N.
Nonindependent Variables
1.
Partial differentiation with nonindependent variables.
Up to now in calculating partial derivatives of functions like w
=
f
(x, y) or w
=
f
(x, y, z),
we have assumed the variables x, y (or x, y,z) were independent. However in realworld
applications this is frequently not so. Computing partial derivatives then becomes confusing,
but it is better to face these complications now while you are still in a calculus course,
than wait to be hit with them at the same time that you are struggling to cope with the
thermodynamics or economics or whatever else is involved.
For example, in thermodynamics, three variables that are associated with a contained
gas are its
p
=
pressure,
v
=
volume,
T
=
temperature,
and you can express other thermodynamic variables like the internal energy
U
and entropy
S
in terms of p, v, and
T.
However, p, v, and
T
are not independent variables. If the gas is a socalled "ideal gas",
they are related by the equation
(I)
pv
=
nRT
(n,
R
constants).
To see what complications this produces, let's consider first a purely mathematical example.
dw
Example
1.
Let w
=
x2
+
y2
+
z2, where z
=
x2
+
y2.
Calculate

.
dx
Discussion.
aw
(a) If we think of x and y as the independent variables, then we can calculate

ax
by
two different methods:
(i) using z
=
x2
+
Y2
to get rid of z, we get
w
=
x2
+
y2
+
(x2
+
y2)2
=
x2
+
y2
+
x4
+
2x2y2
+
y4;
(ii) or by using the chain rule, remembering z is a function of x and y,
w
=
x 2 + y 2 + z 2
so the two methods agree.
(b)
On the other hand, if we think of x and z as the independent variables, using say
method (i) above, we get rid of y by using the relation
y2
=
z

x2, and get
w
=
x2
+
y2
+
z2
=
z2
+
(2

x2)
+
z2
=
Z
+
z2;