Normal (geometry) - Hypersurfaces in n-dimensional Space

Hypersurfaces in n-dimensional Space

The definition of a normal to a surface in three-dimensional space can be extended to -dimensional hypersurfaces in a -dimensional space. A hypersurface may be locally defined implicitly as the set of points satisfying an equation, where is a given scalar function. If is continuously differentiable then the hypersurface is a differentiable manifold in the neighbourhood of the points where the gradient is not null. At these points the normal vector space has dimension one and is generated by the gradient

The normal line at a point of the hypersurface is defined only if the gradient is not null. It is the line passing through the point and having the gradient as direction.