Can all double summations be rewritten as a single sum?
For example, can this, $\displaystyle \sum_{i=-1}^{1}\sum_{j=0}^{2}(2i+3j)$, be rewritten as a single sum, and if so, how?
The answer isn't important this is just an example.
Well, I don't know of any general method, but I would attack the above double sum as follows:
$\displaystyle \sum_{i=-1}^{1}\sum_{j=0}^{2}(2i+3j)$
$\displaystyle =\sum_{i=-1}^{1}\left(2i\sum_{j=0}^{2}1+3\sum_{j=0}^{2}j\rig ht)$
$\displaystyle =\sum_{i=-1}^{1}\left((2i)(3)+3\frac{(2)(2+1)}{2}\right)$
$\displaystyle =\sum_{i=-1}^{1}6i+9$
Depending on the sums you're dealing with, similar strategies could be possible.
Well it's not possible in general to rewrite a double integral as a single integral, without trying to compute the inner one (before or after a change in order of integration), right? So I expect you may be looking for a method that does not exist.
Edit: I stand corrected. But I probably won't be of much further help in this discussion.. I am familiar with Kronecker Delta but don't see how it relates to the question, also it's been years since I used Green's Theorem and I don't really remember it.