Stationary Points of an implicitly defined function?by Mr.Brown Tags: defined, function, implicitly, points, stationary 

#1
Mar807, 08:01 AM

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 :) 



#2
Mar807, 08:50 AM

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P: 8,287

I don't understand what you're doing here. [tex]\nabla g=\frac{\partial g}{\partial x}\bold{i}+\frac{\partial g}{\partial y}\bold{j}+\frac{\partial g}{\partial z}\bold{k}[/tex]. But this is identically zero, if g=0 (unless I'm missing something obvious).




#3
Mar807, 11:48 AM

P: 239

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. 



#4
Mar807, 11:55 AM

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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.




#5
Mar807, 12:02 PM

P: 239

quick example:
g(x,y,z)=x+yz g(x,y,z)=0 grad(g)=i+jk 



#6
Mar807, 12:05 PM

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#7
Mar807, 12:49 PM

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Yes, I thought I was missing something. I don't quite know what I was doing in my last post!




#8
Mar807, 02:15 PM

P: 67





#9
Mar807, 02:21 PM

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Look at ziad1985's example. g(x,y,z)=0 in implicit function definition is not meant to mean g is zero everywhere.




#10
Mar807, 02:34 PM

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 ?




#11
Mar807, 02:37 PM

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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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