@GitGud The notation $x^k$ for coordinates isn't that bad. For example, $x=\sum x^ke_k$ represents an element of $\mathbb R^n$, while $f=\sum f_ke^k$ represents a linear functional on $\mathbb R^n$. I saw that in other books, say, Spivak's Calculus on Manifolds.
– Yai0PhahMar 13 '14 at 14:54