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.
Read more about this topic: Normal (geometry)
Famous quotes containing the word space:
“But alas! I never could keep a promise. I do not blame myself for this weakness, because the fault must lie in my physical organization. It is likely that such a very liberal amount of space was given to the organ which enables me to make promises, that the organ which should enable me to keep them was crowded out. But I grieve not. I like no half-way things. I had rather have one faculty nobly developed than two faculties of mere ordinary capacity.”
—Mark Twain [Samuel Langhorne Clemens] (18351910)