= Vector bundle
{title2=$\pi:E\to B$}
{wiki}
A rank-$r$ <vector bundle> over a <topological space> $B$ is a <fiber bundle> $\pi:E\to B$ whose fibers are <vector spaces> over a fixed <field> $\mathbb F=\mathbb R$ or $\mathbb C$. Each <point> has an <open neighborhood> $U$ and a <vector bundle trivialization> $\pi^{-1}(U)\cong U\times\mathbb F^r$ that is a <linear isomorphism> on each fiber.
On overlaps the <vector bundle trivializations> change by $(x,v)\mapsto(x,g_{ij}(x)v)$, where the invertible <matrices> $g_{ij}(x)$ vary continuously and satisfy the <Čech cocycle condition>. For a smooth <vector bundle>, the base and total space are <smooth manifolds> and these <functions> are smooth. The fiberwise <vector addition> and <scalar multiplication> are therefore globally well-defined.
The <tangent bundle> collects all <tangent spaces> of a <smooth manifold>. Its <transition functions of a vector bundle> are <Jacobian matrices>, so the <chain rule> supplies the cocycle identity. The <Möbius line bundle> supplies another example: every small patch is a product, but no global <frame of a vector bundle> exists. A global fiber-linear product isomorphism is the extra condition defining a <trivial vector bundle>.
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