{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
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{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 
2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 
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0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q1" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "r:=[cos(t),sin(t),cos(t)^2]:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rd:=diff(r,t): # veloci
ty" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "rdd:=diff(rd,t):   # \+
total acceleration" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rddd:
=diff(rdd,t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "mag:=proc(
r) simplify(sqrt(add(r[i]^2,i=1..3))) end:" }}}{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 83 "curv:=sqrt((dot(rd,rd)*
dot(rdd,rdd)-dot(rd,rdd)^2)/(dot(rd,rd)^3)): eval(%,t=Pi/4);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%!\"\"F%#\"\"\"\"\"#F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "cross:=proc(r,q) simplify([r[2]*q[3
]-r[3]*q[2],r[3]*q[1]-r[1]*q[3],r[1]*q[2]-r[2]*q[1]]) end:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "tors:=dot(rd,cross(rdd,rddd))/(dot(
rd,rd)*dot(rdd,rdd)-dot(rd,rdd)^2): eval(%,t=Pi/4);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6##\"\"$\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "speed:=mag(rd):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "sc
alm:=proc(v,s) [seq(simplify(s*v[i]),i=1..3)] end:" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 44 "u:=scalm(rd,1/speed):  # unit tangent vector" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "eval(scalm(diff(u,t),1/curv/
speed),t=Pi/4): simplify(%); # principal normal" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%,$*&\"\"#!\"\"F&#\"\"\"F&F'F$\"\"!" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 2 "Q2" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 23 "restart:   # quite easy" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 63 "A:=int(int(1,y=-sqrt(4-x^2)..sqrt(4-x^2)),x=-1..1/
2): evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nFT0e!\"*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "int(int(x,y=-sqrt(4-x^2)..sq
rt(4-x^2)),x=-1..1/2)/A: evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
$!+.T9sB!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "int(int(y,y=
-sqrt(4-x^2)..sqrt(4-x^2)),x=-1..1/2)/A: evalf(%);  #of course" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"!F$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 80 "M*int(int(x^2+y^2-(x+y)^2/2,y=-sqrt(4-x^2)..sqrt(4-x^
2)),x=-1..1/2)/A: evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&$\"
+b$\\eY(!#5\"\"\"%\"MGF(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q3" 
}{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "int(rho*(1-rho^2),rho=
0..1)*2*Pi;   # Volume" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#!\"
\"%#PiG\"\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "r:=[u,v,
1-u^2-v^2]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "cross:=proc(
r,q);[r[2]*q[3]-r[3]*q[2],r[3]*q[1]-r[1]*q[3],r[1]*q[2]-r[2]*q[1]] end
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "mag:=proc(q);sqrt(add(
q[i]^2,i=1..3))end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "mag(
cross(diff(r,u),diff(r,v)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$,(\"
\"\"F%*&\"\"%F%)%\"uG\"\"#F%F%*&F'F%)%\"vGF*F%F%#F%F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "int(rho*sqrt(1+4*rho^2),rho=0..1)*2
*Pi; # top surface" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#\"\"\",
&*(\"\"&F&\"#7!\"\"F)#F&F%F&#F&F*F+F&%#PiGF&F&" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 56 "int(rho,rho=0..1)*2*Pi;  #bottom surface (circ
ular disk)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#PiG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(%+%%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+ah+s%)!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q
4" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rest
art:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "eq:=x^4*diff(y(x),x
,x)+2*x^2*(1+x)*diff(y(x),x)+y(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#eqG,(*&)%\"xG\"\"%\"\"\"-%%diffG6$-%\"yG6#F(-%\"$G6$F(\"\"#F*F***F4
F*)F(F4F*,&F*F*F(F*F*-F,6$F.F(F*F*F.F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "T:=1/x-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG,&*
&\"\"\"F'%\"xG!\"\"F'F'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
125 "eval(eq,\{diff(y(x),x,x)=diff(y(t),t,t)*diff(T,x)^2+diff(y(t),t)*
diff(T,x,x),\ndiff(y(x),x)=diff(y(t),t)*diff(T,x),y(x)=y(t)\});" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&)%\"xG\"\"%\"\"\",&*&-%%diffG6$-%
\"yG6#%\"tG-%\"$G6$F1\"\"#F(F&!\"%F(*(F5F(-F,6$F.F1F(F&!\"$F(F(F(*(F5F
(,&F(F(F&F(F(F8F(!\"\"F.F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "eq1:=simplify(eval(%,x=1/(t+1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$eq1G,(-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F,\"\"#\"\"\"*&F0F1-F'6$F)F
,F1!\"\"F)F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "therefore     y(
t) = A.exp(t) + B.t.exp(t)      which implies  y(x) = A.exp(1/x-1) + B
.(1/x-1).exp(1/x-1)   equivalent to  A.exp(1/x) + B.exp(1/x) /x" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q5" }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "restart:  # quit
e easy (path independent!!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
93 "findf:=proc(g) local h;h:=int(g[1],x);h:=h+int(g[2]-diff(h,y),y);h
+int(g[3]-diff(h,z),z) end:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 108 "curl:=proc(g) simplify([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 68 "g:=[y*cos(x*y)+z*exp(x*z),x*cos(x*y
)-2*y*z^3, x*exp(x*z)-3*y^2*z^2]:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "curl(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F
$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f:=findf(g);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"fG,(-%$sinG6#*&%\"xG\"\"\"%\"yGF+F+-%$ex
pG6#*&F*F+%\"zGF+F+*&)F,\"\"#F+)F1\"\"$F+!\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 57 "eval(f,\{x=2,y=6,z=-3\})-eval(f,\{x=3,y=-1,z=4
\}):  evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+W=><;!\"%" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q6" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 29 "r:=[v*cos(u),v*sin(u),v^2/2]:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 83 "cross:=proc(r,q);[r[2]*q[3]-r[3]*q[2],r[3]*q[
1]-r[1]*q[3],r[1]*q[2]-r[2]*q[1]] end:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "mag:=proc(q);sqrt(add(q[i]^2,i=1..3))end:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "simplify(mag(cross(diff(r,u),diff(r
,v))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$*&)%\"vG\"\"#\"\"\",&*$F%
F(F(F(F(F(#F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "A:=int(v
*sqrt(1+v^2),v=sqrt(2)..sqrt(6))*2*Pi:  evalf(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+#R%f!z#!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 90 "int(int(r[3]*v*sqrt(1+v^2),u=0..2*Pi),v=sqrt(2)..sqrt(6))/A:  \+
evalf(%);  # the z component" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+cv
(z1#!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "# the x and y c
omponents are clearly equal to 0" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 81 "M/A*int(v^3*sqrt(1+v^2),v=sqrt(2)..sqrt(6))*2*Pi:  ev
alf(%);  # moment of inertia" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&$
\"+6^&f8%!\"*\"\"\"%\"MGF(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "Q7
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rest
art:  #quite easy" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "eq:=x^
2*diff(y(x),x,x)-3*x*diff(y(x),x)+(x^2-5)*y(x)-5+x^2;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#eqG,,*&)%\"xG\"\"#\"\"\"-%%diffG6$-%\"yG6#F(-%
\"$G6$F(F)F*F**(\"\"$F*F(F*-F,6$F.F(F*!\"\"*&,&*$F'F*F*\"\"&F8F*F.F*F*
F<F8F;F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "simplify(eval(e
q,y(x)=x^2*u(x)-1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"xG\"\"#
\"\"\",**&\"\"*F'-%\"uG6#F%F'!\"\"*&F%F'-%%diffG6$F+F%F'F'*&F$F'-F16$F
+-%\"$G6$F%F&F'F'*&F$F'F+F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 39 "# Bessel equation with  n=3.  Solution:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 80 "#  u(x) = A*J_3(x) + B*Y_3(x)   implying  y(x)
 = A*x^2*J_3(x) + B*x^2*Y_3(x) - 1" }}}}{MARK "55 0 0" 21 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }