Editing Inverse function theorem (section)

==Example==
Consider the [[vector-valued function]] <math>F:\mathbb{R}^2\to\mathbb{R}^2\!</math> defined by:
:<math>
F(x,y)=
\begin{bmatrix}
 {e^x \cos y}\\
 {e^x \sin y}\\
\end{bmatrix}.
</math>
The Jacobian matrix of it at <math>(x, y)</math> is:
:<math>
JF(x,y)=
\begin{bmatrix}
 {e^x \cos y} & {-e^x \sin y}\\
 {e^x \sin y} & {e^x \cos y}\\
\end{bmatrix}
</math>
with the determinant:
:<math>
\det JF(x,y)=
e^{2x} \cos^2 y + e^{2x} \sin^2 y=
e^{2x}.
\,\!</math>

The determinant <math>e^{2x}\!</math> is nonzero everywhere.  Thus the theorem guarantees that, for every point {{Mvar|p}} in <math>\mathbb{R}^2\!</math>, there exists a neighborhood about {{Mvar|p}} over which {{Mvar|F}} is invertible. This does not mean {{Mvar|F}} is invertible over its entire domain: in this case {{Mvar|F}} is not even [[injective]] since it is periodic: <math>F(x,y)=F(x,y+2\pi)\!</math>.