112 Functions, limits, derivativesThis equation is read "z equals f of x and y and 0." It happens that the
law of cosines, which involves one of the more important formulas which
should be learned in trigonometry, gives the formula
(3.13) f(x,y,B) _ /x2 + y2 - 2xy cos 0
from which we can compute f(x,y,B) when x, y, B are given numbers. In
spite of the fact that numbers do not move, it is sometimes a convenience
to think of x, y, 0, z as being "variables" and to think of z as being the
"dependent variable" which is a function of the three "independent
variables" x, y, 0.
Many examples are more complicated than this, and we can broaden
our intellectual horizons by thinking briefly about one of them. It is
standard practice to write(3.14) v = f(x,y,z,t)
= fj(x,y,z,t)i + f2(x,y,z,t)j + f3(x,y,z,t)k,where v, a vector, is the velocity of a fluid (which might be air) at the place
having rectangular coordinates x, y, z and at time t. We say that v and
its scalar components are functions of the four variables x, y, z, t. We
mean that when x, y, z are coordinates of a point in the region being con-
sidered and when t is a time (measured in specified units from a specified
zero hour) in the time interval being considered, the velocity v and its
scalar components at that place and time are completely determined and
that f(x,y,z,t) denotes the velocity and fj(x,y,z,t), f2(x,y,z,t), f3(x)y,z,t)
denote the scalar components.
There are two useful and more or less modern ways of attaching mean-
ings to the symbols f and f, appearing in the above example. One is the
dynamic approach and the other is the static approach. In the dynamic
approach, f and fl are regarded as operators or transformers (like machines)
to which we can feed appropriate ordered sets x, y, z, t of numbers. Then
(after mechanical squeaking or electronic flashing or what not) f and f,
produce the required vector f(x,y,z,t) and the required number fi(x,y,z,t).
In the static approach, f is regarded as being the set of ordered quintuples
(x, y, z, t, f(x,y,z,t)) of four numbers and a vector in which the allowable
independent variables come first in the appropriate order and the vector
f(x,y,z,t) comes last. In this static approach, fl is a set of quintuples of
numbers. It is a common but not universal practice to consider these
ideas to be more tangible and useful than the idea that f is a law or rule by
means of which f(x,y,z,t) can be calculated when x, y, z, t are given. A
simpler example may partially clarify these matters. As soon as we know
that the area y of a circular disk is determined by its radius x (x being