Question about the tangent space of a manifold

Question

I am studying manifolds by myself and I am following this note https://www.math.toronto.edu/laithy/367/367.html but i am confused from the definition of a tangent space The definition is showing in the picture

1. what does he mean by $u^{i}$ in the denomator of the formula of $v(f)$ ?is it the coordinate $(u^1,.....,u^{m})$ of $ \mathbb{R}^{m}$ ?,how we can compute the value of $a$ in the fomrula of $v(f)$
2. I have seen many books but I could not find any book gives an example of computing the tangent space I hope that someone can give me an example of computing the tangent space in detail to make it more clear for me.