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

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

Grade:12

1 Answers

SAGAR SINGH - IIT DELHI
879 Points
11 years ago

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

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