Assume we have a global coordinate system ($\vec{X}$) and a local coordinate ($\vec{\zeta}$) at a given point A.

Therefore, we would have: $\vec{X_{AB}}=\vec{X_{A}}+\vec{\zeta}$ and $\vec{X_{AB}}=\vec{X_{B}}-\vec{\zeta}$ in which, AB is the midpoint of the line connecting point A to B when A and B are infinitely close to each other.

$$\vec{\zeta}=\vec{X_B}-\vec{X_A}$$ $$\vec{X_{AB}}=\frac{1}{2}({\vec{X_B}+\vec{X_A}})$$

Now my question is how can I derive the differential operators in two-point correlation technique written bellow:

$${(\frac{\partial }{\partial X_k})}_A =\frac{1}{2}{(\frac{\partial }{\partial X_k})}_{AB}-{(\frac{\partial }{\partial {\zeta}_k})} $$

$${(\frac{\partial }{\partial X_k})}_B =\frac{1}{2}{(\frac{\partial }{\partial X_k})}_{AB}+{(\frac{\partial }{\partial {\zeta}_k})} $$ $k=1,2,3$

In general, when we have: $${(\frac{\partial }{\partial X_A})}={(\frac{\partial }{\partial (X_{AB}-\zeta)})} $$
how can we proceed to split the derivative operator into $(\frac{\partial }{\partial X_{AB}}) $ and $(\frac{\partial }{\partial \zeta}) $?

Thanks a lot


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