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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Sec. 9.7" }{MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Q10:" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "int(y^2+z^2,x=sqrt(y^2+z^2).
.h): factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"hG\"\"\"*$,&
*$)%\"yG\"\"#F&F&*$)%\"zGF,F&F&#F&F,!\"\"F&F(F&" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 58 "2*Pi*int((h-rho)*rho^2*rho,rho=0..h);  #switch
ing to polar" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#5!\"\"%#PiG\"\"
\"%\"hG\"\"&F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Set 9.8" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Q4: Since div
(F) = 10, the answer is 10 times the volume of the cone:" }{MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Pi*3^2*10;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#!*\"\"\"%#PiGF&F&" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 37 "Q8: We start with the dz integration:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "int(x+y
,z=x^2+y^2..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\",(\"
\"%F&*$)F%\"\"#F&!\"\"*$)%\"yGF+F&F,F&F&*&F/F&F'F&F&" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 74 "Then integrate over the upper half disk of radi
us 2, in polar coordinates:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 73 "int(int((4-rho^2)*rho*(cos(theta)+sin(theta))*
rho,rho=0..2),theta=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$G\"
\"#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Set 9.9" }{MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Q2:" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "dot:=proc(r,q);simplify(add(
r[i]*q[i],i=1..3)) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "
mag:=proc(q);sqrt(add(q[i]^2,i=1..3))end:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 52 "plugin:=proc(g,r);eval(g,\{x=r[1],y=r[2],z=r[3]\})e
nd:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "cross:=proc(r,q);[r[
2]*q[3]-r[3]*q[2],r[3]*q[1]-r[1]*q[3],r[1]*q[2]-r[2]*q[1]] end:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "curl:=proc(g);[diff(g[3],y)-
diff(g[2],z),diff(g[1],z)-diff(g[3],x),diff(g[2],x)-diff(g[1],y)]end:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "F:=[y^2,-x^2,0]:  r:=[v
*cos(u),v*sin(u),0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "g:=
curl(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG7%\"\"!F&,&*&\"\"#\"
\"\"%\"xGF*!\"\"*&F)F*%\"yGF*F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 42 "aux:=simplify(cross(diff(r,u),diff(r,v)));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$auxG7%\"\"!F&,$%\"vG!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 47 "-int(int(dot(plugin(g,r),aux),u=0..Pi),v=0..2);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!#K\"\"$" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 9 "Directly:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 69 "r1:=[2*cos(t),2*sin(t),0]: int(dot(plugin(F,r1),di
ff(r1,t)),t=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!#K\"\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "r2:=[t,0,0]: int(dot(plugin(
F,r2),diff(r2,t)),t=-2..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Q6:" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "F:=[y^2,z^2,x^2]:  r:=[cos(u
)*sin(v),sin(u)*sin(v),1+cos(v)]:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "g:=curl(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG
7%,$*&\"\"#\"\"\"%\"zGF)!\"\",$*&F(F)%\"xGF)F+,$*&F(F)%\"yGF)F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "aux:=simplify(cross(diff(r,u
),diff(r,v)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$auxG7%*&-%$cosG6#
%\"uG\"\"\",&F+!\"\"*$)-F(6#%\"vG\"\"#F+F+F+*&-%$sinGF)F+F,F+,$*&-F6F1
F+F0F+F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "int(int(dot(plu
gin(g,r),aux),u=0..Pi),v=Pi/2..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
##!\"%\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Directly:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "r1:=[co
s(t),sin(t),1]: int(dot(plugin(F,r1),diff(r1,t)),t=0..Pi);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6##!\"%\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "r2:=[cos(t),0,1+sin(t)]: int(dot(plugin(F,r2),diff(r2
,t)),t=Pi..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 7 "Review:" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 4 "Q22:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 51 "F:=[-z,5*x,-y]:  r:=[2*cos(t),2*sin(t),2*
cos(t)+2]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "int(dot(plugi
n(F,r),diff(r,t)),t=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&
\"#C\"\"\"%#PiGF&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Q38:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "F:=[sin
(x),z,y]:   r:=[v,2*cos(u),2*sin(u)]:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "aux:=simplify(cross(diff(r,u),diff(r,v)));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$auxG7%\"\"!,$*&\"\"#\"\"\"-%$cosG6#%\"uGF
*F*,$*&F)F*-%$sinGF-F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 
"int(int(dot(plugin(F,r),aux),u=0..Pi/2),v=-1/2..1/2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"\"%" }}}}{MARK "30 0 0" 9 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }