Theory - Anti-symmetry of dWdq
My question is regarding this plot basically (from https://shipjournal.co/index.php/sst/article/download/37/142?inline=1);
As we can see from the blue line, the kernel is symmetric, so wheter a particle is on either side of particle at 0, then the contribution is the same. But for the derivative of the kernel w.r.t. to q, i.e. dWdq (the red line in the plot), it is anti-symmetric, so to me it seems like we have to care about the sign of the contribution.
I have checked a lot of different SPH codes and I can see that many do not do this, i.e. they only care about the absolute size of the contribution and not the direction. Could anyone try to explain to me why, we only care about the absolute value of dWdq and not the sign, even though it is anti-symmetric?