Editing Normal (geometry) (section)

==Varieties defined by implicit equations in ''n''-dimensional space==
A '''[[analytic variety|differential variety]]''' defined by implicit equations in the <math>n</math>-dimensional space <math>\R^n</math> is the set of the common zeros of a finite set of differentiable functions in <math>n</math> variables
<math display=block> f_1\left(x_1, \ldots, x_n\right), \ldots, f_k\left(x_1, \ldots, x_n\right).</math>
The [[Jacobian matrix]] of the variety is the <math>k \times n</math> matrix whose <math>i</math>-th row is the gradient of <math>f_i.</math> By the [[implicit function theorem]], the variety is a [[manifold]] in the neighborhood of a point where the Jacobian matrix has rank <math>k.</math> At such a point <math>P,</math> the '''normal vector space''' is the vector space generated by the values at <math>P</math> of the gradient vectors of the <math>f_i.</math> 

In other words, a variety is defined as the intersection of <math>k</math> 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 <math>P</math> of the variety is the [[affine subspace]] passing through <math>P</math> and generated by the normal vector space at <math>P.</math>

These definitions may be extended {{em|verbatim}} to the points where the variety is not a manifold.

===Example===
Let ''V'' be the variety defined in the 3-dimensional space by the equations 
<math display=block>x\,y = 0, \quad z = 0.</math>
This variety is the union of the <math>x</math>-axis and the <math>y</math>-axis.

At a point <math>(a, 0, 0),</math> where <math>a \neq 0,</math> the rows of the Jacobian matrix are <math>(0, 0, 1)</math> and <math>(0, a, 0).</math> Thus the normal affine space is the plane of equation <math>x = a.</math> Similarly, if <math>b \neq 0,</math> the ''[[normal plane (geometry)|normal plane]]'' at <math>(0, b, 0)</math> is the plane of equation <math>y = b.</math> 

At the point <math>(0, 0, 0)</math> the rows of the Jacobian matrix are <math>(0, 0, 1)</math> and <math>(0, 0, 0).</math> Thus the normal vector space and the normal affine space have dimension 1 and the normal affine space is the <math>z</math>-axis.