Let $B$ be a [topological space](/page/Topological%20Space), let $r\in\mathbb{N}$, and let $\pi:E\to B$ be a rank $r$ real [vector bundle](/page/Vector%20Bundle). Let $U,V\subset B$ be open subsets over which $\pi$ is trivialized by bundle trivializations $\Phi_U:\pi^{-1}(U)\to U\times \mathbb{R}^r$ and $\Phi_V:\pi^{-1}(V)\to V\times \mathbb{R}^r$. Let $g_{VU}:U\cap V\to GL_r(\mathbb{R})$ be the transition function determined by the convention $(\Phi_V\circ \Phi_U^{-1})(b,x)=(b,g_{VU}(b)x)$ for every $b\in U\cap V$ and every $x\in \mathbb{R}^r$.

Then sections $s:U\cup V\to E$ of $\pi$ over $U\cup V$ are in bijective correspondence with pairs of continuous maps $s_U:U\to \mathbb{R}^r$ and $s_V:V\to \mathbb{R}^r$ satisfying $s_V(b)=g_{VU}(b)s_U(b)$ for every $b\in U\cap V$. Under this correspondence, $s_U$ and $s_V$ are the local coordinate representatives of $s$ in the trivializations $\Phi_U$ and $\Phi_V$.