In the infinitesimal neighborhood surrounding a point in an inviscid flow, the small change in pressure, \(d p\), that corresponds to a small change in velocity, \(d V\), is given by the differential relation \(d p=-ho V d V\). 

a. Using this relation, derive a differential relation for the fractional change in density, \(d ho / ho\), as a function of the fractional change in velocity, \(d V / V\), with the compressibility \(\tau\) as a coefficient.

b. The velocity at a point in an isentropic flow of air is \(10 \mathrm{~m} / \mathrm{s}\) (a low speed flow), and the density and pressure are \(1.23 \mathrm{~kg} / \mathrm{m}^{3}\) and \(1.01 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}\), respectively (corresponding to standard sea level conditions). The fractional change in velocity at the point is 0.01 . Calculate the fractional change in density.

c. Repeat part (b), except for a local velocity at the point of \(1000 \mathrm{~m} / \mathrm{s}\) (a high-speed flow). Compare this result with that from part (b), and comment on the differences.