Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PARTIAL DIFFERENTIATION

it exact. Consider the general differential containing two variables,

df=A(x, y)dx+B(x, y)dy.

We see that

∂f

∂x

=A(x, y),

∂f

∂y

=B(x, y)

and, using the propertyfxy=fyx, we therefore require

∂A

∂y

=

∂B

∂x

. (5.9)

This is in fact both a necessary and a sufficient condition for the differential to

be exact.

Using (5.9) show thatxdy+3ydxis inexact.

In the above notation,A(x, y)=3yandB(x, y)=xand so

∂A

∂y

=3,

∂B

∂x

=1.

As these are not equal it follows that the differential is inexact.

Determining whether a differential containing many variablex 1 ,x 2 ,...,xnis

exact is a simple extension of the above. A differential containing many variables

canbewritteningeneralas

df=

∑n

i=1

gi(x 1 ,x 2 ,...,xn)dxi

and will be exact if

∂gi

∂xj

=

∂gj

∂xi

for all pairsi, j. (5.10)

There will be^12 n(n−1) such relationships to be satisfied.

Show that

(y+z)dx+xdy+xdz

is an exact differential.

In this case,g 1 (x, y, z)=y+z,g 2 (x, y, z)=x,g 3 (x, y, z)=xand hence∂g 1 /∂y=1=
∂g 2 /∂x,∂g 3 /∂x=1=∂g 1 /∂z,∂g 2 /∂z=0=∂g 3 /∂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 thatf(x, y, z)=x(y+z)+c.