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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 24 "\011\011\011\011Linear A
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24 "Version .92 for MapleVR7" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 7 "S
ummary" }}{EXCHG {PARA 256 "" 0 "" {TEXT 263 281 " This worksheet cove
rs some examples of:\n       \011tangent planes.\n        tangent line
s.\n        efficient generation of the linear approximation at a poin
t to a \n        function using Maple.\n        comparison of a functi
on from R^2 to R^2 with its linear  \n       \011approximation." }}}}
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256 "" 0 "" {TEXT 258 34 "The tangent plane through (1,1,2)." }}{PARA 
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{TEXT 259 35 "The tangent plane through (-1,0,1)." }}{PARA 0 "> " 0 "
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t2 := polyhedraplot([-1,0,1],polyscale=.1,polytype=hexahedron,color=re
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}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 16 "Transformations." }}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 18 "Load the packages." }}{PARA 4 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 256 "" 0 "" {TEXT 266 38 "Load transforma
tion functions.\nChange " }{TEXT 285 7 "Pacific" }{TEXT 286 13 " below
 to the" }{TEXT 287 21 " name of your machine" }{TEXT 288 1 ":" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "read `/othermaple/transform_plots.t
xt` ;" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 43 "
\011\011\011\011Plot Transformation Package Version 1.0" }}{PARA 6 "" 
1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 58 "This set of function
s aids in transforming two dimensional" }}{PARA 6 "" 1 "" {TEXT -1 35 
"plots by functions from R^2 to R^2." }}{PARA 6 "" 1 "" {TEXT -1 0 "" 
}}{PARA 6 "" 1 "" {TEXT -1 23 "Basic examples include:" }}{PARA 6 "" 
1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 26 "\011\011P1 := plot(x
^2,x=-1..1);" }}{PARA 6 "" 1 "" {TEXT -1 57 "\011\011transform_plot(P1
,[exp(x)*cos(y),exp(x)*sin(y)],[x,y]);" }}{PARA 6 "" 1 "" {TEXT -1 0 "
" }}{PARA 6 "" 1 "" {TEXT -1 57 "to transform the graph of the parabol
a y = x^2 by the map" }}{PARA 6 "" 1 "" {TEXT -1 39 "(x,y) -> [exp(x)*
cos(y),exp(x)*sin(y)]." }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 
1 "" {TEXT -1 3 "and" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "
" {TEXT -1 56 "\011\011transform_rect([2.,2.],[1.,1.],[x^2-y^2,2*x*y],
[x,y]);" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 
53 "to display the rectangle with corners [1,1] and [2,2]" }}{PARA 6 "
" 1 "" {TEXT -1 53 "together with the image of its boundary under the \+
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{EXCHG {PARA 256 "" 0 "" {TEXT 268 41 "A helper function to evaluate a
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{EXCHG {PARA 256 "" 0 "" {TEXT 277 22 " Linear approximation." }}
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\"+LH)fu&Fdu$\"+#yIj?)Fdu7$$\"+([g2t&Fdu$\"+s0T=#)Fdu7$$\"+(ompr&Fdu$
\"+dmNH#)Fdu7$$\"+Ak)=q&Fdu$\"+>OKT#)Fdu7$$\"+oqp(o&Fdu$\"+(f$e_#)Fdu7
$$\"+/^'Gn&Fdu$\"+>NNk#)Fdu7$$\"+[TOecFdu$\"+p2'eF)Fdu7$$\"+#*H=VcFdu$
\"+$z2zG)Fdu7$$\"+L<cGcFdu$\"+50^*H)Fdu7$$\"+&Q4Oh&Fdu$\"+jfP6$)Fdu7$$
\"+]3y)f&Fdu$\"+kJ9B$)Fdu7$$\"+l\\:&e&Fdu$\"++g&RL)Fdu7$$\"+l\"Q&pbFdu
$\"+I([jM)Fdu7$$\"+7.dbbFdu$\"+OHVd$)Fdu7$$\"+ownSbFdu$\"+65Dp$)Fdu7$$
\"+1NUEbFdu$\"+<Cc!Q)FduF\\dmFgdl-%+AXESLABELSG6%Q!6\"Fh[p%(DEFAULTG-%
%VIEWG6$Fj[pFj[p" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 
0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve
 7" "Curve 8" }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 39 "Maple Syntax: \+
An aside on map and map2:" }}{EXCHG {PARA 0 "" 0 "" {TEXT 279 186 "If \+
you want to apply the same function to each entry of a compount object
\nsuch as a list, vector, or matrix, then you need to explictly tell M
aple\nthis, using the functions map or map2. " }}}{EXCHG {PARA 0 "" 0 
"" {TEXT 280 170 "Map applies its first entry (a function) to each ope
rant of its second \nargument (e.g. a matrix) and tacks on any additio
nal arguments to each\nfunction call. For example:" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 32 "aa := [seq(x*y+ Pi/i^2,i=1..3)];" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 14 "map(evalf,aa);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#aaG7%,&*&%\"xG\"\"\"%\"yGF)F)%#PiGF),&F'F)*&#F)\"\"%F)F+F)F),&F'F)*
&#F)\"\"*F)F+F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&*&%\"xG\"\"\"
%\"yGF'F'$\"+aEfTJ!\"*F',&F%F'$\"+N;)R&y!#5F',&F%F'$\"+/&e1\\$F/F'" }}
}{EXCHG {PARA 262 "" 0 "" {TEXT -1 349 "Map2 is like map except that i
t applies its first argument (a function like\nthe substitution functi
on subs) to each operand of its third argument\n (e.g. the list) using
 map2's second argument (e.g. a set of substitutions\}\nas the first a
rgument of each function call.\nThus the example below substitutes x=u
*v and y = u+v in each entry of\nthe list aa." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "map2(subs,\{x=u*v,y=u+v\},aa);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%,&*(%\"uG\"\"\"%\"vGF',&F&F'F(F'F'F'%#PiGF',&F%F'*&#F
'\"\"%F'F*F'F',&F%F'*&#F'\"\"*F'F*F'F'" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 281 63 "You can also use functions defined on the fly with map o
r map2." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "map(t ->evalf(t^2),aa);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*$),&*&%\"xG\"\"\"%\"yGF)F)$\"+a
EfTJ!\"*F)\"\"#F)*$),&F'F)$\"+N;)R&y!#5F)F.F)*$),&F'F)$\"+/&e1\\$F4F)F
.F)" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 62 "Maple Syntax: Another Wa
y to Compute the Linear Approximation:" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT 283 75 "This way is easier, but perhaps more demanding of\ncomfo
rt with programming." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 282 47 "Specify \+
the point where evaluation will happen." }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "pt := [.9,.9];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ptG7$$\"
\"*!\"\"F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 284 83 "The @ sign below i
s composition, so that each entry will be converted to a decimal." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "val := map2(evalf@subs,\{x=pt[1],y=
pt[2]\},F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$valG7$$\"+:(*[%f&!#5
$\"+'=U*\\qF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "L := map2(
evalf@subs,\{x=pt[1],y=pt[2]\},F_deriv);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"LG-%'matrixG6#7$7$$\"+$o*4;i!#5$!+'=U*\\qF,7$$\"+'4pK$yF,$\"
+:(*[%f&F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "L_expn :=eval
m(L &* vector([x-.9,y-.9])+ val); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%'L_expnG-%'vectorG6#7$,(%\"xG$\"+$o*4;i!#5$\"+nz%\\M'F-\"\"\"*&$\"+'
=U*\\qF-F0%\"yGF0!\"\",(F*$\"+'4pK$yF-$\"+W2/N]F-F5*&$\"+:(*[%f&F-F0F4
F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT 271 51 "Images of Tangents to Curves under Trans
formations." }}{EXCHG {PARA 256 "" 0 "" {TEXT 262 143 "Under a functio
n , the tangent line to a curve gets mapped  by the derivative to the \+
 tangent line of the image curve. \nHere's an illustration:" }}}
{EXCHG {PARA 256 "" 0 "" {TEXT 270 69 "First the helix t-> (cos(t),sin
(t),t/20) with  a tangent at  t= Pi/4:" }{MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 52 "Details o
f drawing the helix with its tangent line.\n" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 36 "helix_param := [cos(t),sin(t),t/20];" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 59 "helix_gr := spacecurve(helix_param,t = 0 ..20
,thickness=3):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,helix_paramG7%-%$
cosG6#%\"tG-%$sinGF(,$F)#\"\"\"\"#?" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "helix_point := map(simplify,(map2(subs,t=Pi/4,helix_p
aram)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,helix_pointG7%,$*$-%%sq
rtG6#\"\"#\"\"\"#F,F+F&,$%#PiG#F,\"#!)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "helix_tan_vec := map(simplify,(map2(subs,t=Pi/4,map(d
iff,helix_param,t))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.helix_tan
_vecG7%,$*$-%%sqrtG6#\"\"#\"\"\"#!\"\"F+,$F'#F,F+#F,\"#?" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "helix_tan := spacecurve(evalm(helix
_point+t*helix_tan_vec),t=-1..1,color=red,thickness=3):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "display(\{helix_tan,helix_gr\});" }
}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6%-%'CURVESG6$7T7
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