260 Chapter 9Functions of several variables
If∆xand∆yare small enough, the terms quadratic in ∆are small compared with
the linear terms, and an approximate value of∆zis
(9.16)
This result, valid for allcontinuous functions of two variables, shows that when the
changes∆xand∆yare small enough the total change in zis approximately equal to
the change in zdue to change∆xalone (the first term of the expression on the right of
(9.16)) plus the change in zdue to change∆yalone (the second term). In addition, the
accuracy of the expression improves as∆xand∆yapproach zero. As in Section 4.12,
we consider infinitesimal changes dxand dy, and define the quantity
(9.17)
as the limiting case of (9.16). This quantity is called the total differentialof zwith
respect to xand y.
4The concept of the total differential is readily generalized for functions of any
number of variables; for a function of nvariables,
z 1 = 1 f(x
1,x
2,x
3,1=,x
n)
the total differential is
(9.18)
where, for example,∂z 2 ∂x
1is the partial derivative with respect to variable x
1with all
other variables,x
2,x
3,1=,x
n, kept constant.
EXAMPLES 9.10Find the total differential:
(i)
∂
∂
=,
∂
∂
=− , =
∂
∂
∂
∂
z
xy
z
y
x
y
dz
z
x
dx
z
y
yx
1
2
dy=−
y
dx
x
y
dy
1
2z
x
y
=
dz
z
x
dx
z
x
dx
z
x
=
∂
∂
∂
∂
++
∂
∂
1122nnniniidx
z
x
dx
=
∂
∂
=∑
1dz
z
x
dx
z
y
dy
yx
=
∂
∂
+
∂
∂
∆∆∆z
z
x
x
z
y
y
yx
≈
∂
∂
∂
∂
4The total differential and the equality of the mixed second derivatives were discovered in 1719 by Nicolaus
(II) Bernoulli (1687–1759). The nephew of Johann (I) and Jakob (John and James, or Jean et Jacques), he is not to
be confused with Nicolaus (I), his father, nor with Nicolaus (III), the son of Johann (I) and brother of Daniel and
Johann (II). It is Daniel Bernoulli (1700–1782) who wrote the Hydrodynamicaof 1738 which contains the concept
of ‘Bernoulli’s Theorem’ and a development of the kinetic theory of gases.