unequal_mixed_partials.mws

This function has un equal mixed partial derivatives at (0,0)
JR Jan 15 2002 unequal_mixed_partials.mws

(This can be because the second-order partial deriviatives are not continuous at (0,0).)

> restart:
f:=x*y*(x^2-y^2)/(x^2+y^2);
fx:=simplify(diff(f,x));
fy:=simplify(diff(f,y));

f := x*y*(x^2-y^2)/(x^2+y^2)

fx := y*(x^4-y^4+4*x^2*y^2)/((x^2+y^2)^2)

fy := x*(x^4-y^4-4*x^2*y^2)/((x^2+y^2)^2)

> fxy:=simplify(diff(fx,y));
fyx:=simplify(diff(fy,x));
difference:=fxy-fyx;

fxy := (x^6+9*x^4*y^2-9*y^4*x^2-y^6)/((x^2+y^2)^3)

fyx := (x^6+9*x^4*y^2-9*y^4*x^2-y^6)/((x^2+y^2)^3)

difference := 0

At the origin, one of the second-order mixed pattial derivatives is 1 while the other is -1!

> fx0:=limit((subs(y=0,f)-0)/x,x=0);
fy0:=limit((subs(x=0,f)-0)/y,y=0);
fxy0:=limit((subs(x=0,fx)-fx0)/y,y=0);
fyx0:=limit((subs(y=0,fy)-fy0)/x,x=0);

fx0 := 0

fy0 := 0

fxy0 := -1

fyx0 := 1

> x:=r*cos(theta): y:=r*sin(theta):
plot3d([x,y,f],r=0.001..1,theta=0..2*Pi,grid=[10,51],title="the graph of f");

[Maple Plot]

> plot3d([x,y,fxy],r=0.001..1,theta=0..2*Pi,grid=[10,71],title="the graph of f_xy");

[Maple Plot]

You can see from the graph of f_xy that it is not continuous at (0,0).

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