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)
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