Equations used for the derivative of the comoving volume element

An equation for the calculation of the comoving volume element is given by Carroll, Press & Turner [1], but they give an equation in terms of the proper motion distance. The equation is:

$$dV = \frac{d_M^2}{\left(1+\Omega_K H_0^2 d_M^2\right)^{1/2}} d(d_M) d\Omega$$ (8)

d_M is taken from eq.  4, so it makes use of the w parameter. This implies that for w = -1, eq. 4 for the proper motion distance reduces to eq. 3 so w takes no effect in the calculation of the comoving volume element. As the function calc_vol should use z instead of d_M as a parameter, this equation has to be modified. The resulting equation for the desired derivative $dV/(dz d\Omega)$ is

$$\frac{dV}{dz d\Omega} = \frac{d_M^2}{\left(1+\Omega_K H_0^2 d_M^2\right)^{1/2}} \frac{d(d_M)}{dz}$$ (9)

The numerical approach for the calculation of this derivative is described in section VIF below.