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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "r:=[(2+cos(u))*cos(v),(2+cos
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1 0 63 "int(int(int(r^4*sin(theta)^3,r=0..1),theta=0..Pi),phi=0..2*Pi)
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "curl:=proc(g);[diff(g[3],y)-
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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "dot:=proc(r,q);simplify
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{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG7%*&%\"xG\"\"\",&*$)%\"yG\"\"#F(
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "r:=[sin(v)*cos(u),sin(v)*sin
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" {MPLTEXT 1 0 37 "simplify(cross(diff(r,u),diff(r,v)));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7%*&,&\"\"\"!\"\"*$)-%$cosG6#%\"vG\"\"#F&F&F&-F+
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int(int(int(div(g),z=-sqrt(1-x^2-y^2)..sqrt(1-x^2-y^2)),y=-sqrt(1-x^2)
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\"\"#\"\"\"%#PiGF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "r:=
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{MPLTEXT 1 0 41 "r:=[u,v,u^2+v^2]:     #area of paraboloid" }}}{EXCHG 
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}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,(\"\"\"F%*&\"\"%F%)%\"uG\"\"#F%F%*
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{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "r:=[u,u
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ross(diff(r,u),diff(r,v));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*&\"
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0 68 "int(int(dot(plugin(g,r),cross(diff(r,u),diff(r,v))),u=0..2),v=0.
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" {TEXT -1 2 "3)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "r:=[u,v,1-u-v]:  g:=[x^2,0,3*y^2]:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 27 "cross(diff(r,u),diff(r,v));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7%\"\"\"F$F$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "int(int(dot(plugin(g,r),cross(diff(r,u),diff(r,v))),u
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "5)" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "r:=[(a+b*cos(u))*cos(v),(a+b*cos(u)
)*sin(v),b*sin(u)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "int(
int(plugin(x^2+y^2,r)*mag(cross(diff(r,u),diff(r,v))),u=0..2*Pi),v=0..
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^2/a/b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.\"\"#!\"\"%\"MG\"\"\"*
&)%\"bGF%F(),&%\"aGF(F+F(F%F(#F(F%,&*&\"\"$F(F*F(F(*&F%F()F.F%F(F(F(F-
F&F+F&F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "6)" }{MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plugin:=proc(g,r);eval(g,\{x=r[1],y
=r[2],z=r[3]\})end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "r:=[
(a+rho*cos(u))*cos(v),(a+rho*cos(u))*sin(v),rho*sin(u)]:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(linalg): " }}{PARA 7 "" 1 "" 
{TEXT -1 80 "Warning, the protected names norm and trace have been red
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"det(augment(diff(r,u),diff(r,v),diff(r,rho))):   simplify(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%$rhoG\"\"\",&%\"aGF&*&F%F&-%$cosG
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nt(int(plugin(x^2+y^2,r)*rho*(a+rho*cos(u)),u=0..2*Pi),v=0..2*Pi),rho=
0..b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&**\"\"#\"\"\")%\"aG\"\"$F&
)%#PiGF%F&)%\"bGF%F&F&*&#F)F%F&*(F(F&F*F&)F-\"\"%F&F&F&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "simplify(M*%/int(int(int(rho*(a+rho
*cos(u)),u=0..2*Pi),v=0..2*Pi),rho=0..b));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*(\"\"%!\"\"%\"MG\"\"\",&*&F%F()%\"aG\"\"#F(F(*&\"\"$
F()%\"bGF-F(F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "7)    COM:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "int(int(int(x,x=0..1-y-
z),y=0..1-z),z=0..1)/int(int(int(1,x=0..1-y-z),y=0..1-z),z=0..1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"%" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 16 "MOI w/r [1,1,1]:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 70 "6*M*int(int(int(x^2+y^2+z^2-(x+y+z)^2/3,x=0
..1-y-z),y=0..1-z),z=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#
5!\"\"%\"MG\"\"\"F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "8)" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "r:=[u,v
,-sqrt(1-u^2-v^2)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "mag(
cross(diff(r,u),diff(r,v))):   simplify(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*$,$*&\"\"\"F&,(F&!\"\"*$)%\"uG\"\"#F&F&*$)%\"vGF,F&F&F
(F(#F&F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(h<1);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "2*Pi*int(1/sqrt(1-rho^2)*r
ho,rho=0..sqrt(h*(2-h)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#
\"\"\"%#PiGF&%#h|irGF&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "alter
nately (don't do!!)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "r:=[sin(v)*cos(u),sin(v)*sin(u),cos(v)]:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "mag(cross(diff(r,u),diff(r,v))): si
mplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%%csgnG6#-%$sinG6#%\"
vG\"\"\"F'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "2*Pi*int(si
n(v),v=2*Pi-arccos(1-h)..2*Pi);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
$*(\"\"#\"\"\"%#PiGF&%\"hGF&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
56 "Volume:           z-thickness is   sqrt(1-x^2-y^2)-(1-h)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "2*Pi*int((sqrt(1-rho^2)-1+h)
*rho,rho=0..sqrt(h*(2-h)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"
\"#\"\"\"%#PiGF&,&*&#F&F%F&*$)%#h|irGF%F&F&F&*&#F&\"\"'F&*$)F-\"\"$F&F
&!\"\"F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "9)" }{MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g:=[x+y,2*x-z,y+z]:" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Using Stokes:" }{MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "r:=[u,v,6-3*u-2*v]:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "int(int(dot(plugin(curl(g),
r),cross(diff(r,u),diff(r,v))),u=0..2-2/3*v),v=0..3);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"#@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Direct
ly:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "r
:=evalm([2,0,0]+[-2,3,0]*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG
K%'vectorG6#7%,&\"\"#\"\"\"*&F*F+%\"tGF+!\"\",$*&\"\"$F+F-F+F+\"\"!Q(p
print16\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "int(dot(plugin
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}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "r:=evalm([0,3,0]+[0,-3,6]*t
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\"\"\"*&F+F,%\"tGF,!\"\",$*&\"\"'F,F.F,F,Q(pprint26\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "int(dot(plugin(g,r),[0,-3,6]),t=0..
1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#O" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 29 "r:=evalm([0,0,6]+[2,0,-6]*t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"rGK%'vectorG6#7%,$*&\"\"#\"\"\"%\"tGF,F,\"\"!,&\"\"
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1 0 38 "int(dot(plugin(g,r),[2,0,-6]),t=0..1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#!#;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "10)" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "g:=[y*z
,x*z,x*y]:     curl(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "r:=[3*cos(t),sin(t),0]:
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lugin(g,r),diff(r,t)),t=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
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