0
$\begingroup$

Suppose a surface is given by a level set $f(x,y,z)=0$, and this level set is a graph of a smooth map, i.e $z=g(x,y)$. Let $r(x,y)=(x,y,g(x,y))$. Then is $\nabla f(x,y,z)=(r_x\times r_y)(x,y)$?

I thought this might be true by the implicit function theorem, but I have no idea how to actually get there...

1 Answers 1

1

No, in general not. They may differ by sign and length. Both are normal to the surface though, where ever they are $\neq 0$, and in the codimension one case this means one is a scalar multiple of the other one.

  • 0
    Then why can the gradient be used in example 1 here? http://tutorial.math.lamar.edu/Classes/CalcIII/SurfIntVectorField.aspx – 2017-02-09
  • 0
    Actually I think it's explicitly written they have equal norms – 2017-02-09
  • 0
    @eleventeen You should post the question to which you want an answer, instead of asking something else and come up with follow up questions. It's not even clear to me what you want to ask with the follow up question. You can use each representation of the normal. It seems that in that example a unit normal is needed, so there they divide the gradient by it's norm. Moreover the direction is important, so they multiply by $-1$. This is exactly what my answer points out: direction and length are important. – 2017-02-09
  • 0
    @eleventeen the gradient and the cross product representation _do not_ have equal norms automatically. In order to acchieve that you either need special choices for $f$ / $g$ respectively or have to divide by the norm of the vector. – 2017-02-09
  • 0
    I'm assuming $f,g$ are related in that $g$ expresses the dependence of $z$ on $x,y$ in the level set of $f$. Does that suffice for having equal norms? – 2017-02-09
  • 0
    @eleventeen that is not enough. Please don't use this as discussion forum. Thank you. – 2017-02-09
  • 0
    Can you please explain why or give examples? I understand you have answered the question, but I can't get much out of just "no". – 2017-02-09
  • 0
    If youi look at $f=0$ then also $F:= -2f= 0 $ will describe the same surface, but it's gradient will have double length and point into the other direction. The question is not 'why not', to the contrary it's 'why should they be equal'? The only known thing they have in common is their direction (normal to the surface), length and sign can easily be changed. – 2017-02-09