Stationary Points of an implicitly defined function?

  1. 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 i´d say no g = 0 = constant hence no stationary points but if i do grad(g) = ( 0,0,0) i get stationary points.
    So what´s the answer for this ?
    And if it´s the grad(g) thing how to interpret that kind of stationary point geometricaly.
    Thanks and bye :)
     
  2. jcsd
  3. cristo

    cristo 8,411
    Staff Emeritus
    Science Advisor

    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).
     
  4. 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.
     
  5. Dick

    Dick 25,853
    Science Advisor
    Homework Helper

    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.
     
  6. quick example:
    g(x,y,z)=x+y-z
    g(x,y,z)=0
    grad(g)=i+j-k
     
  7. Dick

    Dick 25,853
    Science Advisor
    Homework Helper

    And the surface g(x,y,z)=0 would be the plane z=(x+y).
     
  8. cristo

    cristo 8,411
    Staff Emeritus
    Science Advisor

    Yes, I thought I was missing something. I don't quite know what I was doing in my last post!
     
  9. yea that´s exactly what i meant. How is that puzzle solved ? :)
     
  10. Dick

    Dick 25,853
    Science Advisor
    Homework Helper

    Look at ziad1985's example. g(x,y,z)=0 in implicit function definition is not meant to mean g is zero everywhere.
     
  11. 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 ?
     
  12. Dick

    Dick 25,853
    Science Advisor
    Homework Helper

    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.
     
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook

Have something to add?