Tangent bundle is always orientable

Hello all, trying to prove that the tangent bundle TM of a smooth n-manifold M is always orientable. To make it easier to write down Im doing n=2.

Let the differentiable structure on M be given by . This then induces a differentiable structure on TM given by where we define i.e. we take as the coordinates of a point (p,w) in TM the coordinates of the point p in U, plus the coordinates of w in the basis

Once you've proved that TM is a smooth 4-manifold (which isn't hard) you then have to show that for every change of coordinate function the determinant of its jacobian matrix is positive, i.e.

for all .

To write out the 4x4 matrix is where I've come unstuck. The first two columns will be the partial derivatives of the 4 component functions of with respect to , i.e. the top-left 2x2 block will just be and the bottom-left will be zero because .

I don't know how to work out the two 2x2 blocks on the right hand side of the matrix. Basically I don't know what to take the partial derivative with respect to, and just how to write it down. Hopefully the top-right will be zero and the bottom-right will be as well, so then we have as a determinant the square of a real number which will always be positive.

Any help anyone could give me would be great!

Cheers,

Atticus