Calculus: Analytic Geometry and Calculus, with Vectors

3.9 Increments and differentials 201'

better. The specific heat o (sigma) of a substance at temperaturex is Q'(x),

Where Q(x) is the number of calories of heat required to raise (or lower) the temperature

of 1 gram of the stuff from 0°C to x°C. For study of this matter, let v* be the

specific heat calculated from the first definition so that

(1) a* =Q(x +1) - Q(x), or = Q'(x)= limQ(x -I- h) - Q(x)

h--.o h

We can see one of the reasons why knowledge of calculus is needed for study of

physical chemistry when we see that v* is a difference quotient and v is a deriva-

tive. Since the graph of Q is never (or not ordinarily) a line, v* and o. are usually

different. The schematic Figure 3.992 illustrates one situation. As is easily

Q(x+1)

v c*

---Q(x)

Figure 3.992

imagined, there are situations in which the graph of Q is "almost straight" over
the interval from x to x + 1 and v is a "good approximation" to or. On the
other hand, v can be dust and ashes when the interval from x to x + 1 straddles
a temperature at which a substance changes from a solid state to a liquid or from
a liquid to a gas.
17 There are reasons why we should conclude with a historical remark. In
the good old days when the "doctrine of limits" was based upon visions of gal-
loping numbers and the "infinitely small infinitesimals" were considered to be
almost the most wonderful products of human thought, differentials were con-
sidered to be the most wonderful. Differentials were the important things, and
the things that we now call derivatives were merely the "differential coefficients"
appearing in formulas like dy = f'(x) dx or dy = 2x dx. Thus differentials have
their origin in old mathematics; it was the fashion to consider them to be "infi-
nitely small" but not quite zero. When at long last the concept of the "infi-
nitely small" was becoming obsolete, attempts were made to salvage differentials
by promoting the idea that they really are not ordinary numbers at all but are
numbers that are in the process of approaching zero.t So far as this course is
concerned, the details of this remark are unimportant. We should, however,
know that differentials have a long and checkered history and that we may
expect to encounter some quite strange concepts as we get around in the world.

t For those who have not peered into old books and consider this to be too incredible to be
true, we quote three passages from W. E. Byerly, "Elements of the Differential Calculus,"
Ginn and Heath, Boston, 1879. Page 149 tells us that ".,1n infinitesimal or infinitely small
quantity is a variable which is supposed to decrease indefinitely; in other words, it is a variable
which approaches the limit zero." Page 185 tells us that "It is to be noted that a differential
is an infinitesimal, and that it differs from an infinitesimal increment by an infinitesimal of a
higher order." Page 186 tells us that "there is a practical advantage ... in regarding the
differential as the main thing, and looking at the derivative as the quotient of two differentials."