I'm embarrassed asking this question. I took a long break. main question is : cartesian coordinate system in $R^n$ space is shown as $(x_1,x_2 ...x_n)$. Show that for $1\le i\le n$ each $x_i:R^n\mapsto R$ function have partial derivatives with respect to its $k.$ variable and $$for \quad 1\le k\le n \quad \frac{\partial x_i}{\partial x_k}(p)=\delta_{ik} $$
Answer in the book was: for $1\le i \le n$
for $1\le i,j \le n$ let $i\neq j$
I dont understand why is $x_i(p_1,p_2...p_{i-1},p_i+s, p_{i+1, ...p_n})=[p_i+s]$ at 1* . and isn't 2* wrong?
It's quite trivial. Let $\{e_1,\dots e_n\}$ then $\{x_1,\dots x_n\}$ is the dual basis. So you understand $1^*$, don't you? $2^*$ is a typo since $p_i-p_i=0$.