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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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2 Answers 2

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.

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So a lower index in the denominator is actually an upper index? If so, why is this the convention? –  Eric Auld Nov 5 '13 at 2:00
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

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.

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