Editing Nash embedding theorems (section)

==''C''<sup>''k''</sup> embedding theorem==
The technical statement appearing in Nash's original paper is as follows: if ''M'' is a given ''m''-dimensional Riemannian manifold (analytic or of class ''C<sup>k</sup>'', 3 ≤ ''k'' ≤ ∞), then there exists a number ''n'' (with ''n'' ≤ ''m''(3''m''+11)/2 if ''M'' is a compact manifold, and with ''n'' ≤ ''m''(''m''+1)(3''m''+11)/2 if ''M'' is a non-compact manifold) and an [[isometric embedding]] ƒ: ''M'' → '''R'''<sup>''n''</sup> (also analytic or of class ''C<sup>k</sup>'').{{sfnm|1a1=Nash|1y=1956}} That is ƒ is an [[Embedding#Differential topology|embedding]] of ''C<sup>k</sup>'' manifolds and for every point ''p'' of ''M'', the [[derivative]] dƒ<sub>''p''</sub> is a [[linear operator|linear map]] from the [[tangent space]] ''T<sub>p</sub>M'' to '''R'''<sup>''n''</sup> which is compatible with the given [[inner product space|inner product]] on ''T<sub>p</sub>M'' and the standard [[scalar product|dot product]] of '''R'''<sup>''n''</sup> in the following sense:
: <math>\langle u,v \rangle = df_p(u)\cdot df_p(v)</math>
for all vectors ''u'', ''v'' in ''T<sub>p</sub>M''. When {{mvar|n}} is larger than {{math|{{sfrac|1|2}}''m''(''m'' + 1)}}, this is an underdetermined system of [[partial differential equation]]s (PDEs).

The Nash embedding theorem is a global theorem in the sense that the whole manifold is embedded into '''R'''<sup>''n''</sup>.  A local embedding theorem is much simpler and can be proved using the [[implicit function theorem]] of advanced calculus in a [[Manifold#Charts|coordinate neighborhood]] of the manifold. The proof of the global embedding theorem relies on Nash's implicit function theorem for isometric embeddings. This theorem has been generalized by a number of other authors to abstract contexts, where it is known as [[Nash–Moser theorem]]. The basic idea in the proof of Nash's implicit function theorem is the use of [[Newton's method]] to construct solutions. The standard Newton's method fails to converge when applied to the system; Nash uses smoothing operators defined by [[convolution]] to make the Newton iteration converge: this is Newton's method with postconditioning.  The fact that this technique furnishes a solution is in itself an [[existence theorem]] and of independent interest. In other contexts, the [[Kantorovich theorem|convergence of the standard Newton's method]] had earlier been proved by [[Leonid Kantorovitch]].