Funktion af flere variable Preben Alsholm, 15/8 2007

f:=(x,y)->sqrt(y-x^2)/(x^2+(y-1)^2); plot3d(f(x,y),x=-2..2,y=x^2..4,view=0..5,axes=framed,title="Eksempel 1"); f:=(x,y)->sqrt(y-x^2)/(1+x^2+(y-1)^2); plot3d(f(x,y),x=-2..2,y=x^2..4,axes=framed,title="A: Eksempel 1, modificeret"); f:=(x,y)->sqrt(abs(y-x^2))/(1+x^2+(y-1)^2); plot3d(f(x,y),x=-2..2,y=-4..4,axes=framed,title="B: Eksempel 1, modificeret"); f:=(x,y)->abs(y-x^2)/(1+x^2+(y-1)^2); plot3d(f(x,y),x=-2..2,y=-4..4,axes=framed,title="C: Eksempel 1, modificeret"); f:=(x,y)->(y-x^2)/(1+x^2+(y-1)^2); plot3d(f(x,y),x=-2..2,y=-4..4,axes=framed,title="D: Eksempel 1, modificeret");

with(plots): contourplot(f(x,y),x=-2..2,y=-4..4,contours=40,grid=[40,40]); contourplot3d(f(x,y),x=-2..2,y=-4..4,contours=40,grid=[40,40]);

f(x,y); Partielle afledede som funktioner findes ved brug af D. S\303\245ledes findes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRInhGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== p\303\245 denne m\303\245de, hvor 1-tallet betyder den f\303\270rste variable: D[1](f); Og LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRInlGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== findes s\303\245ledes: D[2](f); Som vi ser har vi f\303\245et funktioner. Disse kan anvendes i et punkt. Vi finder 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 s\303\245ledes: D[1](f)(0,0); og vi finder 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 s\303\245ledes: D[2](f)(0,0); Den partielle afledede 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 er h\303\246ldningen for tangenten til kurven z = f( x, 0) i punktet x = 0, n\303\245r denne kurve tegnes i et xz-diagram. Nedenfor har vi i stedet tegnet kurven med parameterfremstillingen x = t, y = 0, z= f(t,0) og dennes tangent i punktet (0, 0, f(0,0) ). Desuden tegnes grafen for f: p1:=spacecurve([t,0,f(t,0)],t=-2..2,color=red): p3:=spacecurve([s,0,s*D[1](f)(0,0)],s=-2..2,color=blue): p2:=plot3d(f(x,y),x=-2..2,y=-4..4,style=patchnogrid,axes=framed,title="D: Example 1, modificeret"): display(p1,p2,p3); Her f\303\270lger en lignende grafisk illustration af betydningen af 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: p1:=spacecurve([0,t,f(0,t)],t=-4..4,color=red): p3:=spacecurve([0,s,s*D[2](f)(0,0)],s=-2..2,color=blue): p2:=plot3d(f(x,y),x=-2..2,y=-4..4,style=patchnogrid,axes=framed,title="D: Example 1, modificeret"): display(p1,p2,p3,orientation=[30,110]);

f:=(x,y)->(x^2-y^2)/(x^2+y^2); plot3d(f(x,y),x=-2..2,y=-2..2,axes=framed); Alle niveaukurver ser ud til at "g\303\245 igennem" (0,0): contourplot3d(f(x,y),x=-2..2,y=-2..2,axes=framed);

Vi viser ikke alle de simple regninger, men kun hvordan grafen ser ud: f:=(x,y)->x^3*y/(x^2+y^2); D[1](f)(0,y); D[2](f)(x,0); plot3d(f(x,y),x=-1..1,y=-1..1,axes=framed); De blandede afledede er ens v\303\246k fra (0,0), men i (0,0) er de forskellige f12:=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]);

plot3d(-sqrt(abs(x*y)),x=-2..2,y=-2..2,axes=framed);

f:=(x,y)->x^2+y^2; p:=1,2: L:=f(p)+D[1](f)(p)*(x-p[1])+D[2](f)(p)*(y-p[2]); plot3d([f(x,y),L],x=-4..4,y=-4..4,axes=framed); mtaylor(f(x,y),[x=p[1],y=p[2]],2);

f:='f': mtaylor(f(x,y),[x=a,y=b],2); f:=(x,y)->x^2*exp(x-y); L:=mtaylor(f(x,y),[x=1,y=1],2); plot3d({f(x,y),L},x=0..2,y=0..2);

with(plots): gradplot( x^2-y^2,x=-2..2,y=-2..2); gradplot(sin(x*y),x=-Pi..Pi,y=-Pi..Pi,arrows=THICK); contourplot(sin(x*y),x=-Pi..Pi,y=-Pi..Pi); display(%,%%); plot3d(sin(x*y),x=-Pi..Pi,y=-Pi..Pi,axes=framed); gradplot(x^2+2*y^2+1,x=-1..1,y=-1..1,grid=[15,15], color = x^2+2*y^2+1);

f:=(x,y)->sin(x)*exp(-y^2); p1:=gradplot(f(x,y),x=-4..4,y=-2..2): p2:=contourplot(f(x,y),x=-4..4,y=-2..2,grid=[40,40]): display(p1,p2);