Problem: Prove the identity $V=\sum V[x_i]U_i$, where $x_1, x_2, x_3$ are the natural coordinate functions. (Hint: evaluate $V=\sum v_i U_i $ on $x_j$)
Elementary differential geometry written by Barrett O'neill 1.3-4, p.15
I just don't know what the right term means. I just know that it is a linear combination of $V[x_i]$, a function from $\mathbb R^3$ to $\mathbb R$, and $U_i$, a natural frame field. What is the definition or meaning of $V[x_i]$?