Theory - Derivation of the Symmetric Pressure Gradient
I have been reading Monaghan's paper from 1982, where the approach is stated as such:
Equation 3.1, 3.2 and 3.3 are very clear to me and I derive the exact same result in these cases. The problem arises at 3.4, since when I plug the expressions in 3.2 and 3.3 into 3.1 I get (in my slightly different notation, but should still be clear);
So I am a bit confused with how 3.4 is actually constructed - does anyone happen to have a more thorough walk through?
The only one I have been able to find is done by Cossins (https://arxiv.org/abs/1007.1245), in which he uses the identity (page 8);
And is able to construct an operator very similar to 3.4;
Which is equivivalent to 3.4, since one divides by density and replaces f with rho.
So basically I would really like to understand how Monaghan derived 3.4 - maybe what Cossins did it the approach that one fact should use?