Linearity of everything May I ask for details about how can I prove "linearity of everything" for the following step?
$(f^*dx_i)(Y) = \sum_{j = 1}^lY^j (f^*dx_i)(\frac{\partial}{\partial y^j}) =  \sum_{j = 1}^lY^jdx_i(f_*(\frac{\partial}{\partial y^j}))$.

Check out my comments to my answer to this question, clearly the one to which you refer.
Feel free to leave comments if you need more info.  Cheers, RKL.
