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 $$\frac{\partial}{\partial s}, \frac{\partial}{\partial t}\,\,\text{and}\,\,\frac{\partial}{\partial u}.$$(adsbygoogle = window.adsbygoogle || []).push({});

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!

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# Chain rule and multivariable calculus

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