ABSTRACT
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)
=
(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
(a) Total differential dp= ∂p ∂T
dT + ∂p ∂V
dV
Since pV =kT then p = kT V
hence ∂p ∂T
= k V
and ∂p ∂V
= −kT V 2
Thus dp = k V
dT − kT V 2
dV
Since pV = kT,k = pV T
Hence dp =
( pV T
) V
dT −
( pV T
) T
V 2 dV
i.e. dp= p T
dT − p V
dV
(b) Total differential dT = ∂T ∂p
dp+ ∂T ∂V
dV
Since pV =kT,T = pV k
hence ∂T ∂p
= V k
and ∂T ∂V
= p k
Thus dT = V k
dp+ p k
dV and substituting k= pV T
gives:
dT = V( pV T
) dp + p( pV T
) dV
i.e. dT = T p
dp+ T V
dV
Now try the
Practice Exercise 251 Further problems on the total differential (answers on page 1139)
In Problems 1 to 5, find the total differential dz
1. z= x3+ y2
2. z=2xy− cosx
3. z= x − y x + y
4. z= x ln y
5. z= xy+ √
x
y −4
6. If z= f (a,b,c) and z=2ab−3b2c+abc, find the total differential, dz
7. Given u= ln sin(xy) show that du= cot(xy)(ydx + x dy)
Sometimes it is necessary to solve problems in which different quantities have different rates of change. From equation (1), the rate of change of z, dz
dt is given by:
dz dt
= ∂z ∂u
du dt
+ ∂z ∂v
dv dt
+ ∂z ∂w
dw dt
+ ··· (2)
Problem 4. If z= f (x, y) and z=2x3 sin 2y, find the rate of change of z, correct to 4 significant figures, when x is 2 units and y is π/6 radians and when x is increasing at 4 units/s and y is decreasing at 0.5 units/s
Using equation (2), the rate of change of z,
dz dt
= ∂z ∂x
dx dt
+ ∂z ∂y
dy dt
Since z=2x3 sin2y, then
∂z
∂x = 6x2 sin2y and ∂z
∂y = 4x3 cos2y
is
and since y is decreasing at 0.5 units/s, dy dt
=−0.5
Hence dz dt
=(6x2 sin2y)(+4)+ (4x3 cos2y)(−0.5) =24x2 sin 2y − 2x3 cos2y
When x =2 units and y= π 6
radians, then dz dt
= 24(2)2 sin[2(π/6)]− 2(2)3 cos[2(π/6)] = 83.138− 8.0
Hence the rate of change of z, dz dt
=75.14units/s, correct to 4 significant figures.
