I am having some trouble finishing a proof that the tangent bundle of any manifold is orientable.
What I've done so far is calculate the transition function between two standard charts on the bundle. i.e.
$\bar{\phi} \bar{\psi^{-1}} : (x^1,...,x^n,v^1,...,v^n)\mapsto(y^1,...,y^n,v^{i}\frac{\partial y^1}{\partial x^i},...,v^{i}\frac{\partial y^n}{\partial x^i})$.
Now I want the differential(push-forward) matrix of this map. This breaks up naturally into 4 blocks. Where the first and last are the differentials for $\phi\psi^{-1}$, but I am stuck calculating the other two blocks. i.e. what is $\frac{\partial y^1}{\partial v^i}$ or $\frac{\partial v^i}{\partial x^1}\frac{\partial y^1}{\partial x^i}$?
Thanks
You don't need to calculate this part of the matrix. Since you have an upper left block all of zeros, the determinant will be the product of the determinants of the "main diagonal blocks".