598
B Review of Traditional Calculus
∂z
∂u
=
∂z
∂x
∂x
∂u
+
∂z
∂y
∂y
∂u
,
(B.1.6a)
∂z
∂v
=
∂z
∂x
∂x
∂v
+
∂z
∂y
∂y
∂v
.
(B.1.6b)
If x and y both depend upon the same single parameter t, the Eqs. (B.1.6a)
and (B.1.6b) reduce to
dz
dt
=
∂z
∂x
dx
dt
+
∂z
∂y
dy
dt
(B.1.6c)
giving the derivative of z with respect to t.
(ii) Differentials and the Total Differential.
The usual statement for a differential, as seen in single-variable calculus, is
y = f (x)
⇒
dy = f
(x))x = f
(x)dx ,
(B.1.7)
sometimes also expressed in the form
dy
dx
dx
dt
dt
, or as
dy = y
(x)x
(t) dt ,
(B.1.8)
when f is a composite function, i.e., when y = f (x) and x can also be
expressed as a function of t as x = g(t).
If we have a function of two variables, z = f (x, y), then the total differential, dz, of (the dependent variable) z is given by the linear combination
dz =
∂z
∂x
y
dx +
∂z
∂y
x
dy ,
(B.1.9)
in terms of the partial derivatives with respect to the independent variables x,
y, and their respective (independent) differentials dx and dy. We have placed
parentheses with subscripts around the partial derivatives, as is commonly
done in the physical sciences, especially thermodynamics (where a plethora
of choices of independent and dependent variables can be made). This
physical science convention often proves to be useful for reminding us which
physical variables are being considered to be independent.
(iii) In mathematics, the functional relation f (x, y, z) = 0 represents an implicit
function of x, y, z. Sometimes we are able to isolate one of the variables in
terms of the others: for example, if we could isolate z from f (x, y, z) = 0 in
the form z = F (x, y), then because z can be expressed as an explicit function
of x and y, we may identify z as the dependent variable, in which case, x and
y are the independent variables. We shall restrict ourselves in the following to
functions of three variables: note, however, that even when we cannot express