Let $E \longrightarrow N$ be a rank $r$ smooth vector bundle, let $f:M \longrightarrow N$ be smooth, and let $(s_1,\dots,s_r)$ be a local frame for $E$ over an [open set](/page/Open%20Set) $V\subset N$. Then $(f^*s_1,\dots,f^*s_r)$ is a local frame for $f^*E$ over $f^{-1}(V)$.