DesignMat Uge 9Preben Alsholm, 26/10 2010Pakkerrestart;if ssystem("hostname")=[0,"PC-PKA1"] then
libname:="C:/Documents and Settings/alsholm/Dokumenter/Kopi af DropBox/DMat",libname
elif ssystem("hostname")=[0,"pc-pka"] then
libname:="F:/DMat",libname
elif ssystem("hostname")=[0,"alsholm-PC"] then
libname:="F:/DMat",libname
else print("Fremmed computer. Skriv selv libname.")
end if;with(DMat):with(plots):Partielle aflededeg:=(x,y)->sin(2*x+3*y);Definitionen p\303\245 den partielle afledede af f mht. x i punktet (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):Limit('(g(x[0]+h,y[0])-g(x[0],y[0]))/h',h=0);Alts\303\245 i dette konkrete tilf\303\246lde:%;Gr\303\246nsev\303\246rdien ervalue(%);Det var da voldsomt! Men det er jo blotcombine(%);Direkte i MapleD[1](g)(x[0],y[0]);eller ved brug af diff:diff(g(x,y),x);I det konkrete punkt (0,0):D[1](g)(0,0);Den partielle afledede af g mht. y (variabel nummer 2):D[2](g)(a,b);Vi betragtede tidligere funktionen f givet vedf:=(x,y)->-6*y/(2+x^2+y^2);Grafen ser s\303\245ledes ud:plot3d(f(x,y),x=-2..2,y=-4..4,axes=framed);Det kunne v\303\246re interessant at finde de punkter (x,y) i hvilke begge partielle afledede er nul. S\303\245danne punkter vil blive kaldt station\303\246re punkter for funktionen f.Vi forlanger alts\303\245 f\303\270lgende ligninger opfyldt:{diff(f(x,y),x)=0, diff(f(x,y),y)=0};De punkter, der opfylder disse to ligninger, ersolve(%, [x,y],explicit);F\303\270rst kigger vi p\303\245 punktet (0,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==):x0:=0: y0:=sqrt(2):
p1x:=spacecurve([x,y0,f(x,y0)],x=-2+x0..2+x0,thickness=3,color=red):p3x:=spacecurve([x,y0,f(x0,y0)+(x-x0)*D[1](f)(x0,y0)],x=-2+x0..2+x0,thickness=3,color=blue):p2:=plot3d(f(x,y),x=-2+x0..2+x0,y=-4+y0..4+y0,style=patchnogrid,axes=framed,transparency=0.5):display(p1x,p2,p3x);p1y:=spacecurve([x0,y,f(x0,y)],y=-4+y0..4+y0,thickness=3,color=red):p3y:=spacecurve([x0,y,f(x0,y0)+(y-y0)*D[2](f)(x0,y0)],y=-4+y0..4+y0,thickness=3,color=blue):display(p1y,p2,p3y);display(p1x,p1y,p2,p3x,p3y);Punktet (0,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbXNxcnRHRiQ2Iy1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==) er et minimumspunkt.Dern\303\246st kigger vi p\303\245 punktet (0, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2JS1GLDYtUSomdW1pbnVzMDtGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjIyMjIyMmVtRicvRkVGTC1JJm1zcXJ0R0YkNiMtSSNtbkdGJDYkUSIyRidGL0YvRitGLw==):x0:=0: y0:=-sqrt(2):
p1x:=spacecurve([x,y0,f(x,y0)],x=-2+x0..2+x0,thickness=3,color=red):p3x:=spacecurve([x,y0,f(x0,y0)+(x-x0)*D[1](f)(x0,y0)],x=-2+x0..2+x0,thickness=3,color=blue):p2:=plot3d(f(x,y),x=-2+x0..2+x0,y=-4+y0..4+y0,style=patchnogrid,axes=framed,transparency=0.5):display(p1x,p2,p3x);p1y:=spacecurve([x0,y,f(x0,y)],y=-4+y0..4+y0,thickness=3,color=red):p3y:=spacecurve([x0,y,f(x0,y0)+(y-y0)*D[2](f)(x0,y0)],y=-4+y0..4+y0,thickness=3,color=blue):display(p1y,p2,p3y);display(p1x,p1y,p2,p3x,p3y);Punktet (0, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2JS1GLDYtUSomdW1pbnVzMDtGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjIyMjIyMmVtRicvRkVGTC1JJm1zcXJ0R0YkNiMtSSNtbkdGJDYkUSIyRidGL0YvRitGLw==) er et maksimumspunkt.TangentplanVi tegner kurver og tangenter som ovenfor, men i et tilf\303\246ldigt valgt punkt:x0:=1/2: y0:=-1:
p1x:=spacecurve([x,y0,f(x,y0)],x=-2+x0..2+x0,thickness=3,color=red):p3x:=spacecurve([x,y0,f(x0,y0)+(x-x0)*D[1](f)(x0,y0)],x=-2+x0..2+x0,thickness=3,color=blue):p2:=plot3d(f(x,y),x=-2+x0..2+x0,y=-4+y0..4+y0,style=patchnogrid,axes=framed,transparency=0.5):display(p1x,p2,p3x);p1y:=spacecurve([x0,y,f(x0,y)],y=-4+y0..4+y0,thickness=3,color=red):p3y:=spacecurve([x0,y,f(x0,y0)+(y-y0)*D[2](f)(x0,y0)],y=-4+y0..4+y0,thickness=3,color=blue):display(p1y,p2,p3y);display(p1x,p1y,p2,p3x,p3y);Normalen til tangentplanen vises nu ogs\303\245 (sort):pn:=spacecurve([x0+s*D[1](f)(x0,y0),y0+s*D[2](f)(x0,y0),f(x0,y0)-s],s=-2..2,thickness=3,color=black):display(p1x,p1y,p2,p3x,p3y,pn,scaling=constrained);Animering (tag et billede af gangen):q1:=display(p2,p1x):
q2:=display(q1,p3x):
q3:=display(q2,p1y):
q4:=display(q3,p3y):
q5:=display(q4,pn):
display(p2,q1,q2,q3,q4,q5,insequence=true,scaling=constrained);Ligningen for tangentplanen er z = 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 hvor h\303\270jresiden er lineariseringen af f i punktet (x0,y0).I vores tilf\303\246lde er lineariseringen:L:=f(x0,y0)+D[1](f)(x0,y0)*(x-x0)+D[2](f)(x0,y0)*(y-y0);Eller ved brug af mtaylor:mtaylor(f(x,y),[x=x0,y=y0],2);plot3d([f(x,y),L],x=-2..2,y=-4..4,axes=framed,style=[patchnogrid,patch],shading=[XYZ,Z],transparency=[.1,.9],scaling=constrained);display(%,q5);Partielle afledede af h\303\270jere ordeng(x,y);Diff(g(x,y),x,x): %=value(%);Diff(g(x,y),x,y): %=value(%);Diff(g(x,y),y,x): %=value(%);Diff(g(x,y),y,y): %=value(%);D[1,1](g)(a,b);'D[1,1](g)(a,b)'=D[1,1](g)(a,b);D[1,2](g)(a,b);D[2,2](f)(0,2);Eksempel p\303\245 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Bem\303\246rkning: I dette afsnit betegnes partiel afledet mht. f\303\270rste variabel med LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2JC1JJW1zdWJHRiQ2JS1JI21pR0YkNiVRImZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRictRiM2JC1JI21uR0YkNiRRIjFGJ0YvRi8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YvRitGLw== osv.f:=(x,y)->x^3*y/(x^2+y^2);f(0,0):=0:De partielle afledede af f mht. x og y eksisterer for alle punkter forskellige fra (0,0), men faktisk ogs\303\245 i (0,0):Limit(('f(x,0)-f(0,0)')/x,x=0);vi f\303\245r simpelthen%;Tilsvarende:Limit(('f(0,y)-f(0,0)')/y,y=0);%;S\303\245 de partielle afledede af f\303\270rste orden i (0,0) er begge 0.For at se, at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtRiM2KUYrLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEjMTJGJ0YvLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJ0YvLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRiw2LVEwJkFwcGx5RnVuY3Rpb247RidGL0YyRjVGN0Y5RjtGPUY/RkFGRC1GIzYnLUYsNi1RIihGJ0YvL0YzRlFGNS9GOEZRRjlGO0Y9Rj8vRkJRLDAuMTY2NjY2N2VtRicvRkVGZW8tRiM2Jy1GVzYkRmluRi8tRiw2LVEiLEYnRi9GMi9GNkZRRjdGOUY7Rj1GP0ZBL0ZFUSwwLjMzMzMzMzNlbUYnRmlvRlpGLy1GLDYtUSIpRidGL0Zib0Y1RmNvRjlGO0Y9Rj9GZG8vRkVRLDAuMTExMTExMWVtRidGWkYvRitGWkYvRitGWkYv eksisterer betragter vi gr\303\246nsev\303\246rdienLimit( '(D[1](f)(0,y)-D[1](f)(0,0))'/y,y=0);Vi ved, at 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 = 0, s\303\245 vi skal blot betragteLimit( 'D[1](f)(0,y)'/y,y=0);%;Vi ser, at 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 = 0.Tilsvarende skal vi se, at 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 eksisterer:Limit( '(D[2](f)(x,0)-D[2](f)(0,0))'/x,x=0);Vi ved, at 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 = 0, s\303\245 vi skal blot betragteLimit( 'D[2](f)(x,0)'/x,x=0);%;Vi ser, at 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 = 1. Dermed er de to blandede afledede i (0,0) alts\303\245 forskellige!S\303\245dan ser grafen for vores funktion ud:plot3d(f(x,y),x=-1..1,y=-1..1,axes=framed,caption="Grafen for selve funktionen");plot3d(D[1](f)(x,y),x=-1..1,y=-1..1,axes=framed,caption="Den partielle afledede af f mht. x"):
plot3d(D[2](f)(x,y),x=-1..1,y=-1..1,axes=framed,caption="Den partielle afledede af f mht. y"):
display(Array([%%,%]));De blandede afledede er ens v\303\246k fra (0,0), men i (0,0) er de forskelligef12:=normal(D[1,2](f)(x,y));Grafen for den blandede afledede:plot3d(f12,x=-1..1,y=-1..1,axes=framed,grid=[50,50],style=patchnogrid,caption=typeset("Grafen for ",diff('f'(x,y),x,y)));Ikke-eksistens af tangentplan (men nok af partielle afledede!)plot3d(-sqrt(abs(x*y)),x=-2..2,y=-2..2,axes=framed);K\303\246dereglen: Abstraktf:='f':g:=t->f(X(t),Y(t));Diff(g(t),t): %=value(%);G:=(u,v)->f(X(u,v),Y(u,v));Diff(G(u,v),u): %=value(%);K\303\246dereglen: Halvkonkrete eksemplerF\303\270rst lader vi X og Y v\303\246re konkrete og f ikke:Diff( f(cos(t),sin(t)),t): %=value(%);Diff( f(t^3,arctan(t)),t): %=value(%);Nu lader vi omvendt f v\303\246re konkret og X og Y ikke:f:=(x,y)->ln(x)*(y^2+x);X:='X': Y:='Y':Diff(f(X(t),Y(t)),t): %=value(%);collect(%,[diff(X(t),t),diff(Y(t),t)]);Vi genkender koefficienterne til X' og Y' som de partielle afledede af f.Gradient og niveaukurvef:=(x,y)->(x+y^2)*ln(x);plot3d(f(x,y),x=0..10,y=-5..5,axes=boxed);contourplot(f(x,y),x=0..10,y=-5..5,contours=40);p1:=%:p2:=gradplot(f(x,y),x=0..10,y=-5..5,arrows=SLIM,fieldstrength=maximal(3)):display(p1,p2,scaling=constrained,caption=typeset("Niveaukurver og gradientfelt for ",f(x,y)));