Orientation of submanifold

Hello,

I try to solve this problem:

Let M, N be orientable Manifolds (of dimension m,n) and f:M->N a smooth submersion. => is orientable.

Ok i have a proof, but it can't be correct, since the same "proof" would also work for any embedded submanifold.:

We know that is an embedded submanifold. And therefore we have an canonical atlas for G. If is an atlas for M, then is an atlas for G, whereas is the projection on the first (m-n) coordinates.

Now we have to show that det

And this follows, since the f_i are orientable.(?) This can't be. (because not every submanifold is orient.)

But why is it >0?? We know also that the projection is in some nbh. equal to the map:

whereas is a coordinate map of N and a coordinate maß of M.

Therefore we can put this forumula to the one above and get:

det

Do you see, why this is greater than 0?

Regards