Normals to Coordinate Hypersurfaces as Weight Minus-One Covariant Base Vectors

This paper reviews the well established development of tangents to curves at a point as a linear (vector) space [1], and contributes an analogous development for normals to hypersurfaces containing the point. It is shown that, for any allowable coordinate system, the tangents to the coordinate curves serve as (weight zero) contravariant base vectors for the tangent space, and the normals to the coordinate hypersurfaces serve as weight minus-one covariant base vectors for the normal space.