Mathematical Methods for Physics and Engineering : A Comprehensive Guide

5.3 EXACT AND INEXACT DIFFERENTIALS

this partial derivative account must be taken only ofexplicitappearances ofx 1 in

the functionf,andnoallowance must be made for the knowledge that changing

x 1 necessarily changesx 2 ,x 3 ,...,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 usingx 1 in the above argument applies equally well to any other of

thexi, with the appropriate consequent changes.

Find the total derivative off(x, y)=x^2 +3xywith respect tox, given thaty=sin−^1 x.

We can see immediately that

∂f

∂x

=2x+3y,

∂f

∂y

=3x,

dy

dx

=

1

(1−x^2 )^1 /^2

and so, using (5.8) withx 1 =xandx 2 =y,

df

dx

=2x+3y+3x

1

(1−x^2 )^1 /^2

=2x+3sin−^1 x+

3 x

(1−x^2 )^1 /^2

.

Obviously the same expression would have resulted if we had substituted foryfrom 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/∂yin thex-andy- directions (see (5.5)). Sometimes, however, we wish

to reverse the process and find the functionfthat 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 isdf=

xdy+ydxis simplyf(x, y)=xy+c,wherecis a constant. Differentials such as

this, which integrate directly, are calledexact differentials, whereas those that do

not areinexact differentials. For example,xdy+3ydxis 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 differentialxdy+3ydxis inexact.

On the one hand, if we integrate with respect toxwe conclude thatf(x, y)=3xy+g(y),
whereg(y) is any function ofy. On the other hand, if we integrate with respect toywe
conclude thatf(x, y)=xy+h(x)whereh(x) is any function ofx. These conclusions are
inconsistent for any and every choice ofg(y)andh(x), and therefore the differential is
inexact.

It is naturally of interest to investigate which properties of a differential make