Editing Implicit function theorem (section)

== Statement of the theorem ==
Let <math>f: \R^{n+m} \to \R^m</math> be a [[continuously differentiable function]], and let <math>\R^{n+m}</math> have coordinates <math>(\textbf{x}, \textbf{y})</math>. Fix a point <math>(\textbf{a}, \textbf{b}) = (a_1,\dots,a_n, b_1,\dots, b_m)</math> with <math>f(\textbf{a}, \textbf{b}) = \mathbf{0}</math>, where <math>\mathbf{0} \in \R^m</math> is the zero vector. If the [[Jacobian matrix]] (this is the right-hand panel of the Jacobian matrix shown in the previous section):
<math display="block">J_{f, \mathbf{y}} (\mathbf{a}, \mathbf{b}) = \left [ \frac{\partial f_i}{\partial y_j} (\mathbf{a}, \mathbf{b}) \right ]</math>
is [[invertible]], then there exists an open set <math>U \subset \R^n</math> containing <math>\textbf{a}</math> such that there exists a unique function <math>g: U \to \R^m</math> such that {{nowrap|<math> g(\mathbf{a}) = \mathbf{b}</math>,}} and {{nowrap|<math> f(\mathbf{x}, g(\mathbf{x})) = \mathbf{0} ~ \text{for all} ~ \mathbf{x}\in U</math>.}} Moreover, <math>g</math> is continuously differentiable and, denoting the left-hand panel of the Jacobian matrix shown in the previous section as:
<math display="block">
J_{f, \mathbf{x}} (\mathbf{a}, \mathbf{b}) = \left [ \frac{\partial f_i}{\partial x_j} (\mathbf{a}, \mathbf{b}) \right ],
</math>
the Jacobian matrix of partial derivatives of <math>g</math> in <math>U</math> is given by the [[matrix product]]:<ref>{{Cite journal |first=Oswaldo |last=de Oliveira |title=The Implicit and Inverse Function Theorems: Easy Proofs |journal=Real Anal. Exchange |volume=39 |issue=1 |year=2013 |doi=10.14321/realanalexch.39.1.0207 |pages=214–216 |s2cid=118792515 |arxiv=1212.2066 }}</ref> 
<math display="block">
\left[\frac{\partial g_i}{\partial x_j} (\mathbf{x})\right]_{m\times n} =- \left [ J_{f, \mathbf{y}}(\mathbf{x}, g(\mathbf{x})) \right ]_{m \times m} ^{-1} \, \left [ J_{f, \mathbf{x}}(\mathbf{x}, g(\mathbf{x})) \right ]_{m \times n}
</math>

For a proof, see [[Inverse function theorem#Implicit_function_theorem]]. Here, the two-dimensional case is detailed.

===Higher derivatives===
If, moreover, <math>f</math> is [[analytic function|analytic]] or continuously differentiable <math>k</math> times in a neighborhood of <math>(\textbf{a}, \textbf{b})</math>, then one may choose <math>U</math> in order that the same holds true for <math>g</math> inside <math>U</math>. <ref>{{Cite book |first1=K. | last1=Fritzsche |first2=H. |last2=Grauert |year=2002 |url=https://books.google.com/books?id=jSeRz36zXIMC&pg=PA34 |title=From Holomorphic Functions to Complex Manifolds |publisher=Springer |page=34 |isbn=9780387953953 }}</ref> In the analytic case, this is called the '''analytic implicit function theorem'''.