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-1 -1 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 269 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 429 41 "Math 2331 Fall 2002 Pr actice Final Exam" }}{PARA 0 "" 0 "" {TEXT 256 2 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 430 125 "Questions 1 - 9 are v ery similar to questions from the first three exams. Questions 10-20 a re very similar to the following" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 257 16 "10. Calculate " }{XPPEDIT 265 1 "int( 2*y*dx+ x*dy+2*z,z=C.. ``)" "-%$intG6$,(*(\"\"#\"\"\"%\"yGF(%#dxGF(F(*&%\"xGF(%#dyGF(F(*&F'F( %\"zGF(F(/F/;%\"CG%!G" }{TEXT 266 11 ". where " }{TEXT 356 1 "C" } {TEXT 357 43 " is the oriented curve parameterized by " }{XPPEDIT 359 1 "r(t)=[cos(t),sin(t),sqrt(t)]" "/-%\"rG6#%\"tG7%-%$cosG6#F&-%$si nG6#F&-%%sqrtG6#F&" }{TEXT 358 4 ", " }{XPPEDIT 361 1 "t=0..Pi" "/% \"tG;\"\"!%#PiG" }{TEXT 360 1 "." }}{PARA 0 "" 0 "" {TEXT 431 5 "\na) \+ " }{XPPEDIT 442 1 "0" "\"\"!" }{TEXT 432 20 " b) " } {XPPEDIT 443 1 "Pi/4" "*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 433 19 " \+ c) " }{XPPEDIT 444 1 "Pi/2" "*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 434 20 " d) " }{XPPEDIT 445 1 "Pi" "I#PiG%*protectedG " }{TEXT 435 20 " e) " }{XPPEDIT 446 1 "2*Pi" "*&\"\"# \"\"\"%#PiGF$" }{TEXT 436 11 " \nf) " }{XPPEDIT 447 1 "Pi/3" "*& %#PiG\"\"\"\"\"$!\"\"" }{TEXT 437 18 " g) " }{XPPEDIT 448 1 "Pi/6" "*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT 438 19 " h ) " }{XPPEDIT 449 1 "2*Pi/3" "*(\"\"#\"\"\"%#PiGF$\"\"$!\"\"" }{TEXT 439 17 " i) " }{XPPEDIT 450 1 "3*Pi/2" "*(\"\"$\"\"\"%#Pi GF$\"\"#!\"\"" }{TEXT 440 18 " j) " }{XPPEDIT 451 1 "3*P i/4" "*(\"\"$\"\"\"%#PiGF$\"\"%!\"\"" }{TEXT 441 5 " \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 531 8 "Solutio n" }{TEXT 532 6 ": (c)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "r := t -> [cos(t),sin(t),sqrt(t)]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG:6#%\"tG6\"6$%)operatorG%&ar rowGF(7%-%$cosG6#9$-%$sinGF/-%%sqrtGF/F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 93 "Int( 2*r(t)[2]*diff(r(t)[1],t)+ r(t)[1]*diff(r(t)[ 2],t)+ 2*r(t)[3]*diff(r(t)[3],t),t=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,(*$-%$sinG6#%\"tG\"\"#!\"#*$-%$cosGF*F,\"\"\" F1F1/F+;\"\"!%#PiG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value( \");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 373 22 "11. The vector field " }{TEXT 456 1 "F " }{TEXT 457 13 "(x,y) = (y+1)" }{TEXT 452 1 "i" }{TEXT 453 8 " + (x+2 )" }{TEXT 454 1 "j" }{TEXT 455 1 " " }}{PARA 3 "" 0 "" {TEXT 377 42 " \+ is the gradient field of some function " }{TEXT 378 2 "f(" }{TEXT 380 3 "x,y" }{TEXT 381 1 ")" }{TEXT 379 13 ". Calculate " }{TEXT 376 1 " " }{TEXT 374 15 "f(1,2) - f(0,1)" }{TEXT 375 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 382 74 "a) 2 b) 4 c) 5 d) 8 e) " }{TEXT 386 2 "1 0" }{TEXT 389 3 " " }{TEXT 388 18 " " }}{PARA 0 "" 0 "" {TEXT 383 35 "f) 12 g) 14 " }{TEXT -1 1 " \+ " }{TEXT 387 80 "h) 16 i) 18 j) Cannot be calcula ted from given information." }{TEXT -1 2 " " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Solution" }}{PARA 3 "" 0 "" {TEXT 533 8 "Solution" }{TEXT 534 6 ": (c)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "F := vecto r([y+1,x+2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'VECTORG6#7$, &%\"yG\"\"\"F+F+,&%\"xGF+\"\"#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "curl(vector([y+1,x+2,0]),[x,y,z]); #F is conservative " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%\"\"!F'F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eqn := u(x,y) = int(y+1,x) + g(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%\"uG6$%\"xG%\"yG,& *&,&F*\"\"\"F.F.F.F)F.F.-%\"gG6#F*F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "diff(eqn,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%d iffG6$-%\"uG6$%\"xG%\"yGF+,&F*\"\"\"-F%6$-%\"gG6#F+F+F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "F[2] = rhs(\");" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,&%\"xG\"\"\"\"\"#F&,&F%F&-%%diffG6$-%\"gG6#%\"yGF/F &" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "diff(g(y),y) = solve( \",diff(g(y),y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"gG 6#%\"yGF*\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn2 := s ubs(g(y)=2*y+C,eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%\"u G6$%\"xG%\"yG,(*&,&F*\"\"\"F.F.F.F)F.F.F*\"\"#%\"CGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "subs( \{x=1,y=2\}, rhs(eqn2)) - sub s(\{x=0,y=1\}, rhs(eqn2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 258 17 "12. \+ Calculate " }{XPPEDIT 362 1 "int( z*dx+dy+x,z=C..``)" "-%$intG6$,(*& %\"zG\"\"\"%#dxGF(F(%#dyGF(%\"xGF(/F';%\"CG%!G" }{TEXT 363 11 " whe re " }{TEXT 364 1 "C" }{TEXT 365 42 " is the oriented curve paramet erized by " }{XPPEDIT 367 1 "r(t)=[sqrt(t/2), tan(Pi*t/8),exp(t^2-2*t) ]" "/-%\"rG6#%\"tG7%-%%sqrtG6#*&F&\"\"\"\"\"#!\"\"-%$tanG6#*(%#PiGF,F& F,\"\")F.-%$expG6#,&*$F&F-F,*&F-F,F&F,F." }{TEXT 366 4 ", " } {XPPEDIT 368 1 "t=0..2" "/%\"tG;\"\"!\"\"#" }{TEXT 262 6 ". \n\n" } {TEXT 261 71 "a) 0 b) 1 c) 2 d) 3 \+ e) 4" }{TEXT 371 21 " " }}{PARA 0 "" 0 "" {TEXT 263 34 "f) 5 g) 6 " }{TEXT -1 1 " " } {TEXT 372 38 "h) 7 i) 8 j) 9" }{TEXT -1 2 " \+ " }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Solution" }}{PARA 3 "" 0 "" {TEXT 535 8 "Solution" }{TEXT 536 6 ": \+ (c)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "r := (t) -> [sqrt(t /2),tan(Pi*t/8),exp(t^2-2*t)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(r(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 " " 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 " " {TEXT -1 33 "Warning, new definition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "F := vector([z,1,x]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'VECTORG6#7%%\"zG\"\"\"%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "curl(F,[x,y,z]); #F is conservative" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%\"\"!F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eqn := u(x,y,z) = int(z,x) + g(y,z) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%\"uG6%%\"xG%\"yG%\"zG,& *&F+\"\"\"F)F.F.-%\"gG6$F*F+F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "diff(eqn,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-% \"uG6%%\"xG%\"yG%\"zGF+-F%6$-%\"gG6$F+F,F+" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "1 = rhs(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\" \"\"-%%diffG6$-%\"gG6$%\"yG%\"zGF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn2 := subs(g(y,z)=y+h(z),eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%\"uG6%%\"xG%\"yG%\"zG,(*&F+\"\"\"F)F.F.F*F.- %\"hG6#F+F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x = diff(rhs (eqn2),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG,&F$\"\"\"-%%diffG 6$-%\"hG6#%\"zGF-F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "diff (h(z),z) = solve( \" , diff(h(z),z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"hG6#%\"zGF*\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eqn3 := subs(h(z)= C, eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/-%\"uG6%%\"xG%\"yG%\"zG,(*&F+\"\"\"F)F.F.F*F.% \"CGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "evalm( grad(rhs(e qn3),[x,y,z]) - F ); #Check" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VEC TORG6#7%\"\"!F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "subs( \+ \{x=1,y=1,z=1\}, rhs(eqn3)) - subs(\{x=0,y=0,z=1\}, rhs(eqn3));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 259 80 "13. 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Let \+ S be the hemisphere " }{XPPEDIT 19 1 "\{x^2+y^2+z^2=1,z>` `*0\}" "<$/,(*$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$%\"zGF'F(F(2*&%$~~~GF(\"\"!F(F," }{TEXT -1 24 " , C the unit circle " }{XPPEDIT 19 1 "\{x^2+y^2=1,z= 0\}" "<$/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(/%\"zG\"\"!" }{TEXT -1 70 " \+ traversed counterclockwise. Vector calculus tells us that\n\n \+ " }{BITMAP 168 39 39 1 "?TMwIMB:n@>?>Ov>F:k:;::::yyyyyy:::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::>:yay A:<:ryxI:::::::::::::::::::::::::::::::::::::::<:;:vYyA::::::::::::::: ::::::::::::::::::::::::;Z:jywYZ::yA:::::::::::::::::::::::::::::::::: ::::;:xIJ:Zy=::::::::::::::::::::::::::::::::::::::Z::yAB:jy;::::::::: ::::::::::::::::::::::::::::<:vY:Z::yA::::::::::::::::::::::::::::::ra yvyB:vqyxyyyY::y>>:xExI::Z:jywY:::::::::::::::::::::::::::::::::vMJ:ri ry::<:yqy;:::::::::::::::::::::::::::::::::<:;:Zyuy::;Z::jywY::>:ry::y >>:xsIwyy=:::::::::::::::::::::::::::::;Z::jywY:Z::yA::::::::::::::::: ::::::::::::::::rYyuyravqytavyy=:y=Zy:xEy_YlyryyA::::::: :::::::::::::::::::::jyB:vMxEryyA:<:yqy;:::::::::::::::::::::::::::::: ::::jIsavyy=::ravMxyy;>:ry:::y]ykyyIvMxAxI:::::::::::::::::::::::::::: :ZYlIsyyA:J:Zy=:>:ry::::::::::::::::::::::::::::::::y^y[y=::ra:yAJ:Zy= >:ry:::y_:jIsyyA:::::::::::::::::::::::::::::xCy=yA:J:ryxI:J:Zy=:::::: :::::::::::::::::::::::::vIxAyyy:::xAy]yyY:;:xIJ:B:vYyA::B:>:xiy=::Zy< <:yuYxEyyyxC;ZYlyyI>:xEy_YL:B:Zy;xyy;::::::::::::::ZykyryyA::<:yqy;J:B :vYyA:;Z::::xiy=:::::::::vQ<:;ZykYsyyqI;y^yyYB:>::jywY:::::::vIxAyyy:: :xAy]yyY:B:>:jywy::yA::ra>:xCyyy:::xA;:ryxCy_YL::y]yyYy]:J:ravyy=y_y;x Eyyy:::::::::::::::xAy]yyY:::;Z::jywy::yA:ri>:<:rivyy=::::::::::y]:jIk ywMxEy^::xAyyyJ:B:Zyuy::::::::y]ykyyI::ZykyryyA:::::::xE;Zy[y=::B:>:xi y=vMxC;Zy[y=y>vQxyy;ZyL:rYry:::::::::::::::xAy]yyY:::::::::;Z::xiy=::: :::::::>:<:yqy;xCy^:jyjy;>:<:ryxI:::::::jyB:jy;::xAZy=:::::::>:<:yqy;: :<:;Zyuy:rYZ:jywYvQ:yA::;Z:jywY:::Z:J::::::yqy;::::xAZy=::::::::>:<:ry xI::::::::::J:B:vYyA:y=>:xiy=>:<:ryxI:::::::J:B:vYyA::B:>:xiy=:::::::v IjYsyyA::xA:xIZyL:B:vQxyywQ<:yqy;:xEB:vYyA::::::::::::::B:>:xiy=:::::: ::>:<:ryxIJ:Zy=:::ZykyryyA::jyB:vQxyy;vQ<:;ZylyyI>:<:ryxI:::::::J:B:vY yA::B:>:xiy=:::::::vMjIsyyA::xE;ZykyyI:vI<:y>yA:::<:;ZykyyI::::::::::: :::>:<::yqy;:::::::J:B:Zyuyri:xI::::xAy]yyY:::y?vIvY:rY>:xCxI>:<:ryxI: ::::::J:B:vYyA::B:>:xiy=:::::::vQjyryyA::jyB:vQxyy;vQ<:y]yyY::riZ:jyjy ;:::::::::::::Z:jywy:J:ryxY:Zy=:::::::>:<:ryxIxC;:xI:::::::::xA;ZylyyI ZyL:rYvyy=;Z::xEyyy:::::::J:B:vYyA::B:>:xiy=:::::::Z:J:ryxI:::xA;:xIjY C:>:xEyyy:jYC:>:xAyyy:::::<:;:::::vYyA:::B:jy=:;Zyuy;:xI:::::::J:B:::Z yuy::::::::::y]::yAriZ:jYsyyQ:B:ZYlyyiYlyryyA:::::J:B:vYyA::B:>:xiy=:: :::::ZyK:riry:::jyrYvQvYy]:J:ravyy=ri>:<:y]yyY::::::::::::::::;Z::jywY ::::::::>:<:ryxIxC;:xI::::::::::xAy]ylyyiyK:B:vMxyy?:<:ravQxAryxI::::: J:B:vYyA::B:>:xiy=::::::ZYL:B::Zyuy:::ravIjyra>:jyyIZyK::yyy:J:rivyy=: ::::::::::::B:>:xiy=::::::::>:<:ryxIxE;:xI::::::::::jIsY>:xAy^::<:;:Zy uyy>vIxyy;:::::>:<:yqy;::<:;Zyuy::::::::raZ:jywY::::::>:<:yqy;<:;Zylyy I:>:Zy=:::::::::::::B:>:xiy=::::::::>:<:ryxIJ:Zy]::yA:::::::::::B:>:xi y=:::::::::>:xAxI::Z::vY:::::::::y]:jIsyyA:::::jy:y?yA<:;Zyuy:ri>:xCxI :::::::::::::Z:jyryyA::::::::J:B:Zyuy::y^::yA:::::::::::rY>:xEyyy::::: ::::jyrYvyy=::rYvIxyy;::::::::vQ<:y=yQ::xI::::vMjIsyyAxErYvQxyywQ<:;Zy kyyI:::::::::::::ZykyryyA::::::::xEB:Zyuy:ra>:<:yqy;:::::::::::xCravyy =:::::::::y]ykyyI::ZykyryyA:::::::::xC;:vQjywY::::vQJ:ryxIjYsY>:<:y]yl YC:vYyA:::::::::::::rYvIxyy;:::::::>:<::::::yqy;:::::::::::xEB:vYyA::: ::::::y]YlyyI::Zy;vY::::::::::y?>:xCxI:::Zy<<:ryxI:::::::::::::::::::x Ajy;:::::::::::::::::::::::::vQJ::yqy;::::::::jy:xI::Zy;vY:::::::::::: ::::::::::::::::::::::::::rY:yA:::::::::::::::::::::::::::::::::::::jy rivyy=::rY:yA::::::::::::::::::::::::::::::::::::::xAjy;:::::::::::::: :::::::::::::::::::::::vMxEy^:jyryyA:xCy?>:Zy=:::::::::::::::::::::::: ::::::::::::ravQJ::xI::::::::::::::::::::::::::::::::::::Z:J::ryxI:J:B ::vYyA:::::::::::::::::::::::::::::::::::::;Z::jywY::::::::::::::::::: :::::::::::::::::ZyL:B:vMxyy;:vQJ:rary:::::::::::::::::::::::::::::::: :::::jY;;ZY\\y=::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::1:" } {TEXT -1 4 " " }}{PARA 260 "" 0 "" {TEXT -1 20 "for some function \+ " }{TEXT 421 1 "f" }{TEXT -1 9 ". If " }{TEXT 467 1 " " } {XPPEDIT 468 1 "F=z^3*i+x^3*j+y^3*k" "/%\"FG,(*&%\"zG\"\"$%\"iG\"\"\"F )*&%\"xGF'%\"jGF)F)*&%\"yGF'%\"kGF)F)" }{TEXT -1 1 " " }{TEXT 466 1 " \+ " }{TEXT -1 2 " " }{TEXT 420 16 " then what is " }{XPPEDIT 19 1 "f( 1/sqrt(3),1/sqrt(3),1/sqrt(3))" "-%\"fG6%*&\"\"\"F&-%%sqrtG6#\"\"$!\" \"*&F&F&-F(6#F*F+*&F&F&-F(6#F*F+" }{TEXT 469 4 " ?\n" }}{PARA 0 "" 0 "" {TEXT 423 4 "a) " }{XPPEDIT 19 1 "1/sqrt(3)" "*&\"\"\"F#-%%sqrtG6# \"\"$!\"\"" }{TEXT 470 1 " " }{TEXT 471 17 " b) " } {XPPEDIT 19 1 "2/sqrt(3)" "*&\"\"#\"\"\"-%%sqrtG6#\"\"$!\"\"" }{TEXT 475 1 " " }{TEXT 476 19 " c) " }{XPPEDIT 19 1 "sqrt(3) " "-%%sqrtG6#\"\"$" }{TEXT 477 1 " " }{TEXT 478 18 " d) \+ " }{XPPEDIT 19 1 "4/sqrt(3)" "*&\"\"%\"\"\"-%%sqrtG6#\"\"$!\"\"" } {TEXT 479 1 " " }{TEXT 480 16 " e) " }{TEXT 426 1 " " } {XPPEDIT 19 1 "5/sqrt(3)" "*&\"\"&\"\"\"-%%sqrtG6#\"\"$!\"\"" }{TEXT 472 1 " " }{TEXT 473 4 " " }{TEXT 474 17 " " }} {PARA 0 "" 0 "" {TEXT 424 4 "f) " }{XPPEDIT 19 1 "2*sqrt(3)" "*&\"\"# \"\"\"-%%sqrtG6#\"\"$F$" }{TEXT 481 1 " " }{TEXT 482 1 " " }{TEXT 483 16 " g) " }{XPPEDIT 19 1 "sqrt(6)" "-%%sqrtG6#\"\"'" } {TEXT 484 1 " " }{TEXT 485 1 " " }{TEXT 486 12 " " }{TEXT -1 1 " " }{TEXT 425 4 "h) " }{XPPEDIT 19 1 "2*sqrt(6)" "*&\"\"#\"\"\" -%%sqrtG6#\"\"'F$" }{TEXT 487 1 " " }{TEXT 488 1 " " }{TEXT 489 39 " \+ i) 1 j) " }{TEXT -1 3 "0 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Solution" }} {PARA 3 "" 0 "" {TEXT 541 8 "Solution" }{TEXT 542 6 ": (c)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "wi th(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "curl([z^3,x^ 3,y^3],[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%,$*$ %\"yG\"\"#\"\"$,$*$%\"zGF*F+,$*$%\"xGF*F+" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 60 "curlVector := subs(\{x=1/sqrt(3),y=1/sqrt(3),z=1/sq rt(3)\},\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+curlVectorG-%'VECTO RG6#7%\"\"\"F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "n := ve ctor([1/sqrt(3),1/sqrt(3),1/sqrt(3)] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG-%'VECTORG6#7%,$*$\"\"$#\"\"\"\"\"##F-F+F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "dotprod(curlVector,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"\"$#\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 272 10 "16. If " }{XPPEDIT 347 1 "f(x,y,z)=x^3 - 2*x*y + z^2" "/-% \"fG6%%\"xG%\"yG%\"zG,(*$F&\"\"$\"\"\"*(\"\"#F,F&F,F'F,!\"\"*$F(F.F," }{TEXT 346 17 " then calculate " }{TEXT 350 1 " " }{TEXT 348 12 "div( grad(f(1" }{TEXT 490 1 "," }{TEXT 491 1 "2" }{TEXT 492 1 "," }{TEXT 351 3 "3))" }{TEXT 349 2 ". " }{TEXT 273 2 " \n" }{TEXT 271 142 "\na) \+ 1 b) 2 c) 3 d) 4 e) 5 \nf ) 8 g) 12 h) 15 i) 16 j) 20 " }{TEXT 422 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Solution" }}{PARA 3 "" 0 "" {TEXT 543 8 "Solution " }{TEXT 544 6 ": (f)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f := (x,y,z) -> x^3 - 2*x*y + z^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"fG:6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,(*$9$\"\"$\"\"\"*&F0 F29%F2!\"#*$9&\"\"#F2F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "grad(f(x,y,z),[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECT ORG6#7%,&*$%\"xG\"\"#\"\"$%\"yG!\"#,$F)F-,$%\"zGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diverge(\",[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"'\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(\{x=1,y=2,z=3\},\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 274 38 "17. Calcul ate the surface integral of " }{TEXT 277 1 " " }{XPPEDIT 278 1 "f(x,y, z)=z" "/-%\"fG6%%\"xG%\"yG%\"zGF(" }{TEXT 276 33 " over that part of \+ the surface " }{TEXT 279 1 " " }{XPPEDIT 280 1 "z=sqrt (x^2+y^2) " "/ %\"zG-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF*F+" }{TEXT 275 4 " \n" } {TEXT 326 27 "that lies above the disk " }{XPPEDIT 494 1 "x^2+y^2<1 " "2,&*$%\"xG\"\"#\"\"\"*$%\"yGF&F'F'" }{TEXT 493 2 " ." }{TEXT 328 1 "\n" }{TEXT 327 23 "a) 0 b) " }{XPPEDIT 496 1 "Pi/2" "* &%#PiG\"\"\"\"\"#!\"\"" }{TEXT 495 22 " c) " } {XPPEDIT 336 1 "sqrt(2)/2*Pi" "*(-%%sqrtG6#\"\"#\"\"\"F&!\"\"%#PiGF'" }{TEXT 329 15 " d) " }{XPPEDIT 337 1 "sqrt(3)/3*Pi" "*(-%%s qrtG6#\"\"$\"\"\"F&!\"\"%#PiGF'" }{TEXT 330 13 " e) " } {XPPEDIT 338 1 "1/2" "*&\"\"\"F#\"\"#!\"\"" }{TEXT 331 7 " \nf) " } {XPPEDIT 339 1 "sqrt(2)*Pi" "*&-%%sqrtG6#\"\"#\"\"\"%#PiGF'" }{TEXT 332 14 " g) " }{XPPEDIT 340 1 "2*sqrt(3)/3*Pi" "**\"\"#\"\" \"-%%sqrtG6#\"\"$F$F(!\"\"%#PiGF$" }{TEXT 333 12 " h) " } {XPPEDIT 341 1 "2*sqrt(2)*Pi" "*(\"\"#\"\"\"-%%sqrtG6#F#F$%#PiGF$" } {TEXT 334 12 " i) " }{XPPEDIT 342 1 "sqrt(3)*Pi" "*&-%%sqrtG6# \"\"$\"\"\"%#PiGF'" }{TEXT 335 14 " j) " }{XPPEDIT 343 1 "3/ 2*Pi" "*(\"\"$\"\"\"\"\"#!\"\"%#PiGF$" }}{PARA 3 "" 0 "" {TEXT -1 0 " " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Solution" }}{PARA 3 "" 0 "" {TEXT 545 8 "Solution" }{TEXT 546 6 ": (c)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "vectorLength := v -> sqrt(sum(v[i]^2,i=1..vectdim(v)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-vectorLengthG:6#%\"vG6\"6$%)operatorG%&ar rowGF(-%%sqrtG6#-%$sumG6$*$&9$6#%\"iG\"\"#/F6;\"\"\"-%(vectdimG6#F4F(F (" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "r := (R,theta) -> vect or([R*cos(theta),R*sin(theta),R]); #surface parameterization" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG:6$%\"RG%&thetaG6\"6$%)operatorG%&arro wGF)-%'vectorG6#7%*&9$\"\"\"-%$cosG6#9%F3*&F2F3-%$sinGF6F3F2F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "rR := map(z->diff(z,R),r(R,t heta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rRG-%'VECTORG6#7%-%$cosG 6#%&thetaG-%$sinGF+\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "rtheta := map(z->diff(z,theta),r(R,theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'rthetaG-%'VECTORG6#7%,$*&%\"RG\"\"\"-%$sinG6#%&theta GF,!\"\"*&F+F,-%$cosGF/F,\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "assume(R>0); interface(showassumed=0);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "crossprod(rR,rtheta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%,$*&%\"RG\"\"\"-%$cosG6#%&thetaGF*!\"\",$ *&F)F*-%$sinGF-F*F/,&*&F+\"\"#F)F*F**&F2F6F)F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "surfaceAreaFactor := simplify(vectorLength(\" ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2surfaceAreaFactorG*&\"\"##\" \"\"F&%#R|irGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f := (x, y,z) -> z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6%%\"xG%\"yG%\"z G6\"6$%)operatorG%&arrowGF*9&F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Int(Int(f(x,y,z)*surfaceAreaFactor*R,R=0..1),theta=0. .2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$*(%\"zG\"\"\" \"\"##F*F+%#R|irGF+/F-;\"\"!F*/%&thetaG;F0,$%#PiGF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(z=R,\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$*&%#R|irG\"\"$\"\"##\"\"\"F+/F);\"\"!F-/% &thetaG;F0,$%#PiGF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value (\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#PiG\"\"\"\"\"##F&F'F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 282 8 "18. Let " }{TEXT 315 1 "S" } {TEXT 316 142 " be that part of the boundary of the tetrahedron with \+ vertices (0,0,1), (0,0,0), (1,0,0), (0,1,0) that consists of the three triangles with " }{XPPEDIT 563 1 "x>0" "2\"\"!%\"xG" }{TEXT 497 7 ". 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"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 322 8 "Solution" }{TEXT 323 6 ": (e)\n" }}{PARA 0 "" 0 "" {TEXT -1 91 "Using Stokes's Theorem \+ the required surface integral can be calculated by the line integral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " " } {TEXT 324 5 " " }{XPPEDIT 325 1 "Int(z*dx+x*dy+y^3,z)" "-%$IntG6$, (*&%\"zG\"\"\"%#dxGF(F(*&%\"xGF(%#dyGF(F(*$%\"yG\"\"$F(F'" }}{PARA 0 " " 0 "" {TEXT -1 174 "\nintegrated over the triangle in the yz plane tr aversed from (0,1,0) to (0,0,1) to (0,0,0). Notice that x = 0 and dx \+ = 0 in the yz-plane. 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:::::::::::::Ajysy:::::::::::::::::::::::::::Z:J:ryvY<:;Zymy;B:>:xIyA: >:<:yay=J:B:vYxI:::B:>:xIyA:>:<:yay=:::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::J:vYxI::::::::::::::::::::: ::::::B::xIyA:>:<:yay=J:B:vYxI:::B:>:xIyA:>:<:yay=:: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::J:vYxI:::::::::::::::::::::::::::B::xIyA:>:<:ya y=J:B:vYxI:::B:>:xIyA:>:<:yay=:::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::J:vYxI::::::::::::::::::::::::::: B::xIyA:>:<:yay=J:B:vYxI::;:vY<:;Zymy;:;Z:jysy:::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::VEry::::::::::::::::::::::::::::>:<:yA>:xIyAZ:J:ryvY:J:B:vYxI:;Z:jys y::>:<:yA>:xIyAZ:J:ryvY::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::J?[y=:::::::::::::::::::::::::::J:B:vYJ ::yayM:B:Zymy;:;Z::xIya:J::yay=::<:;:Zymy;B:>:xIyA:::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::>:yay=:::::: :::::::::::::::::::::J:>Z:jysy>:<:::yay=::<:;::Zymy;:::;Z::xIyAZ:J:ryv Y::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::J:vYxI::::::::::::::::::::::::::::B::ry:::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::>:ry:::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::>:ry:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: >:ry:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::NLyay=:: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::J:Zy=:::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::J:Zy=:::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::J:Zy=:::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::J:VZ:J:3:" }}{PARA 0 "" 0 "" {TEXT -1 104 "Note that y = 0 on the side labelled 3 and dz = 0 on the side l abelled 1. Our integral comes down to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 " " }{TEXT 500 1 " " } {XPPEDIT 501 1 "Int(y^3,z=L..``)" "-%$IntG6$*$%\"yG\"\"$/%\"zG;%\"LG%! G" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "wher e L is the side labelled 2. We may parameterize it by " } {XPPEDIT 19 1 "y=1-t,z=t,` `*t=0..1" "6%/%\"yG,&\"\"\"F&%\"tG!\"\"/% \"zGF'/*&%#~~GF&F'F&;\"\"!F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "int((1-t)^3,t=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"%" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 262 "" 0 "" {TEXT 284 27 "19. Let S be the cylinder " }{XPPEDIT 19 1 "x^2+y^2=1,0Nw;>w=Z:>_KG:MZ=V[QZ:FxPZ:^: VZ;>;E:Sb:kxy=muyIF;J:>_:>:==K;=JF>:;::::yyyyyy:: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::Z:ryvy:J::yqy;:::::::::::::::::::::::::::::::::::::::xiy=>:ry:::::::::::::::::::::::::::::::::::::::::::::J:Z y=>:ry::::::::::::::::::::::::::::::::::::::::::xsIyI<:ywiyyyyA::J:ryx I:::::::::::::::::::::::::::::::::::::::::Z:J::ryxI::<:vY::::::::::::: :::::::::::::::::::::::::::::rYyuyraynYyeYlyyI:>:xiy=::::::::::::::::: :::::::::::::::::::::::::::y^YlyyIZ::yA::::::::::::::::::::::::::::::: :::::::::::::jIsYvyy=B:jy;<:vY:::::::::::::::::::::::::::::::::::::::: :::y]ykyyIZ::yAB:>:xiy=:>:<:::ryxI::::::::::<:;::vYyA;Z::jywYvQ<:;ZywI 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;:::::::y]ykyyI::::::Zy<<:ryxI:xC;Z:jywY::::::::::::::::::ZY:<::::::yqy;::::::::::::::::::xE B:vYyArivI<:;ZykYsi>:xiy=:::::::y]Y\\y=::::::::::::::::::::::::::::::: y?>:jywY:::::::::::::y=ry::::::::::::::::::::::::::::::::::::::::::::: ::vIxEyyy:::::::::::::::::::::::::::::::::::::::::::::::vMxEy^:jyryyA: ::::::::::::::::::::::::::::::::::::::::::::Z:J::ryxI::::::::::::::::: ::::::::::::::::::::::::::::ZyL:B:vMxyy;:::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::A:;5:" }{TEXT -1 9 " " }}{PARA 262 "" 0 "" {TEXT 503 8 " where " }{TEXT 504 1 "n" }{TEXT 505 35 " is the out ward normal on S and " }{TEXT 507 1 " " }{XPPEDIT 508 1 "F=x*i+(z^2- z)*k" "/%\"FG,&*&%\"xG\"\"\"%\"iGF'F'*&,&*$%\"zG\"\"#F'F,!\"\"F'%\"kGF 'F'" }{TEXT 506 2 " ." }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 262 " " 0 "" {TEXT 303 4 "a) " }{XPPEDIT 19 1 "0" "\"\"!" }{TEXT 304 22 " \+ b) " }{XPPEDIT 19 1 "Pi/4" "*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 305 19 " c) " }{XPPEDIT 19 1 "Pi/2" "*&%#PiG\"\" \"\"\"#!\"\"" }{TEXT 306 18 " d) " }{XPPEDIT 19 1 "Pi" " I#PiG%*protectedG" }{TEXT 307 19 " e) " }{XPPEDIT 19 1 "2*Pi" "*&\"\"#\"\"\"%#PiGF$" }{TEXT 308 11 " \nf) " }{XPPEDIT 19 1 "-Pi/3" ",$*&%#PiG\"\"\"\"\"$!\"\"F'" }{TEXT 309 16 " g ) " }{XPPEDIT 19 1 "-Pi/2" ",$*&%#PiG\"\"\"\"\"#!\"\"F'" }{TEXT 310 15 " h) " }{XPPEDIT 19 1 "-Pi" ",$%#PiG!\"\"" }{TEXT 311 16 " i) " }{XPPEDIT 19 1 "-3*Pi/2" ",$*(\"\"$\"\"\"%#PiGF% \"\"#!\"\"F(" }{TEXT 312 14 " j) " }{XPPEDIT 19 1 "-2*Pi" ", $*&\"\"#\"\"\"%#PiGF%!\"\"" }{TEXT 313 4 " " }{TEXT 302 1 "\n" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Solution" }}{PARA 3 "" 0 "" {TEXT 548 8 "Solution" }{TEXT 549 6 ": (d)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "F := vector([x,0,z^2-z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"FG-%'VECTORG6#7%%\"xG\"\"!,&*$%\"zG\"\"#\"\"\"F-!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "surface_eqn := x^2 + y^2 = 1;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%,surface_eqnG/,&*$%\"xG\"\"#\"\"\"*$ %\"yGF)F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "N := grad(lh s(surface_eqn),vector([x,y,z]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"NG-%'VECTORG6#7%,$%\"xG\"\"#,$%\"yGF+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "dotprod(F,N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,$*$%\"xG\"\"#F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Integr and := subs(\{x = cos(theta),y = sin(theta)\},x^2);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%*IntegrandG*$-%$cosG6#%&thetaG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "int(int(Integrand,z=0..1), theta = \+ 0 .. 2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#PiG" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 550 52 "Alternative calculati on using the Divergence Theorem" }}{PARA 3 "" 0 "" {TEXT -1 1 "\n" } {TEXT 551 23 "Note that on the disks " }{XPPEDIT 552 1 "x^2+y^2<1,z=0 " "6$2,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(F(/%\"zG\"\"!" }{TEXT 553 8 " an d " }{XPPEDIT 554 1 "x^2+y^2<1,z=1" "6$2,&*$%\"xG\"\"#\"\"\"*$%\"yGF 'F(F(/%\"zGF(" }{TEXT 555 10 " we have " }{TEXT 556 6 "n = -k" } {TEXT 557 6 " and " }{TEXT 558 5 "n = k" }{TEXT 559 25 " respectively . Therefore " }{XPPEDIT 19 1 "F.n=z^2-z" "/(%\"FG%\"nG,&*$%\"zG\"\"#\" \"\"F(!\"\"" }{TEXT 560 21 ". which means that " }{XPPEDIT 19 1 "F.n =0" "/(%\"FG%\"nG\"\"!" }{TEXT 561 3 " ." }}{PARA 269 "" 0 "" {TEXT -1 146 "Thus, the required surface integral is the same as the one ove r the entire boundary of the cylinder. We mat therefore use the Diverg ence Theorem.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diverge(F ,[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%\"zG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "int(int(r*int(2*z,z=0..1),r=0..1),t heta=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%#PiG" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 287 50 "20. Let S be the surface made up of the cylinder " }{XPPEDIT 530 1 "x^2+y^2=1,0Nw;>w=Z:>_KG:MZ=V[QZ :FxPZ:^:VZ;>;E:Sb:kxy=muyIF;J:>_:>:==K;=JF>:;:::: yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::Z:ryvy:J::yqy;::::::::::::::::::::::::::::::::: :::::::::::Z:J:ryxIJ:Zy=:::::::::::::::::::::::::::::::::::::::::::::; :xIJ:Zy=:::::::::::::::::::::::::::::::::::::::::rayvyB:vqyxyyyY:::;Zy uy::::::::::::::::::::::::::::::::::::::::::<:;:Zyuy::B:jy;::::::::::: :::::::::::::::::::::::::::HvyyyyyI::rYyuyraynYyeYlyyI:>:xiy=::::::::: :::::::::::::::::::::::::::::Zlyyyyyy=:::jIsavyy=B:jy;:::::::::::::::: :::::::::::::::::::::::TqyyyyyI::::y^ykyyIZ::yAB:jy;:::::::::::::::::: :::::::::::::::::::Hlyyiy=:::jyrYvyy=B:jy;<:;Zyuy::;Z::::xiy=::::::::: Z:J:::yqyK:B::vYyAy_:J:rYymYsyyqIC:vMxyy[:jYsivM<:;:vIjyyI:::::::::y]y kyyI:>:<:ryxY:Zy=:ZyL:B:ZylyyI:::::::::::>:<:ryxIJ:B:Zyuy:y]:jIsyyqIsi vM<:rYvyyuY>:<:y^yyYrivIjYsyyA:::Toyyiy=:::jyrYvyy=:::::::;Z::xiy=:::: :::::::>:<:ryxIJ:B:Zyuy:;Z:jywYZYlIC:vIvYraZy\\y=:y_:jyjy;::::::::y]:j ywY:::::::>:<:ryxI:::::::::::J:B:Zyuy:;Z::xiy=>:<:yqy;jy:;Zyuyri:xI::< :;Zyuy::::::::J:B:vYyA:::::::;Z::xiy=>:ry::::rYvIxyy;:::>:<:ryxIJ:B:Zy uy:y]:jYsyyAjYC:>:xEyyyxE;Zyuy::y?>:xiy=::ZYwyyyY::::>:<:yqy;::::::J:B :Zyuyri:xI::::xAy]yyY::::;Z::xiy=>:<:ryxIjY;y=yA:xA;ZYlyyI:::>:<:y]yyY :::R]yyyyyy::::>:<:yqy;::::::J:B:Zyuyra>:ry::::::::::;Z::xiy=>:<:ryxI: xA;ZylyyIZyL:rYvyy=::y?>:xAxI:::Zdwyyyyy=:::J:B:vYyA:::::::;Z::::xiy=: ::::::::>:<:ryxIJ:B:Zyuy::y]::yAriZ:jYsyyA:xE;Z:jyryyA::::H:<:ryxIxCZy=:::::::::>:<:ravyyuiZ:jyjy;:ZykyrivyyuY>:<:y>yA jYC:>:xAyyy:::::H:<:ryxIxEZy=::::::::Z:J::rivI<: ;ZykyyI:::vMxA;:vM<:ryyAjyB:ZyyY:B:vQxyy;:Zfyyyyyy=:::J:B:vYyA:::::::; Z::xiy=>:ryB:jy;::::::::::::::::::<:;Zyuy:;Z:jYky;Z::vY::r\\yyyxI::::; Zy[y=:::::::;Z::xiy=:vM:yA::::::::::::::::::rY>:xExI>:<:yqy;jYC:vMxyy; :ZElyyI::::y]ykyyI::::::ZyL:B:Zyuy:ra>:<:yqy;::::::::::::::::::xCravyy =y?vIxEyyyxE;Z:jyryyA::`oyyiy=:::jyrYvyy=::::::;Z::::::jywY::::::::::: :::::::Zy<<:yqy;xEy]:J:rYvQxE;Zyuy::BjyyyxI::::y]Y\\y=:::::::::::::::: :::::::::::::::y?>:jywY:::::::Z\\yyyuy::::vIZy=::::::::::::::::::::::: ::::::::::::::::::R[yyyxI::::y]ylyyI:::::::::::::::::::::::::::::::::: :::::::ZJwyyuy::::vMxEy^:jyryyA::::::::::::::::::::::::::::::::::::::: :ljyyiy=::::<:;:Zyuy::::::::::::::::::::::::::::::::::::::::r=yqy;:::: xE;Z:jIsyyA::::::::::::::::::::::::::::::::::::::::\\ryyyyyI:::::::::: :::::::::::::::::::::::::::::::::::::B[yyyyyy::::::::::::::::::::::::: ::::::::::::::::::::::Z[yyyyyy=::::::::::::::::::::::::::::::::::::::: ::::::::@wyyiy=:::::::::::::::::::::::::::::::::::::::::::::::H:1:" }{TEXT -1 6 " " }}{PARA 0 "" 0 "" {TEXT -1 6 " where" }{TEXT 525 2 " n" }{TEXT 526 35 " is the outwa rd normal on S and " }{TEXT 528 1 " " }{XPPEDIT 529 1 "F=x^3/3*i+z^2 *j +y^2*k" "/%\"FG,(*(%\"xG\"\"$F'!\"\"%\"iG\"\"\"F**&%\"zG\"\"#%\"jGF *F**&%\"yGF-%\"kGF*F*" }{TEXT 527 2 " ." }{TEXT -1 3 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT 286 4 "a) " }{XPPEDIT 19 1 "0" "\"\"!" }{TEXT 288 15 " \+ b) " }{XPPEDIT 19 1 "Pi/12" "*&%#PiG\"\"\"\"#7!\"\"" }{TEXT 289 19 " c) " }{XPPEDIT 19 1 "Pi/8" "*&%#PiG\"\"\"\"\")!\" \"" }{TEXT 290 19 " d) " }{XPPEDIT 19 1 "Pi/6" "*&%#PiG \"\"\"\"\"'!\"\"" }{TEXT 291 18 " e) " }{XPPEDIT 19 1 "P i/4" "*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 292 11 " \nf) " }{XPPEDIT 19 1 "Pi/3" "*&%#PiG\"\"\"\"\"$!\"\"" }{TEXT 293 19 " g) \+ " }{XPPEDIT 19 1 "Pi/2" "*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 294 17 " \+ h) " }{XPPEDIT 19 1 "Pi" "I#PiG%*protectedG" }{TEXT 295 18 " i) " }{XPPEDIT 19 1 "2*Pi" "*&\"\"#\"\"\"%#PiGF$" } {TEXT 296 18 " j) " }{XPPEDIT 19 1 "3*Pi" "*&\"\"$\"\"\" %#PiGF$" }{TEXT 297 4 " " }{TEXT 285 1 "\n" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Solution" }}{PARA 3 "" 0 "" {TEXT 564 8 "Solution" } {TEXT 565 8 ": ( e )\n" }}{PARA 0 "" 0 "" {TEXT -1 28 "Using the Diver gence Theorem" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "F := vector([x^3/3,z^2,y^2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'VECTORG6#7%,$*$%\"xG\"\"$#\"\"\"F,*$%\"zG\"\"# *$%\"yGF1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "diverge(F,[x,y ,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$%\"xG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Int(Int(r*Int(diverge(F,[x,y,z]),z= 0..1),r=0..1),theta=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$In tG6$-F$6$*&%\"rG\"\"\"-F$6$*$%\"xG\"\"#/%\"zG;\"\"!F*F*/F)F2/%&thetaG; F3,$%#PiGF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(x=r*cos (theta),\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$*&%\"rG \"\"\"-F$6$*&F)\"\"#-%$cosG6#%&thetaGF./%\"zG;\"\"!F*F*/F)F5/F2;F6,$%# PiGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#\"\"\"\"\"%" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "11" 0 }{VIEWOPTS 1 1 0 1 1 1803 }