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


$key = q/w({svectorx}'',f);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="$ w( {\svector x}'' ,f)$">|; 

$key = q/s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$ s$">|; 

$key = q/-i2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$ - i 2 \pi$">|; 

$key = q/displaystyleDeltaG({svectorx},{svectorx}'',f)=iiint8pi^2f^2s({svectorx}')G({svecG({svectorx}',{svectorx}'',f)Deltas({svectorx}')d{svectorx}',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="530" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$\displaystyle \Delta G ({\svector x}, {\svector x}'' ,f)=
\iiint 8 \pi^2 f^2 s(...
...( {\svector x}' , {\svector x}'' ,f)\Delta s( {\svector x}' ) d {\svector x}' ,$">|; 

$key = q/displaystyle8pi^2f^2s({svectorx}')frac{u({svectorx},{svectorx}')u({svectorx}',{stau({svectorx}',{svectorx}'')-tau({svectorx},{svectorx}'')]},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="581" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="$\displaystyle 8 \pi^2 f^2 s ( {\svector x}' )
\frac{ u ({\svector x}, {\svector...
... ( {\svector x}' , {\svector x}'' )- \tau ({\svector x}, {\svector x}'' )] \} ,$">|; 

$key = q/displaystyle-2s({svectorx}')frac{u({svectorx},{svectorx}')u({svectorx}',{svectorectorx}',{svectorx}'')+tau({svectorx},{svectorx}'')],{rmand~};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="535" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$\displaystyle -2 s ( {\svector x}' )
\frac{ u ({\svector x}, {\svector x}' )u (...
...tor x}' , {\svector x}'' )+ \tau ({\svector x}, {\svector x}'' )] , {\rm and&nbsp;}$">|; 

$key = q/displaystyle-2s({svectorx}')u({svectorx},{svectorx}')u({svectorx}',{svectorx}_0)-tau({svectorx}',{svectorx}_0)+tau({svectorx},{svectorx}_0)].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="495" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="$\displaystyle -2 s( {\svector x}' ) u ({\svector x}, {\svector x}' )u ( {\svect...
...au ( {\svector x}' , {\svector x}_0) + \tau ( {\svector x}, {\svector x}_0) ] .$">|; 

$key = q/u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="$ u$">|; 

$key = q/displaystylefrac{partialtheta({svectorx},{svectorx}'',f)}{partials({svectorx}')};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="$\displaystyle \frac{\partial \theta ({\svector x}, {\svector x}'' ,f)}{\partial s( {\svector x}' )}$">|; 

$key = q/displaystyle{rmor}~~4pi^2f^2s({svectorx})^2p({svectorx},f)+nabla^2p({svectorx},f);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="$\displaystyle {\rm or }&nbsp;&nbsp; 4 \pi^2 f^2 s({\svector x})^2 p ( {\svector x}, f )+ \nabla^2p ( {\svector x}, f )$">|; 

$key = q/displaystyle=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="$\displaystyle =$">|; 

$key = q/displaystylefrac{partialG({svectorx},{svectorx}'',f)}{partials({svectorx}')},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="$\displaystyle \frac{\partial G ({\svector x}, {\svector x}'' ,f)}{\partial s( {\svector x}' )} ,$">|; 

$key = q/displaystyletheta({svectorx},{svectorx}'',f);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$\displaystyle \theta ({\svector x}, {\svector x}'' ,f)$">|; 

$key = q/displaystylelogG({svectorx},{svectorx}'',f),{~rmand};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="$\displaystyle \log G ({\svector x}, {\svector x}'' ,f), {&nbsp; \rm and }$">|; 

$key = q/{svectorx}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="$ {\svector x}' $">|; 

$key = q/displaystylev({svectorx})^{-2}ddot{p}({svectorx},t)-nabla^2p({svectorx},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="$\displaystyle v({\svector x})^{-2} \ddot{p} ( {\svector x}, t )
- \nabla^2p ( {\svector x}, t )$">|; 

$key = q/displaystylefrac{partialG({svectorx},{svectorx}'',f)}{partials({svectorx}')}=8pix}')G({svectorx},{svectorx}',f)G({svectorx}',{svectorx}'',f).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="401" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="$\displaystyle \frac{\partial G ({\svector x}, {\svector x}'' ,f)}{\partial s( {...
...x}' ) G ({\svector x}, {\svector x}' ,f)G ( {\svector x}' , {\svector x}'' ,f).$">|; 

$key = q/p({svectorx},f);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="$ p ( {\svector x}, f )$">|; 

$key = q/displaystyle{rmso~that}~~p({svectorx},f);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="$\displaystyle {\rm so&nbsp; that }&nbsp;&nbsp; p ( {\svector x}, f )$">|; 

$key = q/displaystyleiiintfrac{partialtheta({svectorx},{svectorx}'',f)}{partials({svector)}G({svectorx},{svectorx}'',f)w({svectorx}'',f)d{svectorx}'',;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="340" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="$\displaystyle \iiint
\frac{\partial \theta ({\svector x}, {\svector x}'' ,f)}{\...
... )} G ({\svector x}, {\svector x}'' ,f)w( {\svector x}'' ,f) d {\svector x}'' ,$">|; 

$key = q/{svectornabla};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="$ {\svector \nabla}$">|; 

$key = q/displaystylefrac{partialG({svectorx},{svectorx}'',t)}{partials({svectorx}')};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$\displaystyle \frac{\partial G ({\svector x}, {\svector x}'' ,t)}{\partial s( {\svector x}' )}$">|; 

$key = q/displaystyle-w({svectorx},f).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$\displaystyle - w ( {\svector x}, f ).$">|; 

$key = q/displaystylephi({svectorx},f);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="$\displaystyle \phi ( {\svector x}, f )$">|; 

$key = q/displaystyle8pi^2f^2s({svectorx}')frac{G({svectorx},{svectorx}',f)G({svectorx}',{svectorx}'',f)}{G({svectorx},{svectorx}'',f)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="$\displaystyle 8 \pi^2 f^2 s( {\svector x}' ) \frac{G ({\svector x}, {\svector x...
...)G ( {\svector x}' , {\svector x}'' ,f)}{G ({\svector x}, {\svector x}'' ,f)} .$">|; 

$key = q/displaystyleiiint8pi^2f^2s({svectorx}')G({svectorx},{svectorx}',f)G({svectorx}',{svectorx}'',f)w({svectorx}'',f)d{svectorx}''.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="418" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$\displaystyle \iiint 8 \pi^2 f^2 s( {\svector x}' ) G ({\svector x}, {\svector ...
...)G ( {\svector x}' , {\svector x}'' ,f)w( {\svector x}'' ,f) d {\svector x}'' .$">|; 

$key = q/displaystyleleft|{svectornabla}v({svectorx})right|ll2pileft|fright|.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="$\displaystyle \left \vert {\svector \nabla}v ({\svector x}) \right\vert \ll 2 \pi \left\vert f \right\vert .$">|; 

$key = q/displaystyleG({svectorx},{svectorx}',t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="$\displaystyle G ({\svector x}, {\svector x}' ,t)$">|; 

$key = q/displaystyle{rmwhere~~}frac{partialtheta({svectorx},{svectorx}'',f)}{partials({svectorx}')};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="$\displaystyle {\rm where&nbsp;&nbsp;}
\frac{\partial \theta ({\svector x}, {\svector x}'' ,f)}{\partial s( {\svector x}' )}$">|; 

$key = q/displaystyleintfrac{partialtheta({svectorx},{svectorx}_0,t-t')}{partials({svectorx}')}p({svectorx},t')dt';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="$\displaystyle \int
\frac{\partial \theta ( {\svector x}, {\svector x}_0, t - t')}{\partial s( {\svector x}' )} p ({\svector x},t') dt'$">|; 

$key = q/displaystyleequiv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="$\displaystyle \equiv$">|; 

$key = q/displaystyleiiintG({svectorx},{svectorx}',f)w({svectorx}',f)d{svectorx}'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$\displaystyle \iiint G ({\svector x}, {\svector x}' ,f)w ( {\svector x}' ,f ) d {\svector x}' .$">|; 

$key = q/displaystyleiiint-2s({svectorx}')u({svectorx},{svectorx}')u({svectorx}',{svectorx}',{svectorx}'')+tau({svectorx},{svectorx}'')]d{svectorx}''.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="599" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="$\displaystyle \iiint -2 s ( {\svector x}' ) u ({\svector x}, {\svector x}' )u (...
...}' , {\svector x}'' )+ \tau ({\svector x}, {\svector x}'' )] d {\svector x}'' .$">|; 

$key = q/displaystylenabla^2u({svectorx},{svectorx}');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="$\displaystyle \nabla^2u ({\svector x}, {\svector x}' )$">|; 

$key = q/displaystyle4pi^2f^2s({svectorx})^2DeltaG({svectorx},{svectorx}',f)+nabla^2DeltaG({svectorx},{svectorx}',f);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$\displaystyle 4 \pi^2 f^2 s({\svector x})^2 \Delta G ({\svector x}, {\svector x}' ,f)+ \nabla^2\Delta G ({\svector x}, {\svector x}' ,f)$">|; 

$key = q/displaystylefrac{partialp({svectorx},f)}{partials({svectorx}')};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="$\displaystyle \frac{\partial p ( {\svector x}, f )}{\partial s( {\svector x}' )}$">|; 

$key = q/displaystylew({svectorx},t);;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$\displaystyle w ( {\svector x}, t );$">|; 

$key = q/displaystyleiiintfrac{partialG({svectorx},{svectorx}'',f)}{partials({svectorx}')}w({svectorx}'',f)d{svectorx}'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$\displaystyle \iiint
\frac{\partial G ({\svector x}, {\svector x}'' ,f)}{\partial s( {\svector x}' )} w( {\svector x}'' ,f) 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="|."$dir".q|img7.png"
 ALT="$ {\svector x}$">|; 

$key = q/G;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$ G$">|; 

$key = q/displaystylefrac{partialtheta({svectorx},{svectorx}'',f)}{partials({svectorx}')}G({svectorx},{svectorx}'',f);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="$\displaystyle \frac{\partial \theta ({\svector x}, {\svector x}'' ,f)}{\partial s( {\svector x}' )} G ({\svector x}, {\svector x}'' ,f)$">|; 

$key = q/{svectorx}'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="$ {\svector x}'' $">|; 

$key = q/displaystylefrac{partialp({svectorx},t)}{partials({svectorx}')};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="$\displaystyle \frac{\partial p ( {\svector x}, t )}{\partial s( {\svector x}' )}$">|; 

$key = q/displaystyleintiiintfrac{partialG({svectorx},{svectorx}'',t)}{partials({svectorx}')}w({svectorx}'',t-t')d{svectorx}''dt';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$\displaystyle \int\iiint
\frac{\partial G ({\svector x}, {\svector x}'' ,t)}{\partial s( {\svector x}' )} w ( {\svector x}'' , t - t' ) d {\svector x}'' dt'$">|; 

$key = q/p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$ p$">|; 

$key = q/u({svectorx},{svectorx}');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="$ u ({\svector x}, {\svector x}' )$">|; 

$key = q/displaystylep({svectorx},t)=iiintu({svectorx},{svectorx}')w[{svectorx}',t-tau({svectorx},{svectorx}')]d{svectorx}'.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="$\displaystyle p ( {\svector x}, t )= \iiint u ({\svector x}, {\svector x}' )w [ {\svector x}' , t - \tau ({\svector x}, {\svector x}' )] d {\svector x}' .$">|; 

$key = q/displaystyle-delta({svectorx}-{svectorx}');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="$\displaystyle - \delta ( {\svector x}- {\svector x}' )$">|; 

$key = q/f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$ f$">|; 

$key = q/displaystyle{rm~and~}2{svectornabla}tau({svectorx},{svectorx}')cdot{svectornabla}u({svectorx},{svectorx}');MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="$\displaystyle {\rm&nbsp; and&nbsp; }
2 {\svector \nabla}\tau ({\svector x}, {\svector x}' )\cdot {\svector \nabla}u ({\svector x}, {\svector x}' )$">|; 

$key = q/displaystylelogp({svectorx},f);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="$\displaystyle \log p ( {\svector x}, f )$">|; 

$key = q/displaystyle-2s({svectorx}')u({svectorx},{svectorx}')u({svectorx}',{svectorx}'')-tau({svectorx}',{svectorx}'')+tau({svectorx},{svectorx}'')].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="492" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="$\displaystyle -2 s ( {\svector x}' )
u ({\svector x}, {\svector x}' )u ( {\svec...
...tau ( {\svector x}' , {\svector x}'' )+ \tau ({\svector x}, {\svector x}'' )] .$">|; 

$key = q/displaystylefrac{partialphi({svectorx},f)}{partials({svectorx}')};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="$\displaystyle \frac{\partial \phi ( {\svector x}, f )}{\partial s( {\svector x}' )}$">|; 

$key = q/w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$ w$">|; 

$key = q/displaystyleu({svectorx},{svectorx}')exp[-i2piftau({svectorx},{svectorx}')].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="$\displaystyle u ({\svector x}, {\svector x}' )\exp [ -i 2 \pi f \tau ({\svector x}, {\svector x}' )] .$">|; 

$key = q/displaystylefrac{partialtheta({svectorx},{svectorx}'',t)}{partials({svectorx}')};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="$\displaystyle \frac{\partial \theta ({\svector x}, {\svector x}'' ,t)}{\partial s( {\svector x}' )}$">|; 

$key = q/displaystylew({svectorx},f)=w(f)delta({svectorx}-{svectorx}_0),{rm~~then~}p({svectorx},f)=G({svectorx},{svectorx}_0,f)w(f).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="482" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="$\displaystyle w ( {\svector x}, f )= w(f) \delta ( {\svector x}- {\svector x}_0), {\rm&nbsp;&nbsp; then&nbsp;}
p ( {\svector x}, f )= G ({\svector x},{\svector x}_0,f) w(f) .$">|; 

$key = q/tau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="$ \tau$">|; 

$key = q/displaystyle-8pi^2f^2s({svectorx})G({svectorx},{svectorx}',f)Deltas({svectorx}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$\displaystyle -8 \pi^2 f^2 s({\svector x}) G ({\svector x}, {\svector x}' ,f)\Delta s({\svector x}),$">|; 

$key = q/ddot{w}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="$ \ddot{w}(t)$">|; 

$key = q/displaystyle-2s({svectorx}')frac{u({svectorx},{svectorx}')u({svectorx}',{svector)-tau({svectorx}',{svectorx}_0)+tau({svectorx},{svectorx}_0)];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="509" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="$\displaystyle -2 s( {\svector x}' ) \frac{ u ({\svector x}, {\svector x}' )u ( ...
...\tau ( {\svector x}' , {\svector x}_0) + \tau ( {\svector x}, {\svector x}_0) ]$">|; 

$key = q/displaystylefrac{iiint8pi^2f^2s({svectorx}')G({svectorx},{svectorx}',f)G({svecto{svectorx},{svectorx}''',f)w({svectorx}''',f)d{svectorx}'''}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="409" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="$\displaystyle \frac
{\iiint 8 \pi^2 f^2 s( {\svector x}' ) G ({\svector x}, {\s...
... ({\svector x}, {\svector x}''' ,f)w( {\svector x}''' ,f) d {\svector x}''' } .$">|; 

$key = q/v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="$ v$">|; 

$key = q/displaystyleu({svectorx},{svectorx}')delta[t-tau({svectorx},{svectorx}')];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="$\displaystyle u ({\svector x}, {\svector x}' )\delta[ t - \tau ({\svector x}, {\svector x}' )]$">|; 

$key = q/displaystyle+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$\displaystyle +$">|; 

$key = q/displaystylenabla^2G({svectorx},{svectorx}',f)=-delta({svectorx}-{svectorx}'),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$\displaystyle \nabla^2G ({\svector x}, {\svector x}' ,f)=
- \delta ( {\svector x}- {\svector x}' ),$">|; 

$key = q/displaystyleu({svectorx},{svectorx}')nabla^2tau({svectorx},{svectorx}')=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="$\displaystyle u ({\svector x}, {\svector x}' )\nabla^2\tau ({\svector x}, {\svector x}' )= 0 .$">|; 

$key = q/displaystyleleft|{svectornabla}tau({svectorx},{svectorx}')right|^2=s({svectorx})^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="$\displaystyle \left \vert {\svector \nabla}\tau ({\svector x}, {\svector x}' )\right \vert^2 = s ({\svector x})^2 ,$">|; 

$key = q/displaystyle4pi^2f^2s({svectorx})^2G({svectorx},{svectorx}',f);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$\displaystyle 4 \pi^2 f^2 s({\svector x})^2 G ({\svector x}, {\svector x}' ,f)$">|; 

$key = q/nabla^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="$ \nabla^2$">|; 

$key = q/displaystyle{rm~or~}G({svectorx},{svectorx}',f);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="$\displaystyle {\rm&nbsp; or&nbsp; }
G ({\svector x}, {\svector x}' ,f)$">|; 

1;