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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 30 "Kommentar \+
till labb-uppgift 2:" }{TEXT 288 59 "\nM\345nga kunde inte skriva upp \+
ekvationen f\366r str\344ckan fr\345n  " }{TEXT 257 1 "r" }{TEXT 258 
11 "(0)  till  " }{TEXT 259 1 "r" }{TEXT 260 1 "(" }{XPPEDIT 18 0 "sqr
t(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 2 ") " }{TEXT 279 4 "d\344r " }
{TEXT -1 1 " " }{TEXT 261 1 "r" }{TEXT 281 1 "(" }{TEXT 262 1 "t" }
{TEXT 282 1 ")" }{TEXT -1 2 "  " }{TEXT 280 19 "\344r torusknuten  ( (
" }{TEXT 268 1 "x" }{TEXT 273 1 "," }{TEXT 269 1 "y" }{TEXT 274 1 "," 
}{TEXT 270 1 "z" }{TEXT 275 3 ") =" }{TEXT -1 1 " " }{TEXT 266 1 "r" }
{TEXT -1 4 "(0)+" }{TEXT 267 1 "t" }{TEXT -1 1 "(" }{TEXT 263 1 "r" }
{TEXT 264 1 "(" }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT 
-1 2 ")-" }{TEXT 265 1 "r" }{TEXT 283 7 "(0)) , " }{TEXT 298 1 "t" }
{TEXT 299 14 " fr\345n 0 till 1" }{TEXT 297 4 "!!)." }{TEXT -1 1 " " }
{TEXT 276 8 "F\366r att " }{TEXT 271 5 "maple" }{TEXT 277 59 " skall r
\344kna med vektorer (koordinatvis) skall man anv\344nda " }{TEXT 272 
5 "evalm" }{TEXT 278 81 " (matrisr\344kning, matte del B!), f\366r ska
l\344rprodukten \344r det s\344krast att sriva in \"" }{TEXT 284 10 "o
rthogonal" }{TEXT 285 17 "\", annars r\344knar " }{TEXT 286 5 "maple" 
}{TEXT 287 161 " komplext (med komplexkonjugatet av andra faktorn). Sj
\344lva kraftf\344ltet \"oscillerade\" f\366r de flesta kurvorna f\366
r kraftigt s\345 att det inte gick att f\345 ut arbetet. " }}{PARA 
256 "" 0 "" {TEXT -1 97 "TIPS: L\345t maple skriva ut alla mellanresul
tat d\345 du f\345r felmedelande, d\345 hittar du oftast felet. " }}
{PARA 257 "" 0 "" {TEXT -1 96 "F\366r arbetet skulle du ritat ut \"int
egranden\", d\345 skulle du sett varf\366r du inte fick n\345got svar.
 " }}{PARA 258 "" 0 "" {TEXT -1 102 "Vill du f\345 maple att integrera
r numeriskt (utan att f\366rst f\366s\366ka att f\345 en exakt l\366sn
ing) skriv f\366rst " }{TEXT 289 3 "Int" }{TEXT -1 11 "..., sedan " }
{TEXT 290 5 "evalf" }{TEXT -1 1 "." }}{PARA 259 "" 0 "" {TEXT -1 68 "H
\344r f\345r du tv\345 typiska l\366sningar f\366r vanliga (\"sv\345ra
\") personnummer:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(l
inalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "a:=7:b:=8:q:=8:p
:=2:c:=9:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "u:=t->[(a+b*co
s(q*t))*cos(p*t),(a+b*cos(q*t))*sin(p*t),c*sin(q*t)];" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"uGR6#%\"tG6\"6$%)operatorG%&arrowGF(7%*&,&%\"a
G\"\"\"*&%\"bGF0-%$cosG6#*&%\"qGF09$F0F0F0F0-F46#*&%\"pGF0F8\"\"\"F0*&
F.F=-%$sinGF:F0*&%\"cGF0-F@F5F0F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "ut:=diff(u(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#utG7%,&*&-%$sinG6#,$%\"tG\"\")\"\"\"-%$cosG6#,$F,\"\"#F.!#k*&,&\"\"
(F.-F0F*F-F.-F)F1F.!\"#,&*&F(\"\"\"F9F=F4*&F6F=F/F=F3,$F8\"#s" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ds:=sqrt(sum(ut[k]^2,k=1..3)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dsG*$-%%sqrtG6#,(*$),&*&-%$si
nG6#,$%\"tG\"\")\"\"\"-%$cosG6#,$F2\"\"#F4!#k*&,&\"\"(F4-F6F0F3F4-F/F7
F4!\"#F9\"\"\"F4*$),&*&F.FAF?FAF:*&F<FAF5FAF9F9FAF4*$)F>F9FA\"%%=&FA" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Int(ds,t=0..2*Pi):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "langd:=evalf(%);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%&langdG$\"+bWq>W!\"(" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 48 "F:=(x,y,z)->[sin(z+y),cos(x*y*z),arctan(x+y+z
)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGR6%%\"xG%\"yG%\"zG6\"6$%)
operatorG%&arrowGF*7%-%$sinG6#,&9&\"\"\"9%F4-%$cosG6#*(9$F4F5F4F3F4-%'
arctanG6#,(F:F4F5F4F3F4F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 47 "l:=t->evalm(u(0)+evalm(t*(u(sqrt(2))-u(0)))):  " }{TEXT -1 17 "l
injens ekvation!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "tangl:=
u(sqrt(2))-u(0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "x1:=eva
lf(l(t)[1]);x2:=evalf(l(t)[2]);x3:=evalf(l(t)[3]);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#x1G,&$\"#:\"\"!\"\"\"%\"tG$!+\\%>SS#!\")" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#x2G,$%\"tG$\"+n%3u#H!\"*" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#x3G,$%\"tG$!+>hQ[&)!\"*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 44 "arbl:=dotprod(F(x1,x2,x3),tangl,orthogonal);" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%arblG,(*&-%$sinG6#,$%\"tG$\"+_w(4
i&!\"*\"\"\",&!#:F/*&,&\"\"(F/-%$cosG6#,$*$-%%sqrtG6#\"\"#\"\"\"\"\")F
?F/-F66#,$F9F=F/F/F/!\"\"*(-F66#,$*&,&$\"#:\"\"!F/F+$!+\\%>SS#!\")F/)F
+F=F>$\"+!zhC]#FOF/F3F>-F(FAF/F/*&-%'arctanG6#,&$F1FLF/F+$\"+9s6mHFOF/
-F(F7F/F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "arbete(linje):
=evalf(Int(arbl,t=0..1.41));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%'ar
beteG6#%&linjeG$\"+\")[&pu*!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
19 "Det andra exemplet:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "
C:=t->[(8+2*cos(6*t))*cos(2*t),(8+2*cos(6*t))*sin(2*t),8*sin(6*t)];" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CGR6#%\"tG6\"6$%)operatorG%&arrow
GF(7%*&,&\"\")\"\"\"-%$cosG6#,$9$\"\"'\"\"#F0-F26#,$F5F7F0*&F.\"\"\"-%
$sinGF9F0,$-F>F3F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 
"lin:=evalm(C(0)+t*(C(sqrt(2))-u(0)));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$linG-%'vectorG6#7%,&\"#5\"\"\"*&%\"tGF+,&!#:F+*&,&\"\")F+-%$c
osG6#,$*$-%%sqrtG6#\"\"#\"\"\"\"\"'F;F+-F46#,$F7F;F+F+F+F+*(F-F<F1F<-%
$sinGF?F+,$*&F-F<-FCF5F+F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Arb
etet l\344ngs str\344ckan:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
66 "arb1:=dotprod(F(lin[1],lin[2],lin[3]),C(sqrt(2))-C(0),orthogonal):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Int(arb1,t=0..1):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"+'['>z:!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
35 "Snabbare g\345r det om du skriver in  " }{TEXT 291 1 "x" }{TEXT 
-1 1 "(" }{TEXT 292 1 "t" }{TEXT -1 3 "), " }{TEXT 293 1 "y" }{TEXT 
-1 1 "(" }{TEXT 294 1 "t" }{TEXT -1 3 "), " }{TEXT 296 1 "z" }{TEXT 
-1 1 "(" }{TEXT 295 1 "t" }{TEXT -1 9 ")  sj\344lv:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 11 "xl:=lin[1];" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#xlG,&\"#5\"\"\"*&%\"tGF',&!#:F'*&,&\"\")F'-%$cosG6#,$*$-%%sqr
tG6#\"\"#\"\"\"\"\"'F7F'-F06#,$F3F7F'F'F'F'" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 11 "yl:=lin[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
ylG*(%\"tG\"\"\",&\"\")F'-%$cosG6#,$*$-%%sqrtG6#\"\"#\"\"\"\"\"'F2F'-%
$sinG6#,$F.F2F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "zl:=lin[
3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#zlG,$*&%\"tG\"\"\"-%$sinG6#,
$*$-%%sqrtG6#\"\"#\"\"\"\"\"'F(\"\")" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "arbint:=dotprod(F(xl,yl,zl),C(sqrt(2))-C(0),orthogona
l):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(Int(arbint,t=0..1));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'['>z:!\"*" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 23 "L\344ngden av denna kurva:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "Ct:=diff(C(t),t):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 32 "dsny:=sqrt(sum(Ct[k]^2,k=1..3));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%dsnyG*$-%%sqrtG6#,(*$),&*&-%$sinG6#,$%\"tG\"\"'\"
\"\"-%$cosG6#,$F2\"\"#F4!#7*&,&\"\")F4-F6F0F9F4-F/F7F4!\"#F9\"\"\"F4*$
),&*&F.FAF?FAF:*&F<FAF5FAF9F9FAF4*$)F>F9FA\"%/BFA" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 27 "evalf(Int(dsny,t=0..2*Pi));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"+@`>GB!\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 13 "" 1 "" 
{TEXT -1 0 "" }}}}{MARK "0 0 46" 10 }{VIEWOPTS 0 0 0 1 1 1803 }