Mathematical Methods for Physics and Engineering : A Comprehensive Guide

5.4 USEFUL THEOREMS OF PARTIAL DIFFERENTIATION

5.4 Useful theorems of partial differentiation

So far our discussion has centred on a functionf(x, y) dependent on two variables,

xandy. Equally, however, we could have expressedxas a function offandy,

oryas a function offandx. To emphasise the point that all the variables are

of equal standing, we now replacefbyz. This does not imply thatx,yandz

are coordinate positions (though they might be). Sincexis a function ofyandz,

it follows that

dx=

(

∂x

∂y

)

z

dy+

(

∂x

∂z

)

y

dz (5.11)

and similarly, sincey=y(x, z),

dy=

(

∂y

∂x

)

z

dx+

(

∂y

∂z

)

x

dz. (5.12)

We may now substitute (5.12) into (5.11) to obtain

dx=

(

∂x

∂y

)

z

(

∂y

∂x

)

z

dx+

[(

∂x

∂y

)

z

(

∂y

∂z

)

x

+

(

∂x

∂z

)

y

]

dz. (5.13)

Now if we holdzconstant, so thatdz= 0, we obtain thereciprocity relation

(

∂x

∂y

)

z

=

(

∂y

∂x

)− 1

z

,

which holds provided both partial derivatives exist and neither is equal to zero.

Note, further, that this relationship only holds when the variable being kept

constant, in this casez, is the same on both sides of the equation.

Alternatively we can putdx= 0 in (5.13). Then the contents of the square

brackets also equal zero, and we obtain thecyclic relation
(
∂y
∂z

)

x

(

∂z

∂x

)

y

(

∂x

∂y

)

z

=− 1 ,

which holds unless any of the derivatives vanish. In deriving this result we have

used the reciprocity relation to replace (∂x/∂z)−y^1 by (∂z/∂x)y.

5.5 The chain rule

So far we have discussed the differentiation of a functionf(x, y) with respect to

its variablesxandy. We now consider the case wherexandyare themselves

functions of another variable, sayu. If we wish to find the derivativedf/du,

we could simply substitute inf(x, y) the expressions forx(u)andy(u)andthen

differentiate the resulting function ofu. Such substitution will quickly give the

desired answer in simple cases, but in more complicated examples it is easier to

make use of the total differentials described in the previous section.