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I'm a little confused about fiber bundles. I have a specific example and would appreciate someone clarifying this for me. Let $f:M\times U\rightarrow TM$ be a map from a product space $M\times U\rightarrow$ to the tangent bundle TM of M. Is ($M\times U$,f,TM) a fiber bundle with $M\times U\equiv E$ the total space and TM the base space? If so, which is the fiber space? Or should the base be just M? For example, let $f(x,u)=\dot x=g(x)h(u), x \in M, u \in U$, for some g,h. Is f the projection map of the a fiber bundle? Should it be TM or $T_{x} M$, the tangent space of M at x? Thanks in advance!

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    @Willie, please see the system in matrix notation: $\dot x=[\dot x_{i}]^T, i=1...n, h(u)=[h_{j}(u)]^T, j=1...m, g(x)=n\times m matrix. So, \dot x=g(x)h(u)$2011-12-16

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In the definition of a fiber bundle, the map from the total space $E$ to the base space $M$ is the canonical projection, meaning locally $f:M\times F\to M$ takes $f(x,p)=x$ for fiber $F$. In your example, I guess $f:M\times U\to TM$ could be the projection of a fiber bundle if $TM$ $was$ a trivial bundle, say $M=\mathbb{R}^4$. But also note that most people would write $TM$ as the the entire bundle, meaning $TM=M\times \mathbb{R}^n$ locally, not just $TM=\mathbb{R}^n$.

EDIT: Note that I keep saying "locally". For any fiber bundle $f:E\to M$ with fiber $F$, you can write $f_i:U_i\times F\to M$ for a local covering $\{U_i\}$ of $M$. I won't write down all the details of that, but you should be familiar with it as well.