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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
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