Question 50437: Find a vector normal to the surface z + 2xy = x^2 + y^2 at the point (1,1,0).
Normally I would use gradient g(x,y,z)=0 or f(x,y)= x^2 -2xy + y^2 -z but I can't make it work???
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! Find a vector normal to the surface z + 2xy = x^2 + y^2 at the point...P SAY.. (1,1,0).
Normally I would use gradient g(x,y,z)=0 or f(x,y)= x^2 -2xy + y^2 -z but I can't make it work???
LET US USE SAME METHOD SUGGESTED BY YOU..
F=X^2+Y^2-2XY-Z=0...
GRAD(F)=i(DF/DX)+j(DF/DY)+k(DF/DZ)....WHERE THE DERIVATIVES ARE ALL PARTIAL DERIVATIVES.HOPE YOU KNOW IT...THIS MEANS WE KEEP ALL VARIABLES EXCEPT ONE UNDER CONSIDERATION CONSTANT .WE FIND THEM ALL AT THE GIVEN POINT P SAY X=1,Y=1,Z=0
DF/DX=2X-2Y..=2-2=0
DF/DY=2Y-2X...=0
DF/DZ=-1...=-1
HENCE NORMAL TO THE PLANE IS
N=GRAD(F)= -k
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