Let $A$ be an open set of $\mathbb{R}^d$ and $f:A\to M$ (I am unsure if there is a typo in the lecture notes here, would $f:A\to S$ make more sense? ) be an immersion such that $f(A)$ is an open set of the submanifold $S$ containing $p$. If $a \in A$ is such that $f(a)=p$, the vectors $(p,f_i(a))$ form a basis of $T_pS$ and if $(U,x)$ is a chart for $p$ obtained as the inverse of the restriction of $f$ to some open set for $a$, we have that $(p,f_i(a))=\frac{\partial}{\partial x^i}|_p$. Then $\frac{\partial}{\partial x^i}$ has as vector part $f_i\circ x$ which is just $f_i$ written in terms of $x^1,...,x^d$ instead of $u^1,...,u^d$.
Example :Let's consider for instance the geographic coordinates $\varphi$ and $\theta$ in $S^2\setminus I$ , $I = \{(a,b,c) \in S^2| a \ge 0,b=0)\}$coming from the immersion $f(u,v)=(\cos(u)\sin(v),\sin(u)\sin(v),\cos(v))$ restricted to $A=]0,2\pi[\times]0,\pi[$. Since $f_1(u,v)=(-\sin u \sin v,\cos u \cos v,0)$ and $f_2(u,v)=(-\cos u \cos v,\sin u \cos v,-\sin v)$, then $\partial_1 = \frac{\partial}{\partial\varphi}\equiv (-\sin \varphi \sin \theta,\cos \varphi \cos \theta,0)$ $\partial_2 = \frac{\partial}{\partial\theta}\equiv (-\cos \varphi \cos \theta,\sin \varphi \cos \theta,-sin \theta)$
I am having trouble understanding this paragraph,even with the example is not clear for me.
Why do $(p,f_i(a))$ form a basis of $T_pS$ , why is $(p,f_i(a))=\frac{\partial}{\partial x^i}|_p$ and why does $\frac{\partial}{\partial x^i}$ have as vector part $f_i\circ x$ ? It is claimed that $(p,f_i(a))$ is an applied basis vector at point $p$, whose vector part is $f_i(a)$ I don't get why we need a function $f$ to defined a basis If I wanted a basis for $T_pS$ I would just take the partial derivatives with respect to the coordinates. They also state that $(U,x)$ is a chart for $p$ obtained as the inverse of the restriction of $f$ to some open set for $a$, that is $x:=(f|_A)^{-1}:f(A) \to A, f(A)\subseteq S, A \subset \mathbb{R}^d$. from this I guess I have to prove that $\frac{\partial}{\partial x^i}= (f_i)^k\partial_k$, but I am unable to do it.
Can someone please clear these things up?