# Notation: subscript vs. superscript for coordinate vector fields

Some books write the coordinate vector fields as $$\frac{\partial}{\partial x_i}$$ with a subscript, and some write it as $$\frac{\partial}{\partial x^i}.$$

Is there a conceptual reason for this distinction? I.e. in some texts I have seen, a supercript is to indicate the components of covectors, and a subscript for vectors.

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Usually one wants lower subscripts to denote vectors, probably coming from the common practice of denoting the standard basis of $\mathbb R^n$ as $e_1,\ldots,e_n$. Then to use Einstein summation convention you would want the coordinate functions $x^i$ on $\mathbb R^n$ to have upper indices since they represent covectors: any vector on $\mathbb R^n$ is written as $$v = x^i(v)e_i.$$ In $\frac{\partial}{\partial x^i}$, the $i$ is a lower index (consistent with vectors having lower indices) and arguably this is the better notation. The main advantage, in my opinion, of using $x_i$ instead of $x^i$ is that you don't confuse the index with an exponent and writing a power of a coordinate function is cleaner.
It's a good convention. Einstein summation is consistent with the fact that the basis of vectors $\left\{ \frac{\partial}{\partial x^1}, \ldots, \frac{\partial}{\partial x^n} \right\}$ is dual to the basis of covectors $\left\{ dx^1, \ldots, dx^n \right\}$. –  Sammy Black Nov 5 '13 at 3:03
The "lower index in the denominator is an upper index" (or the reverse) can be seen as logical by calculating the derivative of a function with respect to a coordinate $x^i$, which we expect to be a gradient: $\frac{\partial}{\partial x^i}f(x)=\frac{\partial f}{\partial x^j}\delta^j_i$. Sum over the $j$s (since they are upper and lower now), so the derivative is a 1-form (has lower $i$). –  levitopher Nov 5 '13 at 3:56