T {V_{max}}\}.$">|; 

$key = q/displaystyle<q_1|q_2>_q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="$\displaystyle &lt; q_1 \vert q_2 &gt;_q$">|; 

$key = q/displaystyle-{sum_{r,s}}{{int_0^T}!{int_{Omega}}}{delta({svectorx}-{{svectorx}_r{partialz}}{delta{P_s{({svectorx},t)}}}{~d{svectorx},w(t),dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="591" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img87.png"
 ALT="$\displaystyle - {\sum_{r,s}}{{\int_0^T}\! {\int_{\Omega}}}{\delta ({\svector x}...
...r{\partial z}}{\delta {P_s {({\svector x}, t)}}}{&nbsp;d {\svector x}  w(t)   d t}$">|; 

$key = q/displaystyle-{{int_0^T}!{int_{Omega}}}delta{rho({svectorx})}left[{1over{{c({svec}cdot{svectornabla}right]{q{({svectorx},t)}}{~d{svectorx},dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="582" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img108.png"
 ALT="$\displaystyle - {{\int_0^T}\! {\int_{\Omega}}}\delta {\rho({\svector x})}\left ...
...}\cdot {\svector \nabla}\right ] {q {({\svector x}, t)}}{&nbsp;d {\svector x}  d t}$">|; 

$key = q/T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$T$">|; 

$key = q/displaystyle{deltaJoverdelta{rho({svectorx})}}={int_0^T}left[{1over{{c({svectorx({svectorx},t)}}cdot{svectornabla}right]{q{({svectorx},t)}}dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="548" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img120.png"
 ALT="$\displaystyle {\delta J \over \delta {\rho({\svector x})}} =
{\int_0^T}\left [ ...
...s {({\svector x}, t)}}\cdot {\svector \nabla}\right ] {q {({\svector x}, t)}}dt$">|; 

$key = q/displaystyle{deltaJoverdelta{{f_s}(t)}}=left.{q{({svectorx},t)}}right|_{mathbf{x}=mathbf{0}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img124.png"
 ALT="$\displaystyle {\delta J \over \delta {{f_s}(t)}} =
\left. {q {({\svector x}, t)}}\right \vert _{\mathbf{x} = \mathbf{0}}$">|; 

$key = q/1slashc^2rho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="$1/c^2 \rho$">|; 

$key = q/displaystyle{int_{Omega}}a{{svectornabla}cdot}{svectorb}{~d{svectorx}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img98.png"
 ALT="$\displaystyle {\int_{\Omega}}a {{\svector \nabla}\cdot}{\svector b}{&nbsp;d {\svector x}}$">|; 

$key = q/L_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="$L_2$">|; 

1;