Editing Inverse function theorem (section)

===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>