In my textbook, a formula for estimating error in bulk-volume measurements is derived, but I don't quite follow one step in the derivation. The book writes the following:
The bulk volume of a porous-rock core sample can be measured in two steps, by weighing the sample first in water, $m_1$ (assuming $100$% water saturation), and then in air, $m_2$.
The bulk volume is calculated as follows:
$V_b = \frac{m_2 - m_1}{\rho_w}$
By differentiating this equation, we obtain:
$dV_b = \frac{\partial V_b}{\partial m_2}dm_2 + \frac{\partial V_b}{\partial m_1}dm_1 + \frac{\partial V_b}{\partial \rho_w}d \rho_w$,
$dV_b = \frac{m_2 - m_1}{\rho_w}\Bigg(\frac{dm_2}{m_2 - m_1} - \frac{dm_1}{m_2 - m1} - \frac{d \rho_w}{\rho_w}\Bigg)$
If the density measurement and the measurements of $m_1$ and $m_2$ are considered to be independent, the uncertainty inherent in the bulk-volume value can be written as:
$\Bigg(\frac{\Delta V_b}{V_b}\Bigg)^2 = 2\Bigg(\frac{\Delta m}{(m_2 - m_1)}\Bigg)^2 + \Bigg(\frac{\Delta \rho_w}{\rho_w}\Bigg)^2$
where the error in the weighing of the two masses is considered to be identical, $\Delta m = \Delta m_1 = \Delta m_2$.
OK, so it is this last step here that I don't quite follow. I assume that we are here using that $dV_b = \Delta V_b$. Since we know that:
$dV_b = \frac{m_2 - m_1}{\rho_w}\Bigg(\frac{dm_2}{m_2 - m_1} - \frac{dm_1}{m_2 - m1} - \frac{d \rho_w}{\rho_w}\Bigg)$
I would then assume that we have:
$\frac{\Delta V_b}{V_b} = \frac{\frac{m_2 - m_1}{\rho_w}\Bigg(\frac{dm_2}{m_2 - m_1} - \frac{dm_1}{m_2 - m1} - \frac{d \rho_w}{\rho_w}\Bigg)}{\frac{m_2 - m_1}{\rho_w}}$
$\frac{\Delta V_b}{V_b} = \Bigg(\frac{dm_2}{m_2 - m_1} - \frac{dm_1}{m_2 - m1} - \frac{d \rho_w}{\rho_w}\Bigg)$
$\frac{\Delta V_b}{V_b} = \Bigg(\frac{\Delta m}{m_2 - m_1} - \frac{d\rho_w}{\rho_w}\Bigg)$
So:
$\Bigg(\frac{\Delta V_b}{V_b}\Bigg)^2 = \Bigg(\frac{\Delta m}{m_2 - m_1} - \frac{d\rho_w}{\rho_w}\Bigg)^2$
$\Bigg(\frac{\Delta V_b}{V_b}\Bigg)^2 = \Bigg(\frac{\Delta m}{m_2 - m_1}\Bigg)^2 - 2 \frac{d\rho_w}{\rho_w}\Bigg(\frac{\Delta m}{m_2 - m_1}\Bigg) + \Bigg(\frac{d \rho_w}{\rho_w}\Bigg)^2$
But this is obviously not the same.
So if anyone can please explain to me that last step in the derivation of the formula, I would be very grateful!