= Solution
A smooth rank-$r$ <vector bundle> over a <smooth manifold> $M$ consists of a smooth total space $E$, a smooth projection $\pi:E\to M$, and an $r$-dimensional real <vector space> structure on each fibre $E_p=\pi^{-1}(p)$. Each point has a neighborhood $U$ with a <vector bundle trivialization>
$$
\tau_U:\pi^{-1}(U)\xrightarrow{\sim}U\times\mathbb R^r
$$
that is a <diffeomorphism>, commutes with projection to $U$, and is a <linear isomorphism> on every fibre. The overlap maps have the form $(p,v)\mapsto(p,h(p)v)$ with $h:U\cap V\to GL_r(\mathbb R)$ smooth. For a complex <vector bundle>, replace $\mathbb R$ by $\mathbb C$ and require complex-linear fibre maps.
A <connection on a vector bundle> is an $\mathbb R$-linear map
$$
\nabla:\Gamma(E)\longrightarrow\Omega^1(M;E),\qquad
\nabla(fs)=df\otimes s+f\nabla s,
$$
where $\Gamma(E)$ is the <module of smooth sections>, $\Omega^1(M;E)$ consists of <vector-bundle-valued differential forms> of degree one, and $f$ is a <smooth function>. Equivalently, evaluating on a <vector field> $X$ gives operators $\nabla_X$ with $\nabla_{fX}s=f\nabla_Xs$ and $\nabla_X(fs)=X(f)s+f\nabla_Xs$. A <connection on a vector bundle> is local: its value at a point depends on the germ of the <section of a vector bundle>, so the same definition applies to local sections.
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