Say I have a function of three variables, $F=F(s_{12},s_{23},s_{13}) = F(s,t,-s-t)$, where $s_{12}=s,s_{23}=t$ and $s_{13}=u = -s-t$. I want to compute the differential operators $$\frac{\partial}{\partial s}, \frac{\partial}{\partial t}\,\,\text{and}\,\,\frac{\partial}{\partial u}.$$
I can write $$\frac{\partial}{\partial s} = \frac{\partial s_{12}}{\partial s} \frac{\partial}{\partial s_{12}} + \frac{\partial s_{23}}{\partial s} \frac{\partial}{\partial s_{23}} + \frac{\partial s_{13}}{\partial s} \frac{\partial}{\partial s_{13}} = \frac{\partial}{\partial s_{12}} - \frac{\partial}{\partial s_{13}}$$
Similarly, $$\frac{\partial}{\partial t} = \frac{\partial}{\partial s_{23}} - \frac{\partial}{\partial s_{13}}$$ How should I go about computing $\partial/\partial u$? I can also write $$\frac{\partial}{\partial u} = \frac{\partial s_{12}}{\partial u} \frac{\partial}{\partial s_{12}} + \frac{\partial s_{23}}{\partial u} \frac{\partial}{\partial s_{23}} + \frac{\partial s_{13}}{\partial u} \frac{\partial}{\partial s_{13}}$$ but I am not sure how to simplify the first two terms.
Thanks!
$\partial\over\partial u$ is only defined in terms of a complete set of independent variables including $u$. You have not defined such a set, as $u$ is a function of $s$ and $t$, and is thus not independent. So $\partial\over\partial u$ is not well-defined.
You could consider either $s$ or $t$ to be a function of $u$ and the other. In this case, you could define $\partial\over\partial u$ in that sense, but you would find (in general - I have not checked here) that the meaning of $\partial\over\partial u$ would differ depending on whether $s$ or $t$ was chosen as the other independent variable. And also, $\partial\over\partial s$ or $\partial\over\partial t$ would also change from what it is when $s$ and $t$ are considered independent. (I would not be surprised if in this simple case, though, if these variant definitions actually coincided.)