Read Men of Mathematics Online
Authors: E.T. Bell
Consider a continuous, unlooped curve whose equation is y = f(x) in Cartesian coordinates. It is required to find the area included between the curve, the x-axis and the two perpendiculars
AAâ² BBâ²
drawn to the x-axis from any two points
A, B
on the curve. The distances
OAâ² OBâ²
are
a, b
respectivelyânamely, the coordinates of
Aâ², Bâ²
are (a, 0), (b, 0). We proceed as Archimedes did, cutting the required area into parallel strips of equal breadth, treating these strips as rectangles by disregarding the top triangular bits (one of which is shaded in the figure), adding the areas of all these rectangles, and finally evaluating
the limit of this sum
as the number of rectangles is increased indefinitely. This is all very well, but how are we to calculate the limit? The answer is surely one of the most astonishing things a mathematician ever discovered.
First, find
â«f
(
x
)
dx.
Say the result is
F
(
x
). In this substitute
a
and 6, getting
F
(
a
) and
F
(
b
). Then subtract the first from the second,
F(b)âF(a). This is the required area.
Notice the connection between
y
=
f
(
x
), the equation of the given curve;
which (as seen in the chapter on Fermat) gives the
slope
of the tangent line to the curve at the point (
x, y
); and
â«f
(
x
)
dx
, or
F
(
x
), which is the function whose
rate of change
with respect to x is equal to
f
(
x
). We have just stated that the
area
required, which is a
limiting sum
of the kind described in connection with Archimedes, is given
by
F
(
b
)â
F
(
a
). Thus we have connected
slopes,
or
derivatives,
with
limiting sums,
or, as they are called,
definite integrals.
The symbol
J
is an old-fashioned
S,
the first letter of the word
Summa.
Summing all this up in symbols, we write for the area in question
a
is the
lower limit
of the sum,
b
the
upper limit;
and in which
F
(
b
),
F
(
a
) are calculated by evaluating the
“indefinite integral” fâ«
(
x
)
dx,
namely, by finding that function
F
(
x
) such that its derivative with respect to
x,
is equal to
f
(
x
). This is the fundamental theorem of the calculus as it presented itself (in its geometrical form) to Newton and independently also to Leibniz. As a caution we repeat that numerous refinements demanded in a modern statement have been ignored.
*Â Â *Â Â *
Two simple but important matters may conclude this sketch of the leading notions of the calculus as they appeared to the pioneers. So far only functions of a single variable have been considered. But nature presents us with functions of several variables and even of an infinity of variables.
To take a very simple example, the volume,
V,
of a gas is a function of its temperature,
T,
and the pressure,
P,
on it; say
V
=
F
(
T, P
)âthe actual form of the function
F
need not be specified here. As
T, P
vary,
V
varies. But suppose
only one
of
T, P
varies while the other is held constant. We are then back essentially with a function of
one
variable, and the derivative of
F(T, P)
can be calculated with respect to this variable. If
T
varies while
P
is held constant, the derivative of
F(T, P)
with respect to
T
is called the
partial derivative
(with respect to
T),
and to show that the variable
P
is being held constant, a different symbol,
d,
is used for this partial derivative,
Similarly, if
P
varies while
T
is held constant, we get
Precisely as in the case of ordinary second, third, . . . derivatives, we have the like for partial derivatives; thus
signifies the partial derivative of
with respect to
T.
The great majority of the important equations of mathematical
physics are
partial differential equations.
A famous example is Laplace's equation, or the “equation of continuity,” which appears in the theory of Newtonian gravitation, electricity and magnetism, fluid motion, and elsewhere: