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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 54 "Vector Functions and Spac
e Curves (10.1, 10.2 and10.3)" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "Example: Page 707 example 5. Find
 a vector function that represents the curve of intersection of the cy
linder x^2 +y^2 =1 and the plane y+z=2." }}{PARA 0 "" 0 "" {TEXT -1 9 
"Solution:" }}{PARA 0 "" 0 "" {TEXT -1 63 "We first look at the inters
ection of these two equations below:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "impl
icitplot3d(\{x^2+y^2=1,y+z=2\},x=-5..5, y=-5..5, z=-5..5,axes=BOXED,nu
mpoints=5000);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 182 "After finding \+
the parametric equations for the curve which lies in the intersection \+
(by using x=cos t, y= sin t and z=2-y = 2-sin t, see page 707), we plo
t the following space curve:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 59 "spacecurve([cos(t),sin(t),2-sin(t)],t=0..4*Pi,axes=BOXED);\n" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Example:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 51 "spacecurve([sin(t),cos(t),t],t=0..4*Pi,axes=BOXE
D);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Example: page 710, number \+
21:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "spacecurve([sin(t),c
os(t),t^2],t=0..4*Pi,color=black,axes=BOXED);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Example: Numb
er 8 (comment on the following curve)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 93 "spacecurve([exp(-t)*cos(10*t),exp(-t)*sin(10*t),exp(-
t)],t=0..4*Pi,axes=BOXED,numpoints=500);" }}}{PARA 0 "" 0 "" {TEXT -1 
26 "Example page 710, number 6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 55 "spacecurve([t,t^2,exp(-t)],t=-10*Pi..10*Pi,axes=BOXED);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Example: page 710, number 10" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "spacecurve([cos(t),sin(t),ln
(t)],t=0..4*Pi,axes=BOXED);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Ex
ample: page 710, number 9:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
58 "spacecurve([cos(t),sin(t),sin(5*t)],t=0..2*Pi,axes=BOXED);" }}}
{EXCHG {PARA 18 "" 0 "" {TEXT -1 47 "Derivatives of Vector Functions a
nd T,N, and B." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "r:=t->[ex
p(t),exp(t)*sin(t),exp(t)*cos(t)];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "dr:=t->diff(r(t),t);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "dr(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "T
:=t->dr(t)/sqrt(innerprod(dr(t),dr(t)));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "T(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "si
mplify(T(t),trig);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dT:=t
->diff(T(t),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "dT(t);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "N:=t->dT(t)/sqrt(innerprod
(dT(t),dT(t)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "N(t);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "simplify(N(t),trig);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "B:=t->crossprod(T(t),N(t));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "B(t);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "simplify(B(t),trig);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 66 "kappa1:=sqrt(innerprod(dT(t),dT(t)))/sqrt(in
nerprod(dr(t),dr(t)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The abo
ve kapa1 is the curvature from formula 9 on page 719." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(kappa1);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 73 "Let's see if the formulae 9 and 10 of finding t
he curvature are the same." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "ddr:=t->diff(dr(t),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
36 "vector1:=t->crossprod(dr(t),ddr(t));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "vector1(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 60 "numerator:=simplify(sqrt(innerprod(vector1(t),vector1(t))));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "denominator:=(innerprod(dr(
t),dr(t)))^(3/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "kappa2
:=simplify(numerator/denominator);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "simplify(kappa2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 166 "Note: It seems that the kappa1 and kappa2 do not agree. This i
s because Maple has trouble simplifying the radicals. Let's see if the
 graphs of kappa1 and kapp2 agree." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "plot(\{kappa1(t),kappa2(t)\},t=0..4*Pi);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 79 "This suggests that the curvature is decre
asing from  0  to 4*Pi. Let's see why." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "spacecurve(r(t),t=0..4*Pi,axes=BOXED,numpoints=100);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 7 "Example" }{TEXT -1 41 ": Let'
s see the curvature for a 2D graph." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "r:=t->[t,sin(t),0];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "dr:=t->diff(r(t),t);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "dr(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "T
:=t->dr(t)/sqrt(innerprod(dr(t),dr(t)));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "T(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "si
mplify(T(t),trig);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dT:=t
->diff(T(t),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "dT(t);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "N:=t->dT(t)/sqrt(innerprod
(dT(t),dT(t)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "N(t);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "simplify(N(t),trig);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "B:=t->crossprod(T(t),N(t));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "B(t);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "simplify(B(t),trig);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 66 "kappa1:=sqrt(innerprod(dT(t),dT(t)))/sqrt(in
nerprod(dr(t),dr(t)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The abo
ve kapa1 is the curvature from formula 9 on page 719." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(kappa1);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 73 "Let's see if the formulae 9 and 10 of finding t
he curvature are the same." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "ddr:=t->diff(dr(t),t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
36 "vector1:=t->crossprod(dr(t),ddr(t));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "vector1(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 60 "numerator:=simplify(sqrt(innerprod(vector1(t),vector1(t))));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "denominator:=(innerprod(dr(
t),dr(t)))^(3/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "kappa2
:=simplify(numerator/denominator);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "simplify(kappa2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 166 "Note: It seems that the kappa1 and kappa2 do not agree. This i
s because Maple has trouble simplifying the radicals. Let's see if the
 graphs of kappa1 and kapp2 agree." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "plot(\{kappa1(t),kappa2(t)\},t=0..4*Pi);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 79 "This suggests that the curvature is decre
asing from  0  to 4*Pi. Let's see why." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "spacecurve(r(t),t=0..4*Pi,axes=BOXED,numpoints=100);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(\{sin(t),kappa1(t)
\},t=0..4*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 1 " " }}}}{MARK "83" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }
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