Editing Principal bundle (section)

=== Trivial bundle and sections ===
Over an open ball <math>U \subset \mathbb{R}^n</math>, or <math>\mathbb{R}^n</math>, with induced coordinates <math>x_1,\ldots,x_n</math>, any principal <math>G</math>-bundle is isomorphic to a trivial bundle<blockquote><math>\pi:U\times G \to U</math></blockquote>and a smooth section <math>s \in \Gamma(\pi)</math> is equivalently given by a (smooth) function <math>\hat{s}: U \to G</math> since<blockquote><math>s(u) = (u,\hat{s}(u)) \in U\times G </math></blockquote>for some smooth function. For example, if <math>G=U(2)</math>, the Lie group of <math>2\times 2</math> [[Unitary matrix|unitary matrices]], then a section can be constructed by considering four real-valued functions<blockquote><math>\phi(x),\psi(x),\Delta(x),\theta(x) : U \to \mathbb{R}</math></blockquote>and applying them to the parameterization

<math display="block">\hat{s}(x) = e^{i\phi(x)}\begin{bmatrix}
 e^{i\psi(x)} & 0 \\
 0 & e^{-i\psi(x)}
\end{bmatrix}
\begin{bmatrix}
 \cos \theta(x)  & \sin \theta(x) \\
 -\sin \theta(x) & \cos \theta(x) \\
\end{bmatrix} 
\begin{bmatrix}
 e^{i\Delta(x)} & 0 \\
 0 & e^{-i\Delta(x)}
\end{bmatrix}.
</math>This same procedure valids by taking a parameterization of a collection of matrices defining a Lie group <math>G</math> and by considering the set of functions from a patch of the base space <math>U\subset X</math> to <math>\mathbb{R}</math> and inserting them into the parameterization.