Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Gradient and tangent planes

  1. Aug 29, 2010 #1
    For a tangent plane to a surface, why is the normal vector for this plane equal to the gradient vector? Or is it not?
  2. jcsd
  3. Aug 29, 2010 #2
    You have to be a bit more precise: if a surface is defined by



    [tex]\nabla f\big|_{\mathbf{x}_0}[/tex]

    is a vector tangent to the surface at point x0. This is because it is orthogonal to the velocities of all possible curves that pass through x0:

    [tex]0=df=\nabla f\cdot\mathbf{v}[/tex]
  4. Aug 29, 2010 #3
    Hi, there is a quick proof of this.
    Suppose a surface:

    and a point:
    P(x0,y0,z0) [tex]\in[/tex] surface.

    Let C be a curve on the surface passing through P. This curve can be described by a vector function:


    C lies on the surface this implies that:

    differentiating (if F and r are differentiable) we have:

    [tex]\Rightarrow[/tex] The vector r'(t) (tangent to the surface) is perpendicular to the levele surface.
    Last edited: Aug 29, 2010
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook