Questions about partial derivative notation in Evan's PDE book In the appendix of Evan's PDE book, Evan writes
(1)$\dfrac{\partial^2u}{\partial x_i \partial x_j} = u_{x_i x_j}$.
But, this notation seems strange to me because $u_{x_i x_j}$ means that we differentiate $u$ with respect to $x_i$ and then with respect to $x_j$ while $\dfrac{\partial^2u}{\partial x_i \partial x_j}$ means we differentiate $u$ with respect to $x_j$ and then with respect to $x_i$. Am I missing something? 
(2)Given a multiindex $\alpha$, define
$D^\alpha u(x) := \dfrac{\partial^{|\alpha|}u(x)}{\partial x_1^{\alpha_1}... \partial x_n^{\alpha_n}} = \partial x_1^{\alpha_1}...\partial x_n^{\alpha_n} u$
But, I have a question about this notation. It seems to me that this notation does not specify in which order the partial differentiation is performed.
For example, in $R^2$, if $\alpha = (2, 1)$, Evan's notation  $D^\alpha u(x) $ would mean $\partial x_1^2 \partial x_2^1 u$. But, what does $\partial x_1^2 \partial x_2^1 u$ mean? Does it mean that I differentiate u with resepct x2 once and then with respect to x1, and then with respect to x1? Or does it mean that I differentiate u with respect x1, and then with respect to x2, and then with respect to x1? Or does it mean that I differentiate u with respect x1, and then with respect to x1, and then with respect to x2? 
There are three ways to get the multi index (2,1) and how do I know which one it is?

You are correct, it does not specify any order. Indeed it is intended to be used with functions regular enough for Schwarz' theorem to apply. (For example, if $u$ has continuous partial derivatives up to order $|\alpha|$, the theorem applies)
