calc11

Question

For a tangent plane to a surface, why is the normal vector for this plane equal to the gradient vector? Or is it not?

SAGAR SINGH - IIT DELHI · Accepted Answer

DEAR ANKIT,

You have to be a bit more precise: if a surface is defined by

$LaTeX Code: f(\\mathbf{x})=0$

then

$LaTeX Code: \\nabla f\\big|_{\\mathbf{x}_0}$

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:

$LaTeX Code: 0=df=\\nabla f\\cdot\\mathbf{v}$

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Sagar Singh

B.Tech IIT Delhi

sagarsingh24.iitd@gmail.com

For a tangent plane to a surface, why is the normal vector for this plane equal to the gradient vector? Or is it not?