chapter 37

Total differential, rates of change and small changes

Pages 7

Problem 2. If z = f (u,v,w) and z =3u2 −2v +4w3v2 find the total differential, dz

The total differential

dz = ∂z ∂u

du + ∂z ∂v

dv + ∂z ∂w

dw

∂z

∂u = 6u (i.e. v and w are kept constant)

∂z

∂v = −2 + 8w3v

(i.e. u and w are kept constant) ∂z

∂w = 12w2v2 (i.e. u and v are kept constant)

Hence

dz=6udu + (8vw3 − 2) dv + (12v2w2)dw

Problem 3. The pressure p, volume V and temperature T of a gas are related by pV =kT , where k is a constant. Determine the total differentials (a) dp and (b) dT in terms of p, V and T .