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 curve...

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Bibliographic Details
Published in:SIAM review 1973-04, Vol.15 (2), p.275-282
Main Authors: Scholten, William B., Gaggioli, Richard A.
Format: Article
Language:English
Subjects:
Coordinate systems
Coordinate transformations
Curves
Determinants
Hypersurfaces
Mathematical manifolds
Mathematical vectors
Neighborhoods
Partial derivatives
Tangent function
Tangents
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Summary: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.
ISSN:0036-1445
1095-7200
DOI:10.1137/1015029