Varieties Defined By Implicit Equations in n-dimensional Space
A differential variety defined by implicit equations in the n-dimensional space is the set of the common zeros of a finite set of differential functions in n variables
The Jacobian matrix of the variety is the k×n matrix whose i-th row is the gradient of fi. By implicit function theorem, the variety is a manifold in the neighborhood of a point of it where the Jacobian matrix has rank k. At such a point P, the normal vector space is the vector space generated by the values at P of the gradient vectors of the fi.
In other words, a variety is defined as the intersection of k hypersurfaces, and the normal vector space at a point is the vector space generated by the normal vectors of the hypersurfaces at the point.
The normal (affine) space at a point P of the variety is the affine subspace passing through P and generated by the normal vector space at P.
These definitions may be extended verbatim to the points where the variety is not a manifold.
Read more about this topic: Normal (geometry)
Famous quotes containing the words varieties, defined, implicit and/or space:
“Now there are varieties of gifts, but the same Spirit; and there are varieties of services, but the same Lord; and there are varieties of activities, but it is the same God who activates all of them in everyone.”
—Bible: New Testament, 1 Corinthians 12:4-6.
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—Patricia Meyer Spacks (b. 1929)
“The true colour of life is the colour of the body, the colour of the covered red, the implicit and not explicit red of the living heart and the pulses. It is the modest colour of the unpublished blood.”
—Alice Meynell (1847–1922)
“In bourgeois society, the French and the industrial revolution transformed the authorization of political space. The political revolution put an end to the formalized hierarchy of the ancien regimé.... Concurrently, the industrial revolution subverted the social hierarchy upon which the old political space was based. It transformed the experience of society from one of vertical hierarchy to one of horizontal class stratification.”
—Donald M. Lowe, U.S. historian, educator. History of Bourgeois Perception, ch. 4, University of Chicago Press (1982)