im currently taking up a course on differential geometry and the last topic is topology. id like to ask for help in our homework since im kind of new to this kind of questions which involves proving since im from the physics department and im taking up this course in the math department. thanks in advance
let (M,T,A) be an n-dimensional manifold
let p $\in$ M and (U, $\xi=(x^1,\dots,x^n$)) be a coordinate neighborhood of p.
a. show that $\partial_{x^i}(p) \in T_pM$ for all $i \in$ {$1,\dots,n$}.
b. show that {$\partial_{x^1}(p),\dots,\partial_{x^n}(p)$} is a linearly independent subset of $T_pM$