{VERSION 6 0 "IBM INTEL NT" "6.0" }
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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "r:=[cos(t),sin(t),t]
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%-%$cosG6#%\"tG-%$sinGF(F)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "rd:=diff(r,t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rdG7%,$-%$sinG6#%\"tG!\"\"-%$cosGF)
\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "rdd:=diff(rd,t);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rddG7%,$-%$cosG6#%\"tG!\"\",$-%
$sinGF)F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rddd:=dif
f(rdd,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rdddG7%-%$sinG6#%\"tG,
$-%$cosGF(!\"\"\"\"!" }}}{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 13 "rdm:=mag(rd);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$rdmG*$\"\"##\"\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "arclen:=int(mag(rd),t=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%'arclenG,$*(\"\"#\"\"\"F'#F(F'%#PiGF(F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 51 "dot:=proc(r,q);simplify(add(r[i]*q[i],i=1..3)) e
nd:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dot(rd,rdd)/dot(rd,r
d)*rrdd;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 67 "curv:=sqrt((dot(rd,rd)*dot(rdd,rdd)-dot(rd,r
dd)^2)/(dot(rd,rd)^3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%curvG#\"
\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "cross:=proc(r,
q);[r[2]*q[3]-r[3]*q[2],r[3]*q[1]-r[1]*q[3],r[1]*q[2]-r[2]*q[1]] end:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "tors:=dot(rd,cross(rdd,
rddd))/(dot(rd,rd)*dot(rdd,rdd)-dot(rd,rdd)^2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%torsG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "grad:=proc(f);[diff(f,x),diff(f,y),diff(f,z)] end:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f:=x*y^2/z^3;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"fG*(%\"xG\"\"\"%\"yG\"\"#%\"zG!\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "grad(f);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%*&%\"yG\"\"#%\"zG!\"$,$**F&\"\"\"%\"xGF+F%F+F'F(F+,$*
*\"\"$F+F,F+F%F&F'!\"%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "g:=[x^2+y^2,-x^2-y^2,x^2+y^2-z^2];" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"gG7%,&*$)%\"xG\"\"#\"\"\"F+*$)%\"yGF*F+F+,&F'!\"\"F,F0,(F'F+
F,F+*$)%\"zGF*F+F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "div:=
proc(g);diff(g[1],x)+diff(g[2],y)+diff(g[3],z) end:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 7 "div(g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,(*&\"\"#\"\"\"%\"xGF&F&*&F%F&%\"yGF&!\"\"*&F%F&%\"zGF&F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "curl:=proc(g);[diff(g[3],y)-diff(g[
2],z),diff(g[1],z)-diff(g[3],x),diff(g[2],x)-diff(g[1],y)]end:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "curl(g);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%,$*&\"\"#\"\"\"%\"yGF'F',$*&F&F'%\"xGF'!\"\",&*&F&F'F
+F'F,*&F&F'F(F'F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "curl(g
rad(f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "div(curl(g));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "div
(grad(f));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"#\"\"\"%\"xGF&%
\"zG!\"$F&**\"#7F&F'F&%\"yGF%F(!\"&F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "23 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }