{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 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{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 "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 271 0 "" }{TEXT 272 23 "Calculus IV with Maple\n" }{TEXT 273 32 "Copyright 2002, Dr. J ack Wagner\n" }{TEXT -1 30 "j.wagner@intelligentsearch.com" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 11 "\nLesson 7: " }{TEXT 275 39 "Basic Differ ential Geometry of Surfaces" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 5 "" 0 "" {TEXT -1 8 "Topics: " }{TEXT 276 187 "\011Tange nts to a surface along coordinate curves. Normal to a surface. The ta ngent plane. The first fundamental form. The length of a curve project ed onto a surface. Angle between vectors" }{TEXT 274 2 ".\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "We begin by examining surfaces wh ich we know as the graph of a map, . These are most commonly of the f orm " }{XPPEDIT 18 0 "z = f(x,y);" "6#/%\"zG-%\"fG6$%\"xG%\"yG" } {TEXT -1 264 " . We consider first, the partial derivatives of z at a given point, P. That is, the directional derivatives of z with respec t to each variable. We will then find the tangents to the surface in \+ the direction of each of the variables; the coordinate directions. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT 257 12 "Example 7.1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "restart: with(plots): with(L inearAlgebra): with(VectorCalculus):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f := ;\011\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "The point P on the surface." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "p1 := eval(f, \{x=Pi/4, y=Pi/4\}); \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "We compute the principal parts of the tan gent vectors in the coordinate directions." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 21 "df[x] := diff(f, x);\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 48 "dfx[x, y=Pi/4] := eval(df[x], \{x=Pi/4, y=Pi/4\}); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "df[y] := diff(f, y);\n \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "dfy[x, y=Pi/4] := eva l(df[y], \{x=Pi/4, y=Pi/4\}); \n\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Now normalize each of these tan gent vectors. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "t[1] := N ormalize(dfx[x, y=Pi/4], 2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "t[2] := Normalize(dfy[x, y=Pi/4], 2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 169 "It is not necessary t o normalize for the purpose of drawing the tangent line, but we will n eed the unit tangent vectors later. \011Now we create the tangent line equations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "T[1] := p1+s *t[1]: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "T[2] := p1+s* t[2]:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "\"" }{TEXT 261 10 "spac ecurve" }{TEXT -1 40 "\" will only accept lists as an argument." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "TL[1] := convert(T[1], list) : TL[2] := convert(T[2], list):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "P1 := spacecurve(\{TL[1], TL[2]\}, s=-1..1, thickness=3, color =black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "P2 := plot3d(f, x=0..Pi/2, y=0..Pi/2, color=pink):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "display(P1, P2, axes=boxed, orientation=[-40, 40]); \+ \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "Here we have a pair of tan gents lying in the direction of the coordinate lines on the surface. T hese are really the images of the x any y axes with the origin transp orted to P." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 166 "Now we add a normal line. Remember that the cross pro duct of two vectors is orthogonal to the plane in which they lie and o riented according to the right hand rule. " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 19 "n := t[1] &x t[2];\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "We work with the unit nor mal." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "n := Normalize(n); \011\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Now create the equation for the normal line " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "N := p1+s*n:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "NL := convert(N, list):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 72 "P3 := spacecurve(\{TL[1], TL[2], NL\}, s=-1..1 , thickness=3, color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "display(P1, P2, P3, axes=boxed, orientation=[-25,60]);\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 200 "Now we have a set of unit vectors , \{t1, t2, n\}, which can be used as the basis for a coordinate syst em to investigate phenomena occurring on or near the surface in the ne ighborhood of the point, P= (" }{XPPEDIT 18 0 "Pi/4,Pi/4,f(Pi/4,Pi/4); " "6%*&%#PiG\"\"\"\"\"%!\"\"*&F$F%F&F'-%\"fG6$*&F$F%F&F'*&F$F%F&F'" } {TEXT -1 2 ")." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 203 "We do this in our daily lives. We do not ordinaril y refer our motion to spherical coordinates, but rather North, South, \+ East, West up and down. The tangents we have found, however, are not \+ orthogonal! " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Anticipating a re sult, " }{XPPEDIT 18 0 "cos(theta) = t[1].t[2]/(abs(t[1])*abs(t[2])); " "6#/-%$cosG6#%&thetaG*&-%\".G6$&%\"tG6#\"\"\"&F-6#\"\"#F/*&-%$absG6# &F-6#F/F/-F56#&F-6#F2F/!\"\"" }{TEXT -1 9 " where " }{XPPEDIT 18 0 " theta;" "6#%&thetaG" }{TEXT -1 22 " is the angle between " }{XPPEDIT 18 0 "t[1];" "6#&%\"tG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t[ 2];" "6#&%\"tG6#\"\"#" }{TEXT -1 19 ". Remembering that " }{XPPEDIT 18 0 "t[1]" "6#&%\"tG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t[2 ];" "6#&%\"tG6#\"\"#" }{TEXT -1 26 " are unit tangent vectors:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "theta[rad] := evalf(arccos(t [1].t[2]));\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "theta[deg] := evalf(convert(theta[rad], degrees));\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 142 "It is also a simple matt er to define a tangent plane at a point. We compute the equation of t he tangent plane orthogonal to the normal at P. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "r := :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "TP1 := (r-p1).n=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Represent the tangent plane as \+ an explicit function " }{XPPEDIT 18 0 "z(x,y)" "6#-%\"zG6$%\"xG%\"yG" }{TEXT -1 17 " by solving for z" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "TP := solve(TP1, z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "P4 := plot3d(TP, x=0..2, y=0..2, color=blue, style=wireframe):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "P5 := spacecurve(NL, s=-1 ..1, thickness=3, color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "display(P2, P4, P5, axes=boxed, orientation=[150,70]);\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "It turns out to be more fruitful t o define the tangent plane in terms of the two tangent vectors, " } {XPPEDIT 18 0 "t[1];" "6#&%\"tG6#\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "t[2];" "6#&%\"tG6#\"\"#" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "TP2 := p1+s*t[1]+t*t[2]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "P6 := plot3d(TP2, s=-1..1, t =-1..1, color=blue, style=wireframe):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "display(P1, P2, P6, axes=framed, orientation=[150,70] );\011\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 403 "This is a useful exe rcise because, just as the tangent at a point on a curve is an approxi mation to the curve in a neighborhood of the point, so the tangent pl ane at a point on a surface is an approximation to the surface in a ne ighborhood of the point. The set of vectors, \{t1 t2, n\}, form a co nvenient set of axes to which events on the surface may be referred. \+ Thus, given a surface defined by: " }}{PARA 0 "" 0 "" {TEXT -1 22 " \+ " }{XPPEDIT 18 0 "F(u,v) = [f[1](u,v), f[2](u,v).f [3](u,v)];" "6#/-%\"FG6$%\"uG%\"vG7$-&%\"fG6#\"\"\"6$F'F(-%\".G6$-&F,6 #\"\"#6$F'F(-&F,6#\"\"$6$F'F(" }{TEXT -1 48 "\nand a curve on that sur face parametrized by t: " }{XPPEDIT 18 0 "F(t) = [u(t), v(t)];" "6#/-% \"FG6#%\"tG7$-%\"uG6#F'-%\"vG6#F'" }{TEXT -1 57 ", consider a tangent vector, V, to the curve at a point " }{XPPEDIT 18 0 "p = [u(t[0]), v( t[0])];" "6#/%\"pG7$-%\"uG6#&%\"tG6#\"\"!-%\"vG6#&F*6#F," }{TEXT -1 2 " :" }{XPPEDIT 18 0 "V = F[u];" "6#/%\"VG&%\"FG6#%\"uG" }{TEXT -1 0 " " }{TEXT 262 2 "u'" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "t[0];" "6#&%\"tG6# \"\"!" }{TEXT -1 5 ") + " }{XPPEDIT 18 0 "F[v];" "6#&%\"FG6#%\"vG" } {TEXT -1 3 "v'(" }{XPPEDIT 18 0 "t[0];" "6#&%\"tG6#\"\"!" }{TEXT -1 10 "), where " }{XPPEDIT 18 0 "F[u] = dF/du;" "6#/&%\"FG6#%\"uG*&%#dF G\"\"\"%#duG!\"\"" }{TEXT -1 7 " and " }{TEXT 263 2 "u'" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "t[0];" "6#&%\"tG6#\"\"!" }{TEXT -1 2 ")=" } {XPPEDIT 18 0 "du/dt;" "6#*&%#duG\"\"\"%#dtG!\"\"" }{TEXT -1 91 " and \+ similarly for v. Now, V lies in the tangent plane which has as its co ordinate basis, " }{XPPEDIT 18 0 "F[u];" "6#&%\"FG6#%\"uG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "F[v];" "6#&%\"FG6#%\"vG" }{TEXT -1 35 " and \+ has components in that basis, " }{TEXT 264 2 "u'" }{TEXT -1 1 "(" } {XPPEDIT 18 0 "t[0];" "6#&%\"tG6#\"\"!" }{TEXT -1 6 "), v'(" } {XPPEDIT 18 0 "t[0];" "6#&%\"tG6#\"\"!" }{TEXT -1 57 ") . The quadrati c form, defined by the inner product on " }{XPPEDIT 18 0 "R^3;" "6#*$ %\"RG\"\"$" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "I[p];" "6#&%\"IG6#%\"pG " }{TEXT -1 2 "= " }{XPPEDIT 18 0 "`<,>`(V,V);" "6#-%$<,>G6$%\"VGF&" } {TEXT -1 23 " and expressed in the " }{XPPEDIT 18 0 "F[u]" "6#&%\"FG6 #%\"uG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "F[v]" "6#&%\"FG6#%\"vG" } {TEXT -1 23 " basis is known as the " }{TEXT 265 24 "first fundamental form. " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "I[p];" "6#&%\"IG6#%\"pG" } {TEXT -1 2 "= " }{XPPEDIT 18 0 "`<,>`(V,V);" "6#-%$<,>G6$%\"VGF&" } {TEXT -1 4 " = <" }{XPPEDIT 18 0 "F[u];" "6#&%\"FG6#%\"uG" }{TEXT -1 0 "" }{TEXT 266 2 "u'" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "t[0];" "6#&%\"t G6#\"\"!" }{TEXT -1 5 ") + " }{XPPEDIT 18 0 "F[v];" "6#&%\"FG6#%\"vG " }{TEXT -1 3 "v'(" }{XPPEDIT 18 0 "t[0];" "6#&%\"tG6#\"\"!" }{TEXT -1 3 "), " }{XPPEDIT 18 0 "F[u];" "6#&%\"FG6#%\"uG" }{TEXT -1 0 "" } {TEXT 267 2 "u'" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "t[0];" "6#&%\"tG6#\" \"!" }{TEXT -1 5 ") + " }{XPPEDIT 18 0 "F[v];" "6#&%\"FG6#%\"vG" } {TEXT -1 3 "v'(" }{XPPEDIT 18 0 "t[0];" "6#&%\"tG6#\"\"!" }{TEXT -1 5 ")> = " }{XPPEDIT 18 0 "`<,>`(F[u],F[u])[p];" "6#&-%$<,>G6$&%\"FG6#%\" uG&F(6#F*6#%\"pG" }{XPPEDIT 18 0 "(du/dt*t[0])^2;" "6#*$*(%#duG\"\"\"% #dtG!\"\"&%\"tG6#\"\"!F&\"\"#" }{TEXT -1 5 " + 2 " }{XPPEDIT 18 0 "`<, >`(F[u],F[v]);" "6#-%$<,>G6$&%\"FG6#%\"uG&F'6#%\"vG" }{TEXT -1 1 " " } {XPPEDIT 18 0 "du/dt*t[0];" "6#*(%#duG\"\"\"%#dtG!\"\"&%\"tG6#\"\"!F% " }{TEXT -1 2 " " }{XPPEDIT 18 0 "dv/dt*t[0];" "6#*(%#dvG\"\"\"%#dtG! \"\"&%\"tG6#\"\"!F%" }{TEXT -1 4 " + " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "`<,>`(F[v],F[v]);" "6#-%$<,>G6$&%\"FG6#%\"vG&F'6#F)" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "(dv/dt*t[0])^2;" "6#*$*(%#dvG\"\"\"%#dtG!\"\"&%\"tG6 #\"\"!F&\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "\n Tradi tionally one sets: " }{XPPEDIT 18 0 "E = F[u].F[u];" "6#/%\"EG-% \".G6$&%\"FG6#%\"uG&F)6#F+" }{TEXT -1 7 " " }{XPPEDIT 18 0 "F = \+ F[u].F[v];" "6#/%\"FG-%\".G6$&F$6#%\"uG&F$6#%\"vG" }{TEXT -1 9 " \+ " }{XPPEDIT 18 0 "G = F[v].F[v];" "6#/%\"GG-%\".G6$&%\"FG6#%\"vG&F) 6#F+" }{TEXT -1 37 " \n\011In these terms we have:" } {XPPEDIT 18 0 "ds^2;" "6#*$%#dsG\"\"#" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "abs(V)^2;" "6#*$-%$absG6#%\"VG\"\"#" }{TEXT -1 4 " = " } {XPPEDIT 18 0 "E;" "6#%\"EG" }{XPPEDIT 18 0 "(du/dt*t[0])^2;" "6#*$*(% #duG\"\"\"%#dtG!\"\"&%\"tG6#\"\"!F&\"\"#" }{TEXT -1 5 " + 2 " } {XPPEDIT 18 0 "F;" "6#%\"FG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "du/dt*t[0 ];" "6#*(%#duG\"\"\"%#dtG!\"\"&%\"tG6#\"\"!F%" }{TEXT -1 2 " " } {XPPEDIT 18 0 "dv/dt*t[0];" "6#*(%#dvG\"\"\"%#dtG!\"\"&%\"tG6#\"\"!F% " }{TEXT -1 4 " + " }{XPPEDIT 18 0 "G;" "6#%\"GG" }{TEXT -1 1 " " } {XPPEDIT 18 0 "(dv/dt*t[0])^2;" "6#*$*(%#dvG\"\"\"%#dtG!\"\"&%\"tG6#\" \"!F&\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 58 " \+ We have expressed the element of arc length, " }{TEXT 268 2 "ds" } {TEXT -1 217 ", in terms that are intrinsic to the surface. Many oth er properties of the surface can also be expressed in terms of these i ntrinsic quantities. As an example, the angle between the two coordin ate tangent vectors, " }{XPPEDIT 18 0 "F[u];" "6#&%\"FG6#%\"uG" } {TEXT -1 6 " and " }{XPPEDIT 18 0 "F[v];" "6#&%\"FG6#%\"vG" }{TEXT -1 22 " at p, is found by: " }{XPPEDIT 18 0 "cos(theta) = `<,>`(F[u] ,F[v])/(abs(F[u])*abs(F[v]));" "6#/-%$cosG6#%&thetaG*&-%$<,>G6$&%\"FG6 #%\"uG&F-6#%\"vG\"\"\"*&-%$absG6#&F-6#F/F3-F66#&F-6#F2F3!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "F/sqrt(E*G);" "6#*&%\"FG\"\"\"-%%sqrtG6#*&% \"EGF%%\"GGF%!\"\"" }{TEXT -1 56 ". The coordinate vectors are orthog onal if and only if " }{TEXT 269 1 "F" }{TEXT -1 4 "=0.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 11 "Example \+ 7.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Compute the arc length of the curve " }{XPPEDIT 18 0 "t = x^2; " "6#/%\"tG*$%\"xG\"\"#" }{TEXT -1 28 " projected onto the surface " } {XPPEDIT 18 0 "f(x,y) = [x, y, sin(x*y)];" "6#/-%\"fG6$%\"xG%\"yG7%F'F (-%$sinG6#*&F'\"\"\"F(F." }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "restart: with(plots): with(LinearAlgebra): with(Vecto rCalculus):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f := ;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "u := t: v \+ := t^2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The parabola " }{XPPEDIT 18 0 "y = x^2;" "6#/%\"yG*$%\"xG \"\"#" }{TEXT -1 19 " in the x-y plane. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "C1 := ; \n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Th e parabola projected onto the surface" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "C2 := eval(f, \{x=u, y=v\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "Now we create plot st ructures for the surface, the parabola in the x-y plane and the parabo la projected onto the surface. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "C1L := convert(C1, list): C2L := convert(C2, list):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "P1 := plot3d(f, x=0..Pi/2, y =0..Pi/2, color=pink, style=wireframe):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "S1 := spacecurve(C1L, t=0..1.2, color=blue, thickness =3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "S2 := spacecurve(C2 L, t=0..1.2, color=red, thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "display(S1, S2, P1, axes=boxed, labels=[x, y, z], ori entation=[160,60]);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Now we c ompute the coefficients of the first fundamental form." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "df[x] := diff(f, x); df[y] := diff (f, y);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "E := df[x].df[ x]; \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "F := df[x].df[y]; \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "G := df[y].df[y];\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Now the derivatives of " }{XPPEDIT 18 0 "u(t);" "6#-%\"uG6#%\"tG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "v(t);" "6#-%\"vG6#%\"tG" }{TEXT -1 2 ".." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "du := diff(u, t); \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dv := diff(v, t);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "Now we compute the element of arc length in terms of the coeffici ents of the first fundamental form. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "ds := sqrt(E*du^2+2*F*du*dv+G*dv^2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ds[u, v] := subs(\{x=t, y=t^2\}, ds );\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 " Compute the a rc length of the curve between t=0 and t=1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "arclength := evalf(int(ds[u, v], t=0..1));\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "Once the methodology is in place \+ it is a simple matter to call up the saved worksheet and replace the s urface, the curve and the parameter limits. " }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 12 "Example 6.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Project the plane curve, " }{XPPEDIT 18 0 "y = sin(2*x);" "6#/%\"yG-%$sinG6#* &\"\"#\"\"\"%\"xGF*" }{TEXT -1 20 " onto the surface, " }{XPPEDIT 18 0 "z = log(x+y);" "6#/%\"zG-%$logG6#,&%\"xG\"\"\"%\"yGF*" }{TEXT -1 48 " , and find its length, on the surface, between " }{XPPEDIT 18 0 " Pi/4;" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Pi/2;" "6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "restart: with(plots): with(LinearAlgebra): \+ with(VectorCalculus):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 11 "The surface" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "u := t: v := sin(2*t):\011" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The curve in the pla ne defined parametrically" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "C1 := :\011\011\011\011" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The curve embedded in the surfa ce" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "C2 := subs(\{x=u, y=v \}, f):\011" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "P1 := plot3d (f, x=0..Pi/2, y=0..Pi/2, color=pink, style=wireframe):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "S1 := spacecurve(evalm(C1), t=0..Pi /2, color=blue, thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "S2 := spacecurve(evalm(C2), t=0..Pi/2, color=red, thickness=3): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "display(S1, S2, P1, axe s=framed, labels=[x, y, z]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "df[x] := diff(f, x); df[y] := diff(f, y);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "E := df[x].df[x];\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "F := df[x].df[y];\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "G := df[y].df[y];\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "du := diff(u, t);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dv := diff(v, t);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "ds[1] := sqrt(E*du^2+2*F*du*dv+G*dv^2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ds[2] := subs(\{x=t, y=sin(t)\}, ds [1]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "arclength := eva lf(int(ds[2], t=Pi/4..Pi/2));\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 264 "Note that, in creating a composite plot from separate plot struct ures, options specific to each plot such as the color and thickness ar e specified for each plot structure, while options such as the style o f axes and labeling are specified in the display command. " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 11 "E xample 7.4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "If we were interest ed in working at a point on the surface of Ex. 7.3, say " }{XPPEDIT 18 0 "P = f(Pi/4,Pi/4);" "6#/%\"PG-%\"fG6$*&%#PiG\"\"\"\"\"%!\"\"*&F)F *F+F," }{TEXT -1 123 ", we would like to have a coordinate system wit h origin at P, and with axes tangent to the surface. Proceeding as be fore:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "P := eval(f, \{x=u , y=v\});\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "P := eval(P, t=Pi/4);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "v1 := eval(d f[x], \{x=u, y=v\});\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "v 1 := eval(v1, t=Pi/4);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "v1 := Normalize(v1, 2, inplace);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "v2 := eval(df[y], \{x=u, y=v\});\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "v2 := subs(t=Pi/4, v2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "v2 := Normalize(v2, 2, inplace);\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ax1 := P+s*v1; ax2 := \+ P+s*v2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "S4 := spacecurve (\{evalm(ax1), evalm(ax2)\}, s=-1..1, color=black, thickness=3):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "display(P1, S2, S4, axes=fra med, labels=[x, y, z]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "theta := evalf(arccos(v1.v2));\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "theta := evalf(convert(theta, degrees));\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "In courses on differential geomet ry the equations for curves, imbedded in a surface, are derived with r espect to such coordinate systems. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 8 "Practice" }{TEXT -1 22 "\n1. Plot the surface, " }{XPPEDIT 18 0 "z = x^3+y^3;" "6#/%\"zG,&*$% \"xG\"\"$\"\"\"*$%\"yGF(F)" }{TEXT -1 85 " together with coordinate t angent lines, normal line and tangent plane at the point " }{XPPEDIT 18 0 "x = 0,y = 0;" "6$/%\"xG\"\"!/%\"yGF%" }{TEXT -1 36 " .\n2. For the surface defined by " }{XPPEDIT 18 0 "z = (x^3-y^3)^(1/3);" "6#/% \"zG),&*$%\"xG\"\"$\"\"\"*$%\"yGF)!\"\"*&F*F*F)F-" }{TEXT -1 146 ", f ind the gradient at the point (5, -5) and the plane orthogonal to the \+ gradient through the point, (5, -5, 6.299605249).\n3. Project the curv e, " }{XPPEDIT 18 0 "y = sqrt(x);" "6#/%\"yG-%%sqrtG6#%\"xG" }{TEXT -1 19 " onto the surface " }{XPPEDIT 18 0 "z = ln(sqrt(x*y));" "6#/% \"zG-%#lnG6#-%%sqrtG6#*&%\"xG\"\"\"%\"yGF-" }{TEXT -1 23 " , between \+ the points " }{XPPEDIT 18 0 "x = 0 .. 2,y = 0 .. 2;" "6$/%\"xG;\"\"!\" \"#/%\"yG;F&F'" }{TEXT -1 55 " , and find the length of the projected \+ curve, between " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = 2;" "6#/%\"xG\"\"#" }{TEXT -1 119 ". Plot the curve, and its projection together with the surface. \n4. Project the straight line, y=3x onto the surface " }{XPPEDIT 18 0 "S(x,y) = \+ [x+y, x-y, x*y];" "6#/-%\"SG6$%\"xG%\"yG7%,&F'\"\"\"F(F+,&F'F+F(!\"\"* &F'F+F(F+" }{TEXT -1 59 ", and plot them both on the same set of axes \+ in the range " }{XPPEDIT 18 0 "x = -1 .. 1;" "6#/%\"xG;,$\"\"\"!\"\"F '" }{TEXT -1 43 ". Find the length of the projected line. \n" }}}} {MARK "0 0 2" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }