Calculus Lessons

The natural world is in continuous, never-ending flux. The aim of calculus is to
describe this continuous change mathematically. As modern physicists see it, the
methods of calculus solve two related problems. Given an equation that expresses
a continuous change, what is the equation for the rate of the change? And, conversely,
given the equation for the rate of change, what is the equation for the
change? Newton approached calculus this way, but often with geometrical arguments
that are frustratingly difficult for those with little geometry. I will avoid
Newton’s complicated constructions and present here for future reference a few
rudimentary calculus lessons more in the modern style.
Suppose you want to describe the motion of a ball falling freely from the Tower
in Pisa. Here the continuous change of interest is the trajectory of the ball, expressed
in the equation
gt2
s (1)
2
in which t represents time, s the ball’s distance from the top of the tower, and g
a constant we will interpret later as the gravitational acceleration. One of the
problems of calculus is to begin with equation (1) and calculate the ball’s rate of
fall at every instant.
This calculation is easily expressed in Leibniz symbols. Imagine that the ball
is located a distance s from the top of the tower at time t, and that an instant
later, at time t dt, it is located at s ds; the two intervals dt and ds, called
“differentials” in the terminology of calculus, are comparatively very small. We
have equation (1) for time t at the beginning of the instant. Now write the equation
for time t dt at the end of the instant, with the ball at s ds,s ds
2
g [t2 2tdt (dt)2] (2)
2
gt2 g gtdt (dt)2.
2 2
Notice the term s on the left side of the last equation and the term the right.
gt2
2
According to equation (1), these terms are equal, so they can be canceled from
the last equation, leaving
g ds gtdt (dt)2. (3)
2
In the realm where calculus operates, the time interval dt is very small, and
(dt)2 is much smaller than that. (Squares of small numbers are much smaller
numbers; for example, compare 0.001 with (0.001)2 0.000001.) Thus the term
containing (dt)2 in equation (3) is much smaller than the term containing dt, in
fact, so small it can be neglected, and equation (3) finally reduces to
ds gtdt. (4)
Dividing by the dt factor on both sides of this equation, we have finally
ds
gt. (5)
dt
(As any mathematician will volunteer, this is far from a rigorous account of the
workings of calculus.)
This result has a simple physical meaning. It calculates the instantaneous
speed of the ball at time t. Recall that speed is always calculated by dividing a
distance interval by a time interval. (If, for example, the ball falls 10 meters at
constant speed for 2 seconds, its speed is meters per second.) In equation
10
5
2
(5), the instantaneous distance and time intervals ds and dt are divided to calculate
the instantaneous speed .
ds
dt
The ratio in equation (5) is called a “derivative,” and the equation, like any
ds
dt
other containing a derivative, is called a “differential equation.” In mathematical
physics, differential equations are ubiquitous. Most of the theories mentioned in
this book rely on fundamental differential equations. One of the rules of theoretical
physics is that (with a few exceptions) its laws are most concisely stated
in the common language of differential equations.
The example has taken us from equation (1) for a continuous change to equation
(5) for the rate of the change at any instant. Calculus also supplies the means(4) we write
ds gtdt. (6)
We know that this must be equivalent to equation (1), so we infer that the rules
for evaluating the two “integrals” in equation (6) are
ds s, (7)
and
gt2 gtdt . (8)
2
Integrals and integration are just as fundamental in theoretical physics as differential
equations. Theoreticians usually compose their theories by first writing
differential equations, but those equations are likely to be inadequate for the
essential further task of comparing the predictions of the theory with experimental
and other observations. For that, integrated equations are often a necessity.
The great misfortune is that some otherwise innocent-looking differential equations
are extremely difficult to integrate. In some important cases (including one
Newton struggled with for many years, the integration of the equations of motion
for the combined system comprising Earth, the Moon, and the Sun), the equations
cannot be handled at all without approximations.
A glance at a calculus textbook will reveal the differentiation rule used to
arrive at equation (5), the integration rules (7) and (8), and dozens of others. As
its name implies, calculus is a scheme for calculating, in particular for calculations
involving derivatives and differential equations. The scheme is organized
around the differentiation and integration rules.
Calculus provides a perfect mathematical context for the concepts of mechanics.
In the example, the derivative calculates a speed. Any speed v is calcuds
dt
lated the same way,
ds
v . (9)
dt
If the speed changes with time—if there is an acceleration—that can be expressed
as the rate of change in v, as the derivative . So the acceleration differential
dv
dt
equation isa , (10)
dt
in which a represents acceleration. The freely falling ball accelerates, that is, its
speed increases with time, as equation (5) combined with equation (9), which is
written
v gt, (11)
shows. The constant factor g is the acceleration of free fall, that is, the gravitational
acceleration.
This discussion has used the Leibniz notation throughout. Newton’s calculus
notation was similar but less convenient. He emphasized rates of change with
time, called them “fluxions,” and represented them with an overhead dot notation.
For example, in Newton’s notation, equation (5) becomes
˙s gt,
in which , Newton’s symbol for , is the distance fluxion, and equation (10) is
˙ ds s
dt
a v˙ ,
with representing , the speed fluxion.