Editing Inverse function theorem (section)

== Applications ==
===Implicit function theorem===
The inverse function theorem can be used to solve a system of equations
:<math>\begin{align}
&f_1(x) = y_1 \\
&\quad \vdots\\
&f_n(x) = y_n,\end{align}</math>
i.e., expressing <math>y_1, \dots, y_n</math> as functions of <math>x = (x_1, \dots, x_n)</math>, provided the Jacobian matrix is invertible. The [[implicit function theorem]] allows to solve a more general system of equations:
:<math>\begin{align}
&f_1(x, y) = 0 \\
&\quad \vdots\\
&f_n(x, y) = 0\end{align}</math>
for <math>y</math> in terms of <math>x</math>. Though more general, the theorem is actually a consequence of the inverse function theorem. First, the precise statement of the implicit function theorem is as follows:<ref>{{harvnb|Spivak|1965|loc=Theorem 2-12.}}</ref>
*given a map <math>f : \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^m</math>, if <math>f(a, b) = 0</math>, <math>f</math> is continuously differentiable in a neighborhood of <math>(a, b)</math> and the derivative of <math>y \mapsto f(a, y)</math> at <math>b</math> is invertible, then there exists a differentiable map <math>g : U \to V</math> for some neighborhoods <math>U, V</math> of <math>a, b</math> such that <math>f(x, g(x)) = 0</math>. Moreover, if <math>f(x, y) = 0, x \in U, y \in V</math>, then <math>y = g(x)</math>; i.e., <math>g(x)</math> is a unique solution.
To see this, consider the map <math>F(x, y) = (x, f(x, y))</math>. By the inverse function theorem, <math>F : U \times V \to W</math> has the inverse <math>G</math> for some neighborhoods <math>U, V, W</math>. We then have:
:<math>(x, y) = F(G_1(x, y), G_2(x, y)) = (G_1(x, y), f(G_1(x, y), G_2(x, y))),</math>
implying <math>x = G_1(x, y)</math> and <math>y = f(x, G_2(x, y)).</math> Thus <math>g(x) = G_2(x, 0)</math> has the required property. <math>\square</math>

===Giving a manifold structure===
In differential geometry, the inverse function theorem is used to show that the pre-image of a [[regular value]] under a smooth map is a manifold.<ref>{{harvnb|Spivak|1965|loc=Theorem 5-1. and Theorem 2-13.}}</ref> Indeed, let <math>f : U \to \mathbb{R}^r</math> be such a smooth map from an open subset of <math>\mathbb{R}^n</math> (since the result is local, there is no loss of generality with considering such a map). Fix a point <math>a</math> in <math>f^{-1}(b)</math> and then, by permuting the coordinates on <math>\mathbb{R}^n</math>, assume the matrix <math>\left [ \frac{\partial f_i}{\partial x_j}(a) \right]_{1 \le i, j \le r}</math> has rank <math>r</math>. Then the map <math>F : U \to \mathbb{R}^r \times \mathbb{R}^{n-r} = \mathbb{R}^n, \, x \mapsto (f(x), x_{r+1}, \dots, x_n)</math> is such that <math>F'(a)</math> has rank <math>n</math>. Hence, by the inverse function theorem, we find the smooth inverse <math>G</math> of <math>F</math> defined in a neighborhood <math>V \times W</math> of <math>(b, a_{r+1}, \dots, a_n)</math>. We then have
:<math>x = (F \circ G)(x) = (f(G(x)), G_{r+1}(x), \dots, G_n(x)),</math>
which implies
:<math>(f \circ G)(x_1, \dots, x_n) = (x_1, \dots, x_r).</math>
That is, after the change of coordinates by <math>G</math>, <math>f</math> is a coordinate projection (this fact is known as the [[submersion theorem]]). Moreover, since <math>G : V \times W \to U' = G(V \times W)</math> is bijective, the map
:<math>g = G(b, \cdot) : W \to f^{-1}(b) \cap U', \, (x_{r+1}, \dots, x_n) \mapsto G(b, x_{r+1}, \dots, x_n)</math>
is bijective with the smooth inverse. That is to say, <math>g</math> gives a local parametrization of <math>f^{-1}(b)</math> around <math>a</math>. Hence, <math>f^{-1}(b)</math> is a manifold. <math>\square</math> (Note the proof is quite similar to the proof of the implicit function theorem and, in fact, the implicit function theorem can be also used instead.)

More generally, the theorem shows that if a smooth map <math>f : P \to E</math> is transversal to a submanifold <math>M \subset E</math>, then the pre-image <math>f^{-1}(M) \hookrightarrow P</math> is a submanifold.<ref>{{cite web|website=northwestern.edu|title=Transversality
|url=https://sites.math.northwestern.edu/~jnkf/classes/mflds/4transversality.pdf}}</ref>