Some books write the coordinate vector fields with a subscript as $$\frac{\partial}{\partial x_i}$$ while some write it with a superscript 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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I never really liked this convention myself, but if you ever have to do any heavy duty differential-geometric calculations (such as in a course on general relativity) you have to keep track of a myriad of indicies, some of which are representing (tensor products of) vectors and others which represent (tensor products of) covectors, and distinguishing between the two would be very difficult if the upper indexes were not used. However, in a pure mathematics course on a more abstract coordinate-free approach to differential geometry I don't think the notation is that helpful, and I prefer to just stick to lower indicies.
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.