TITLE: Motivation of vector bundle of a manifold
QUESTION [9 upvotes]: I am studying about vector bundle from M.Lee but not getting the feel of it. Can someone explain me about the importance of vector bundle? Why do we need to study about it?
Thanks!
Also, i have to study about connection of vector bundle so i wanted to know about the  motivation of connection of vector bundle also. If someone can tell me about its importance and motivation then it would be great!
Thanks again!

REPLY [4 votes]: Manifolds are spaces that locally look like $\mathbb{R}^n$. Similarly, vector bundles are spaces that locally look like $\mathbb{R}^n \times V$, where $V$ is some vector space. The simplest kind of vector bundle is a trivial bundle $M \times V$, if $M$ is a manifold, but the need for nontrivial vector bundles is seen immediately from looking at the tangent space.
A section of the bundle $\mathbb{R}^n \times V$ is just a smooth function $\phi:\mathbb{R}^n \rightarrow V$. If you have such a function you can, for instance, take partial derivatives: $\frac{\partial \phi}{\partial x_i}$, and this lets you talk about differential equations. However, it's not so easy on a nontrivial vector bundle: The reason is that if $x$ and $y$ are two points close together on a manifold you can't take $\phi(x) - \phi(y)$- they lie in different vector spaces. A connection is a gadget that allows you to take partial derivatives and formulate PDE's/field theories on a manifold.