On page 50 of John M Lee's Introduction to smooth manifolds about the local coordinates of the pushfoward $F_*$ it says
$$\left( \left. F_*\frac{\partial}{\partial x^i}\right|_{p} \right)f = \frac{\partial(f\circ F)}{\partial x^i}(p) = \frac{\partial f}{\partial y^j}(F(p))\frac{\partial F^j}{\partial x^i}(p) = \left( \left.\frac{\partial F^j}{\partial x^i}(p)\frac{\partial}{\partial y^j}\right|_{F(p)}\right)f.$$
I don't fully understand the very last equation. Could someone elaborate how the last equation comes up?
Prior to posting this question i searched for an answer but couldn't find any, i hope this is not a duplicate.
Thanks for any help.
Well, what you gotta understand is that $$\left( \left.\frac{\partial F^j}{\partial x^i}(p)\frac{\partial}{\partial y^j}\right|_{F(p)}\right)$$ is a tangent vector at $T_{F(p)}N$, and as such it "eats" germs of functions at $F(p)$ and spits out a real number. Tangent vectors are linear operators, so if $v = v^{j} \left.\frac{\partial}{\partial y^{j}}\right|_{F(p)}$, then $v(f) = v^{j} \left.\frac{\partial f}{\partial y^{j}}\right|_{F(p)}$. In your case, the $v^j$ are given by $\frac{\partial F^j}{\partial x^i}(p)$, therefore:
$$\left( \left.\frac{\partial F^j}{\partial x^i}(p)\frac{\partial}{\partial y^j}\right|_{F(p)}\right)f = \frac{\partial f}{\partial y^j}(F(p))\frac{\partial F^j}{\partial x^i}(p)$$
Keep in mind we're using Einstein notation here.