# Stationary Points of an implicitly defined function?

 P: 67 Hi i just got a short question about definition if i got an implicitly defined function g(x,y,z) = 0 and then be asked whether g hast stationary points. How to answer that intuitively iīd say no g = 0 = constant hence no stationary points but if i do grad(g) = ( 0,0,0) i get stationary points. So whatīs the answer for this ? And if itīs the grad(g) thing how to interpret that kind of stationary point geometricaly. Thanks and bye :)
 Mentor P: 8,325 I don't understand what you're doing here. $$\nabla g=\frac{\partial g}{\partial x}\bold{i}+\frac{\partial g}{\partial y}\bold{j}+\frac{\partial g}{\partial z}\bold{k}$$. But this is identically zero, if g=0 (unless I'm missing something obvious).
 P: 240 He is seeing a contradiction between 2 statements: if grad(g) = ( 0,0,0) then there is stationary points and if g(x,y,z) = 0 then there is no stationary points. since g(x,y,z) = 0 then grad(g) = ( 0,0,0) I think that what he means.
 Sci Advisor HW Helper Thanks P: 25,235 Stationary Points of an implicitly defined function? If it is a surface defined by g(x,y,z)=0 (a level surface), there is no need that grad(g)=0. What would be zero is grad(g).t for t a tangent vector to the surface.
 P: 240 quick example: g(x,y,z)=x+y-z g(x,y,z)=0 grad(g)=i+j-k
HW Helper
Thanks
P: 25,235
 Quote by ziad1985 quick example: g(x,y,z)=x+y-z g(x,y,z)=0 grad(g)=i+j-k
And the surface g(x,y,z)=0 would be the plane z=(x+y).
 Mentor P: 8,325 Yes, I thought I was missing something. I don't quite know what I was doing in my last post!
P: 67
 Quote by ziad1985 He is seeing a contradiction between 2 statements: if grad(g) = ( 0,0,0) then there is stationary points and if g(x,y,z) = 0 then there is no stationary points. since g(x,y,z) = 0 then grad(g) = ( 0,0,0) I think that what he means.
yea thatīs exactly what i meant. How is that puzzle solved ? :)
 Sci Advisor HW Helper Thanks P: 25,235 Look at ziad1985's example. g(x,y,z)=0 in implicit function definition is not meant to mean g is zero everywhere.
 P: 67 yea that´s what i understand in a sense it defines a z(x,y) or x(z,y) and so on so when i ask what stationary points does g have i mean what stationary points does z(x,y) have ?
 Sci Advisor HW Helper Thanks P: 25,235 If you want stationary points of z(x,y), then find an expression for z(x,y), take partial derivatives wrt x and y and set them both equal to zero.

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