Men of Mathematics (21 page)

The simplest instances of rates in physics are velocity and acceleration, two of the fundamental notions of dynamics. Velocity is rate of change of
distance
(or “position,” or “space”) with respect to
time; acceleration
is rate of change of
velocity
with respect to
time.

If
s
denotes the distance traversed in the time
t
by a moving particle (it being assumed that the distance is a function of the time), the velocity
at the time
t
is
Denoting this velocity by
v,
we have the corresponding acceleration,

This introduces the idea of a
rate of a rate,
or of a
second derivative.
For in accelerated motion the velocity is not constant but variable, and hence it has a rate of change: the acceleration is the rate of change of the rate of change of distance (both rates with respect to time); and to indicate this
second
rate, or “rate of a rate,” we write
for the acceleration. This itself may have a rate of change with respect to the time; this
third
rate is written
And so on for fourth, fifth, . . . rates, namely for fourth, fifth, . . . derivatives. The most important derivatives in the applications of the calculus to science are the first and second.

*  *  *

If now we look back at what was said concerning Newton's second law of motion and compare it with the like for acceleration, we see that “forces” are proportional to the accelerations they produce. With this much we can “set up” the
differential equation
for a problem which is by no means trivial—that of “central forces”: a particle is attracted toward a fixed point by a force whose direction always passes through the fixed point. Given that the force varies as some function of the distance
s,
say as
F(s),
where
s
is the distance of the particle at the time
t
from the fixed point
O
,

it is required to describe the motion of the particle. A little consideration will show that

the minus sign being taken because the attraction diminishes the velocity. This is the
differential equation
of the problem, so called because it involves a rate (the acceleration), and rates (or derivatives) are the object of investigation in the
differential
calculus.

Having translated the problem into a differential equation we are now required to solve this equation, that is, to find the relation between
s
and
t,
or, in mathematical language, to solve the differential equation by expressing
s
as a function of
t.
This is where the difficulties begin. It may be quite easy to translate a given physical situation into a set of differential equations which no mathematician can solve. In general every essentially new problem in physics leads to types of differential equations which demand the creation of new branches of mathematics for their solution. The particular equation above can however be solved quite simply in terms of elementary functions if
as in Newton's law of gravitational attraction. Instead of bothering with this particular equation, we shall consider a much simpler one which will suffice to bring out the point of importance:

We are given that
y
is a function of
x
whose derivative is equal to
x\
it is required to express
y
as a function of
x.
More generally, consider in the same way

This asks, what is the function
y
(of
x)
whose derivative (rate of change) with respect to
x
is equal
to f(x)?
Provided we can find the function required (or provided such a function exists), we call it the
anti-derivative of f(x)
and denote it by
∫ f
(
x
)
dx
—for a reason that will appear presently. For the moment we need note only that
∫f
(
x
)
dx
symbolizes a function (if it exists)
whose derivative
is equal to
f
(
x
).

By inspection we see that the first of the above equations has the solution ½
x
²
+
c
, where
c
is a constant (number not depending on the variable
x)
; thus
∫x dx
= ½
x
²
+
c.

Even this simple example may indicate that the problem of evaluating
∫f(x)dx
for comparatively innocent-looking functions
f(x)
may be beyond our powers. It does not follow that an “answer” exists at all
in terms of known functions
when an
f(x)
is chosen at random—the odds against such a chance are an infinity of the worst sort (“non-denumerable”) to one. When a physical problem leads to one of these nightmares approximate methods are applied which give the result within the desired accuracy.

With the two basic notions,
and
∫f
(
x
)
dx,
of the calculus we can
now describe
the fundamental theorem of the calculus
connecting them. For simplicity we shall use a diagram, although this is not necessary and is undesirable in an exact account.