CHAPTER 12
### Chapter 12 - Differentials

We have at last arrived at the final chapter of this
book. The goal of this chapter is to extend the systematic approach of
integration to include other types of mathematical functions involving relative
change of variables. The process of identifying which dimension is changing
with respect to another one is the first step in the systematic approach. The
second step involves defining an instantaneous change for the function by
writing what is called a differential. Differentials are of the form:

Dividing both sides by the
infinitely small, *dx*, defines the
derivative or instantaneous rate of change of
with respect to *x*. What is important to understand is
that
is assumed to be
constant over the interval, *dx*. Thus
the infinite sum of these *dx. s* is
given by integrating the function. The net change in the function, *F(x)* , over an interval form *x = a t*o* x = b* is thus:

Let us now look at various
combinations of *f(x)* and* x*, and see how we differentials are
defined for each case. Consider **case 1**,
where some situation, *F* is a function
of two independent conditions* f* and *x,*

If f is a function of x, *f(x)*, then its value changes as *x* changes. The **differential** is defined by converting the independent *x* into an infinitely small *dx*.

The differential is then integrated
to determine how much the situation, *F*, changes as x changes from *x = a *to *x = b*.

Consider **case 2** where F is a function of two independent conditions, *f* and

In this case,
represents a fixed or
constant condition. We can replace *A*
with
to get:

If f is given as a function of x,
then we will first have to re-write *f(x)*
as a function of A. We know that:

If
, then
. Once we have
, we can write our differential;

We can then integrate this function
from some value of
. Keep in mind that,

This approach to writing and
integrating the differential for case 2 functions is confusing and abstract. An
easier and more logical approach is to integrate with respect to *x* only: For example if,

First let
, then
or:

Substituting
this back into the function yields:

If *f *is a constant then from *x =a t*o* x = b*, the change in *F*
is:

The result is the same function we
started with. If *f* were a function of
*x*, then its value would change as *x* goes from a to b. The **differential** can be written as:

Integrating this
function over an interval from x = a to x = b results in

This results in the same numerical
answer as the previously defined method of writing the differential.

To summarize, if
; and *f* is a
function of *x*, then:

Here
, where

The **third case** of writing a differential is for functions of the form:

Once again,
is a constant factor
that defines F. We can let
, then:

Substituting
this back into case three:

Our results can be generalized as
follows. If *F* is a function of two
variables, *f *and *x* such as:

If *A* is some function of *x*,
and *f* is also a function of *x* then both these values change as x
changes. It is important to understand that
is a constant value,
dependent on *x*, that defines *F.* Therefore, the differential is
defined as:

Since *A* is a function of x, then:

Substituting this back into the
differential gives us the general expression:

We can integrate this over any
interval of
to find the net
change in *F *over the interval. Our
results can be generalized for any combination of
and
, where
and
can represent **any** function of *x*. The differential and integral is: