can anyone help me with this problem:
Show that for a manifold $M$, the tangent bundle $TM$ also has the structure of a manifold. If $M$ is an n-manifold, what is the dimension of $TM$?
for the 1st part it is enough to show that it is locally euclidean.
thanx.
The key idea is to come up with coordinate charts; that is, one-one maps between open sets $V\subset M$ and open subsets of $\mathbb{R}^n$. The two-dimensional special case is enough to illustrate the general idea. If we have a chart on $V\subset M$, then we have coordinates $(x,y)$ in the patch $V$. There are a bunch of ways to define $TM$ formally, but in all cases you can visualize elements of $T_p$ as vectors with tails at $p$. Sp a vector $v\in T_p$ with $p\in V$ is fully specified by four numbers: coordinates $(x,y)$ of $p$, and the components $(v_x,v_y)$ of the vector. So $(x,y,v_x,v_y)$ are coordinates for certain elements of $TM$, specifically those belonging to $T_p$ with $p\in V$.
By doing this over all patches $V$ in an atlas for $M$, you get an atlas for $TM$.
Of course there's a lot of crank-turning to make this a formal proof. First you need to show (in the general $n$-dim case) that any $T_p$ is an $n$-dimensional vector space, and give a basis for it --- the basis depending on a coordinate chart for a patch containing $p$. The details will depend on what formal definition of $TM$ you are using (equivalence classes of smooth curves, derivations, etc.) Then you have to show the "smooth patching property", i.e., that if $V_1$ and $V_2$ are coordinate patches on $M$ and $V_1\cap V_2\neq\varnothing$, that the smooth transition function from $V_1$-coordinates to $V_2$-coordinates lifts to a smooth transition function for the corresponding (local) coordinates on patches of $TM$. There's a bunch of stuff to write out, but that's the general idea.