Normal space (to a surface)

at a point $P$

The orthogonal complement $N_PF$ to the tangent space $T_PF$ (see Tangent plane) of the surface $F^m$ in $V^n$ at $P$. The dimension of the normal space is $n-m$ (the codimension of $F$). Every one-dimensional subspace of it is called a normal to $F$ at $P$. If $F$ is a smooth hypersurface, then it has a unique normal at every of its points.

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