If we work (for example) in the category of differentiable manifolds, then i saw that it is standard calculating the transition functions of the tangent bundle of a differentiable manifold. It seems to me that this happens because we can "change chart". I couldn't find such a calculation, for example, for projective varieties.
Let $k=\overline{k}$. Let $V \subset \mathbb{P}^n$ be a projective variety, defined (for simplicity) by an irreducible, homogeneus polynomial $F$ (such that $(F)$ is radical). How can i do to calculate in this case the transition functions of the tangent bundle of $V$? I am sorry if the question is trivial.