Exact and inexact differentials

5.3 EXACT AND INEXACT DIFFERENTIALS
this partial derivative account must be taken only of explicit appearances of x1 in
the function f, and no allowance must be made for the knowledge that changing
x1 necessarily changes x2 , x3 , . . . , xn . The contribution from these latter changes is
precisely that of the remaining terms on the RHS of (5.8). Naturally, what has
been shown using x1 in the above argument applies equally well to any other of
the xi , with the appropriate consequent changes.
Find the total derivative of f(x, y) = x2 + 3xy with respect to x, given that y = sin−1 x.
We can see immediately that
∂f
= 2x + 3y,
∂x
∂f
= 3x,
∂y
dy
1
=
dx
(1 − x2 )1/2
and so, using (5.8) with x1 = x and x2 = y,
1
df
= 2x + 3y + 3x
dx
(1 − x2 )1/2
3x
= 2x + 3 sin−1 x +
.
(1 − x2 )1/2
Obviously the same expression would have resulted if we had substituted for y from the
start, but the above method often produces results with reduced calculation, particularly
in more complicated examples. 5.3 Exact and inexact differentials
In the last section we discussed how to find the total differential of a function, i.e.
its infinitesimal change in an arbitrary direction, in terms of its gradients ∂f/∂x
and ∂f/∂y in the x- and y- directions (see (5.5)). Sometimes, however, we wish
to reverse the process and find the function f that differentiates to give a known
differential. Usually, finding such functions relies on inspection and experience.
As an example, it is easy to see that the function whose differential is df =
x dy + y dx is simply f(x, y) = xy + c, where c is a constant. Differentials such as
this, which integrate directly, are called exact differentials, whereas those that do
not are inexact differentials. For example, x dy + 3y dx is not the straightforward
differential of any function (see below). Inexact differentials can be made exact,
however, by multiplying through by a suitable function called an integrating
factor. This is discussed further in subsection 14.2.3.
Show that the differential x dy + 3y dx is inexact.
On the one hand, if we integrate with respect to x we conclude that f(x, y) = 3xy + g(y),
where g(y) is any function of y. On the other hand, if we integrate with respect to y we
conclude that f(x, y) = xy + h(x) where h(x) is any function of x. These conclusions are
inconsistent for any and every choice of g(y) and h(x), and therefore the differential is
inexact. It is naturally of interest to investigate which properties of a differential make
155
PARTIAL DIFFERENTIATION
it exact. Consider the general differential containing two variables,
df = A(x, y) dx + B(x, y) dy.
We see that
∂f
= A(x, y),
∂x
∂f
= B(x, y)
∂y
and, using the property fxy = fyx , we therefore require
∂B
∂A
=
.
∂y
∂x
(5.9)
This is in fact both a necessary and a sufficient condition for the differential to
be exact.
Using (5.9) show that x dy + 3y dx is inexact.
In the above notation, A(x, y) = 3y and B(x, y) = x and so
∂B
= 1.
∂x
∂A
= 3,
∂y
As these are not equal it follows that the differential is inexact. Determining whether a differential containing many variable x1 , x2 , . . . , xn is
exact is a simple extension of the above. A differential containing many variables
can be written in general as
df =
n
gi (x1 , x2 , . . . , xn ) dxi
i=1
and will be exact if
∂gi
∂gj
=
∂xj
∂xi
for all pairs i, j.
(5.10)
There will be 12 n(n − 1) such relationships to be satisfied.
Show that
(y + z) dx + x dy + x dz
is an exact differential.
In this case, g1 (x, y, z) = y + z, g2 (x, y, z) = x, g3 (x, y, z) = x and hence ∂g1 /∂y = 1 =
∂g2 /∂x, ∂g3 /∂x = 1 = ∂g1 /∂z, ∂g2 /∂z = 0 = ∂g3 /∂y; therefore, from (5.10), the differential
is exact. As mentioned above, it is sometimes possible to show that a differential is exact
simply by finding by inspection the function from which it originates. In this example, it
can be seen easily that f(x, y, z) = x(y + z) + c. 156