{VERSION 4 0 "SGI MIPS UNIX" "4.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {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 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 1 8 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 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 "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 256 12 "241 Lab 7. " }{TEXT 270 20 "J. \+ Ringland 4/4/2001" }{TEXT 271 1 "\n" }{TEXT 265 44 "Specifying 3D Regi ons and doing 3D integrals" }}{PARA 0 "" 0 "" {TEXT 269 4 "READ" } {TEXT -1 282 " THE FOLLOWING!!!\nI have written a Maple command to ena ble you to see if you've specified the region you think you have in 3D space.\nClick the \"+\" box below and execute the cell containing th e definition of the \"region3d\" command.\nThen close the cell again b y clicking the \"-\" box." }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 "\"re gion3d\" and \"integral3d\" commands " }{TEXT 266 65 "JR 11/8&13/2000 &4/4/2001 & 4/17/2001(range sanity checking added)" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 7273 "Plot3d := proc(triple,prange,qrange)\n \+ # print(triple,prange,qrange);\n # This required because a range of the form a..a \n # causes plot3d to throw an error.\n if op(rhs(q range))[1]=op(rhs(qrange))[2] then \n # nothing (is there a better \+ thing to do?)\n else \n if ( op(rhs(prange))[1]=op(rhs(prange))[2 ] ) then\n # nothing (is there a better thing to do?)\n else \n plot3d(triple,prange,qrange,axes=boxed,\n lab els=['axis1','axis2','axis3'],grid=[25,25]);\n fi;\n fi;\nend:\n\n region3d := proc(xyz,range3,range2,range1) # inner variable now comes \+ first 11/13/2000\n\n local v1,v2,v3,v1lo,v1hi,v2lo,v2hi,v3lo,v3hi,\n \+ fudge,v1fudge,v2fudge,\n p1lo,p1hi,p2lo,p2hi,p3lo,p3hi,x ,y,z,jacobian;\n\n x:=xyz[1]; y:=xyz[2]; z:=xyz[3];\n kernelopts(ASS ERT=true);\n ASSERT(whattype(rhs(range1))=`..`,`: 1st variable's rang e is bad`);\n ASSERT(whattype(rhs(range2))=`..`,`: 2nd variable's ran ge is bad`);\n ASSERT(whattype(rhs(range3))=`..`,`: 3rd variable's ra nge is bad`);\n v1:=lhs(range1); v2:=lhs(range2); v3:=lhs(range3);\n \+ v1lo:=op(rhs(range1))[1]; v1hi:=op(rhs(range1))[2];\n v2lo:=op(rhs(r ange2))[1]; v2hi:=op(rhs(range2))[2];\n v3lo:=op(rhs(range3))[1]; v3h i:=op(rhs(range3))[2];\n\n # Sanity checking of variable ranges:\n A SSERT((whattype(v1lo)=float or whattype(v1lo)=integer),\n \+ `: the endpoints of the range of the outer (last) variable must be jus t plain numbers`);\n ASSERT((whattype(v1hi)=float or whattype(v1hi)=i nteger),\n `: the endpoints of the range of the outer (las t) variable must be just plain numbers`);\n ASSERT(numboccur(v2lo,lhs (range3))=0,\n `: the range of the middle variable can't d epend on the inner variable`);\n ASSERT(numboccur(v2hi,lhs(range3))=0 ,\n `: the range of the middle variable can't depend on th e inner variable`);\n\n\n # The following is to work around the diffi culty that occurs\n # for example in parametrizing a hemisphere by ca rtesian coords,\n # where the range endpoint formulas may be exercise d on the boundary\n # of their domain: then with f.p. arithmetic valu es may actually\n # be requested just outside the domain, causing hol es in the surfaces\n # that Maple draws. I therefore stay a tiny bit \+ inside the specified\n # endpoints: a small enough amount that hopefu lly it makes no visible glitch,\n # but large enough to prevent any h oles. \n fudge:=1.0e-9;\n v1fudge:=fudge*(v1hi-v1lo);\n v1lo:=v1lo+ v1fudge;\n v1hi:=v1hi-v1fudge;\n\n v2fudge:=fudge*simplify(v2hi-v2lo );\n v2lo:=v2lo+v2fudge;\n v2hi:=v2hi-v2fudge;\n\n p1lo:=Plot3d(sub s(v1=v1lo,[x,y,z]), \n v2=subs(v1=v1lo,v2lo)..subs(v1=v 1lo,v2hi), \n v3=subs(v1=v1lo,v3lo)..subs(v1=v1lo,v3hi) );\n\n p1hi:=Plot3d(subs(v1=v1hi,[x,y,z]), \n v2=subs( v1=v1hi,v2lo)..subs(v1=v1hi,v2hi), \n v3=subs(v1=v1hi,v 3lo)..subs(v1=v1hi,v3hi));\n\n p2lo:=Plot3d(subs(v2=v2lo,[x,y,z]), \n v1=subs(v2=v2lo,v1lo)..subs(v2=v2lo,v1hi),\n \+ v3=subs(v2=v2lo,v3lo)..subs(v2=v2lo,v3hi));\n\n p2hi:=Plot3d(su bs(v2=v2hi,[x,y,z]), \n v1=subs(v2=v2hi,v1lo)..subs(v2= v2hi,v1hi), \n v3=subs(v2=v2hi,v3lo)..subs(v2=v2hi,v3hi ));\n\n p3lo:=Plot3d(subs(v3=v3lo,[x,y,z]), \n v1=subs (v3=v3lo,v1lo)..subs(v3=v3lo,v1hi), \n v2=subs(v3=v3lo, v2lo)..subs(v3=v3lo,v2hi));\n\n p3hi:=Plot3d(subs(v3=v3hi,[x,y,z]), \+ \n v1=subs(v3=v3hi,v1lo)..subs(v3=v3hi,v1hi), \n \+ v2=subs(v3=v3hi,v2lo)..subs(v3=v3hi,v2hi));\n\n jacobian:=li nalg[matrix](3,3,\n [diff(x,v1),diff(x,v2),diff(x,v3),\n \+ diff(y,v1),diff(y,v2),diff(y,v3),\n diff(z,v1),diff(z, v2),diff(z,v3)]);\n print('jacobian'=abs(simplify(expand(linalg[det]( jacobian)))));\n\n plots[display](p1lo,p1hi,p2lo,p2hi,p3lo,p3hi);\nen d:\n\nintegral3d:= proc(f,xyz,range3,range2,range1) # inner variable n ow comes first 11/13/2000\n local x,y,z,v1,v2,v3,v1lo,v1hi,v2lo,v2hi, v3lo,v3hi,jacobian,answer;\n print(region3d(xyz,range1,range2,range3) );\n x:=xyz[1]; y:=xyz[2]; z:=xyz[3];\n kernelopts(ASSERT=true);\n \+ ASSERT(whattype(rhs(range1))=`..`,`: 1st variable's range is bad`);\n \+ ASSERT(whattype(rhs(range2))=`..`,`: 2nd variable's range is bad`);\n ASSERT(whattype(rhs(range3))=`..`,`: 3rd variable's range is bad`); \n v1:=lhs(range1); v2:=lhs(range2); v3:=lhs(range3);\n v1lo:=op(rhs (range1))[1]; v1hi:=op(rhs(range1))[2];\n v2lo:=op(rhs(range2))[1]; v 2hi:=op(rhs(range2))[2];\n v3lo:=op(rhs(range3))[1]; v3hi:=op(rhs(ran ge3))[2];\n\n # Sanity checking of variable ranges:\n ASSERT((whatty pe(v1lo)=float or whattype(v1lo)=integer),\n `: the endpoi nts of the range of the outer (last) variable must be just plain numbe rs`);\n ASSERT((whattype(v1hi)=float or whattype(v1hi)=integer),\n \+ `: the endpoints of the range of the outer (last) variable m ust be just plain numbers`);\n ASSERT(numboccur(v2lo,z)=0,\n \+ `: the range of the middle variable can't depend on the inner vari able`);\n ASSERT(numboccur(v2hi,z)=0,\n `: the range of t he middle variable can't depend on the inner variable`);\n \n jacobi an:=linalg[matrix](3,3,\n [diff(x,v1),diff(x,v2),diff(x,v3), \n diff(y,v1),diff(y,v2),diff(y,v3),\n diff(z,v1), diff(z,v2),diff(z,v3)]);\n jacobian:=abs(simplify(expand(linalg[det]( jacobian))));\n print('jacobian'=jacobian); \n print(Int(Int(Int(f* jacobian,v3=v3lo..v3hi),v2=v2lo..v2hi),v1=v1lo..v1hi));\n print(Int(I nt(int(f*jacobian,v3=v3lo..v3hi),v2=v2lo..v2hi),v1=v1lo..v1hi));\n pr int(Int(int(int(f*jacobian,v3=v3lo..v3hi),v2=v2lo..v2hi),v1=v1lo..v1hi ));\n answer:=int(int(int(f*jacobian,v3=v3lo..v3hi),v2=v2lo..v2hi),v1 =v1lo..v1hi); print(answer);\n return (answer);\nend:\n\n\npl ain_integral3d:= proc(f,xyz,range3,range2,range1) # inner variable now comes first \n local x,y,z,v1,v2,v3,v1lo,v1hi,v2lo,v2hi,v3lo,v3hi,ja cobian,answer;\n x:=xyz[1]; y:=xyz[2]; z:=xyz[3];\n kernelopts(ASSER T=true);\n ASSERT(whattype(rhs(range1))=`..`,`: 1st variable's range \+ is bad`);\n ASSERT(whattype(rhs(range2))=`..`,`: 2nd variable's range is bad`);\n ASSERT(whattype(rhs(range3))=`..`,`: 3rd variable's rang e is bad`);\n v1:=lhs(range1); v2:=lhs(range2); v3:=lhs(range3);\n v 1lo:=op(rhs(range1))[1]; v1hi:=op(rhs(range1))[2];\n v2lo:=op(rhs(ran ge2))[1]; v2hi:=op(rhs(range2))[2];\n v3lo:=op(rhs(range3))[1]; v3hi: =op(rhs(range3))[2];\n\n # Sanity checking of variable ranges:\n ASS ERT((whattype(v1lo)=float or whattype(v1lo)=integer),\n `: the endpoints of the range of the outer (last) variable must be just \+ plain numbers`);\n ASSERT((whattype(v1hi)=float or whattype(v1hi)=int eger),\n `: the endpoints of the range of the outer (last) variable must be just plain numbers`);\n ASSERT(numboccur(v2lo,z)=0, \n `: the range of the middle variable can't depend on the inner variable`);\n ASSERT(numboccur(v2hi,z)=0,\n `: the range of the middle variable can't depend on the inner variable`);\n \+ \n jacobian:=linalg[matrix](3,3,\n [diff(x,v1),diff(x,v2),d iff(x,v3),\n diff(y,v1),diff(y,v2),diff(y,v3),\n d iff(z,v1),diff(z,v2),diff(z,v3)]);\n jacobian:=abs(simplify(expand(li nalg[det](jacobian))));\n \n return(int(int(int(f*jacobian,v3=v3lo.. v3hi),v2=v2lo..v2hi),v1=v1lo..v1hi));\nend:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 51 "Now proceed with the exercises in the next section." }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Exercises " }}{PARA 0 "" 0 "" {TEXT 258 51 "As you read through the examples bel ow, be sure to " }{TEXT 259 18 "twirl the 3d plots" }{TEXT 260 55 " so you're really sure what these 3d regions are like!\n" }{TEXT -1 1 " \+ " }{TEXT 261 3 "(1)" }{TEXT -1 108 " The simplest 3D region is the rec tangular box. Here's an example,showing how to use the \"region3d\" co mmand:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "region3d([x,y,z],x =0..1,y=0..2,z=0..3);" }}}{PARA 0 "" 0 "" {TEXT 262 3 "(2)" }{TEXT -1 189 " For more interesting regions, we use the facts that\n the ran ge of the second variable can depend on the first variable, and \n \+ the range of the third variable can depend on the first " }{TEXT 257 3 "and" }{TEXT -1 78 " second variables.\nSo we could make the maximum y value depend on x like this:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "region3d([x,y,z],y=0..sqrt(1-x^2),x=0..1,z=0..1);" }}}{PARA 0 "" 0 "" {TEXT 263 3 "(3)" }{TEXT -1 72 " And we could make the maximum z value depend on both x and y like this:" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 57 "region3d([x,y,z],z=0..1+x^2+y^2,y=0..sqrt(1-x^2),x= 0..1);" }}}{PARA 0 "" 0 "" {TEXT -1 112 "Now the previous examples are all in what we could call \"zyx\" format: the endpoints for z can dep end on y and x," }}{PARA 0 "" 0 "" {TEXT -1 167 "the endpoints for y c an depend on x, and the endpoints for x are just numbers. \nBut x,y,z \+ don't have to have those roles. For example, here's a region in \"xyz \" format:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "region3d([x,y, z],x=-2+z^2+y^2..3-z^2-y^2, y=-sqrt(2-z^2)..sqrt(2-z^2),\n \+ z=-1..1);" }}}{PARA 0 "" 0 "" {TEXT 264 3 "(4)" }{TEXT -1 81 " No w it's your turn!\n See if you can figure out how to do the following \+ pyramid:\n" }{GLPLOT3D 400 300 300 {PLOTDATA 3 "6)-%%MESHG6#X.%)anythi ngG6\"6\"^[[[[]\\q^`r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