Pressure and density
Pressure
Pressure is the measure of volume stress. It is defined as the the ratio of the magnitude of the total force \(F\) exerted on a surface to the area \(A\) of the surface:
\[p := \frac{F}{A}\]
The SI unit of pressure is the pascal.
\[\left[p\right] = 1\,\frac{\mathrm{N}}{\mathrm{m}^2} = 1\,\mathrm{Pa}\]
Density
The density of a homogeneous substance is the mass per unit volume:
\[\varrho := \frac{m}{V}.\]
If the object is not homogeneous, we have to define density for a very small volume \(\Delta V\) within which the density can be approximated as constant:
\[\varrho = \lim_{\Delta V \to 0}\frac{\Delta m}{\Delta V} = \frac{\mathrm{d} m}{\mathrm{d} V}.\]
The SI units of density:
\[\left[\varrho\right] = 1\,\frac{\mathrm{kg}}{\mathrm{m}^3} = \frac{1000\,\mathrm{g}}{10^6\,\mathrm{cm}^3} = 10^{-3}\,\frac{\mathrm{g}}{\mathrm{cm}^3}.\]