# LaTeX2HTML 96.1 (Feb 5, 1996)
# Associate images original text with physical files.

$key = q/{displaymath}rmRighthalf:fracx-22leqyleqfrac2-x2hspace.2in0leqxleq2.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=418 HEIGHT=35 ALIGN=BOTTOM ALT="displaymath91" SRC="img2.gif"  > '; 
$key = q/{displaymath}J=|f_u(x,y)g_v(x,y)-f_v(x,y)g_u(x,y)|{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=395 HEIGHT=18 ALIGN=BOTTOM ALT="displaymath119" SRC="img16.gif"  > '; 
$key = q/{displaymath}J=rmabsolutevalueofdetleft[arraycccf_u;SPMamp;f_v;SPMamp;f_wg_u;SPMamp;g_v;SPMamp;g_wh_u;SPMamp;h_v;SPMamp;h_warrayright]{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=408 HEIGHT=68 ALIGN=BOTTOM ALT="displaymath129" SRC="img21.gif"  > '; 
$key = q/{displaymath}I=int_-2^0int_-(2+x)2^(x+2)2(3x+6y)^2dydx+int_0^2int_(x-2)2^(2-x)2(3x+6y)^2dydx{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=488 HEIGHT=45 ALIGN=BOTTOM ALT="displaymath95" SRC="img4.gif"  > '; 
$key = q/{displaymath}intint_R(xy)F(x,y)dA=intint_R(uv)F(f(u,v),g(u,v))JdA{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=450 HEIGHT=39 ALIGN=BOTTOM ALT="displaymath123" SRC="img18.gif"  > '; 
$key = q/{displaymath}x=f(u,v)hspace.2iny=g(u,v){displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=343 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath107" SRC="img10.gif"  > '; 
$key = q/{displaymath}rmAreaofR(xy)=|f_u(x,y)g_v(x,y)-f_v(x,y)g_u(x,y)|(rmAreaofR(uv)){displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=511 HEIGHT=18 ALIGN=BOTTOM ALT="displaymath117" SRC="img15.gif"  > '; 
$key = q/{displaymath}rmLefthalf:frac-2-x2leqyleqfracx+22hspace.2in-2leqxleq0.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=430 HEIGHT=35 ALIGN=BOTTOM ALT="displaymath93" SRC="img3.gif"  > '; 
$key = q/{displaymath}I=int_-2^2int_-2^2(2.25)u^2dudv.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=346 HEIGHT=40 ALIGN=BOTTOM ALT="displaymath137" SRC="img25.gif"  > '; 
$key = q/{displaymath}x=(u+v)2hspace.2iny=(u-v)4.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=367 HEIGHT=18 ALIGN=BOTTOM ALT="displaymath99" SRC="img6.gif"  > '; 
$key = q/{displaymath}I=intint_R(xy)(3x+6y)^2dA{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=348 HEIGHT=39 ALIGN=BOTTOM ALT="displaymath89" SRC="img1.gif"  > '; 
$key = q/{displaymath}f(u,v+dv)=x+f_vdvhspace.2inrmandhspace.2ing(u,v+dv)=y+g_vdv{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=464 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath113" SRC="img13.gif"  > '; 
$key = q/{displaymath}x=(u+v)2=f(u,v),hspace.2iny=(u-v)4=g(u,v),{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=444 HEIGHT=18 ALIGN=BOTTOM ALT="displaymath131" SRC="img22.gif"  > '; 
$key = q/{displaymath}f(u+du,v)=f(u,v)+f_u(u,v)du=x+f_u(u,v)du{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=446 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath109" SRC="img11.gif"  > '; 
$key = q/{displaymath}int_a^bf(x)dx=int_G(a)^G(b)f(g(u))g'(u)du{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=379 HEIGHT=44 ALIGN=BOTTOM ALT="displaymath105" SRC="img9.gif"  > '; 
$key = q/{displaymath}intintint_R(xyz)F(x,y,z)dV={displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=351 HEIGHT=39 ALIGN=BOTTOM ALT="displaymath125" SRC="img19.gif"  > '; 
$key = q/{displaymath}I=intint_R(uv)9u^2(14)dA{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=343 HEIGHT=39 ALIGN=BOTTOM ALT="displaymath135" SRC="img24.gif"  > '; 
$key = q/{displaymath}g(u+du,v)=g(u,v)+g_u(u,v)du=y+g_u(u,v)du{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=444 HEIGHT=17 ALIGN=BOTTOM ALT="displaymath111" SRC="img12.gif"  > '; 
$key = q/{displaymath}rmAreaofR(xy)=|f_u(x,y)g_v(x,y)-f_v(x,y)g_u(x,y)|dud{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=459 HEIGHT=18 ALIGN=BOTTOM ALT="displaymath115" SRC="img14.gif"  > '; 
$key = q/{displaymath}I=intint_R(xy)(3x+6y)^2dAneqintint_R(uv)9u^2dA{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=418 HEIGHT=39 ALIGN=BOTTOM ALT="displaymath103" SRC="img8.gif"  > '; 
$key = q/{displaymath}(3x+6y)^2=(3(u+v)2+6(u-v)4)^2=9u^2{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=426 HEIGHT=22 ALIGN=BOTTOM ALT="displaymath101" SRC="img7.gif"  > '; 
$key = q/{displaymath}J=rmabsolutevalueofdetleft[arrayccf_u;SPMamp;f_vg_u;SPMamp;g_varrayright]{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=387 HEIGHT=48 ALIGN=BOTTOM ALT="displaymath121" SRC="img17.gif"  > '; 
$key = q/{displaymath}f_u=12,hspace.1inf_v=12,hspace.1ing_u=14,hspace.1ing_v=-14.{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=419 HEIGHT=18 ALIGN=BOTTOM ALT="displaymath133" SRC="img23.gif"  > '; 
$key = q/{displaymath}u=x+2yhspace.2inv=x-2y{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=343 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath97" SRC="img5.gif"  > '; 
$key = q/{displaymath}intintint_R(uvw)F(f(u,v,w),g(u,v,w),h(u,v,w))JdV{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=441 HEIGHT=39 ALIGN=BOTTOM ALT="displaymath127" SRC="img20.gif"  > '; 

1;

