Notation for arbitrary indexing of summation, integration, and derivation Suppose I have a multivariate function $f(x_1, x_2, \dots, x_n)$ and I'd like to divide the arguments of this function into two groups.  The indices of these groups can be represented by two sets.  For example, suppose $n=5$ and 
$$ A \in \{2, 4\} $$
$$ B \in \{1, 3, 4\} $$
This means that $x_2, x_4$ belong to one group, and $x_1, x_3, x_5$ belong to the other group (which will always be the complement of the first group).  
Now, I'd like to know how to notate the following three things:


*

*Take the partial derivative of $f$ with respect to the indices in set $A$ or $B$.  i.e. in this example for $A$ we'd write $\frac{\partial^2}{\partial x_2 \partial x_4} f(x_1,x_2,x_3,x_4,x_5)$.  

*Take the integral of $f$ with respect to the indices in either $A$ or $B$.  In this example for $B$ we'd write $\int_{x_5} \int_{x_3} \int_{x_1} f(x_1,x_2,x_3,x_4,x_5) dx_1 dx_3 dx_5$

*Perform a summation of $f$ only with respect to the indices in either $A$ or $B$.  In this example for $A$, we'd write $\sum_{x_2} \sum_{x_4} f(x_1,x_2,x_3,x_4,x_5) $


What is the proper way to notationally express these three possible operations on $f$ for any general function $f$, and any general known set $A$ (and consequently $B$)?

Possibly:
1) $\frac{\partial^{|A|}}{\prod_{i\in A}\partial_{x_i}}f(x_1,x_2,x_3,x_4,x_5)$.
Sorry I don't have any ideas for the 2 and 3. However, you could just define some notation such as a large capital iota which functions like the large sigma for sums and large pi for products. 
