Graphics Reference
In-Depth Information
4.9
Differential Forms
The object of this section, the next section, and Section 8.12 is to try and make some
sense out of differential symbols, such as “dx” and “dxdy”, and to formalize the rules
that the algebra of these differentials satisfies.
Most likely, the reader encountered such symbols very early in his or her calcu-
lus course. A common place is in integrals like
b
Ú
()
fxdx
.
a
What really is the meaning of “dx”? Of course, in the integral above it probably was
just pure notation. One could equally well just have written
b
Ú
f
.
a
The typical reason for including the “dx” is to enable one to apply the chain rule more
easily by reducing it to a formal symbol manipulating process: Given x = g(u) one sub-
stitutes g(u) for x and g¢(u)du for dx.
In two dimensions, things get more complicated. One often writes double inte-
grals as
Ú
X
(
)
f x y dxdy
,
.
But just like the one-dimensional integrals, orientation plays a role and
ÚÚ
(
)
ÚÚ
(
)
f x y dydx
,
=-
f x y dxdy
,
.
X
X
In terms of manipulating differentials, one can express this algebraically by saying
that dydx =-dxdy.
Another place where one encounters the “dx” notation is in expressions such as
∂
∂
f
x
dx
∂
∂
f
y
dy
df
=
+
for functions f(x,y). This again has its uses, like the classical notation dy/dx for the
derivative that is handy for changes of variables, but there come times, as the author
himself found out on occasions as a student, when things get more complicated and
such simple-minded notation involving differentials can get confusing. One ends up
making some transformations that one has seen someone else make and that may be
correct if one is lucky, but which one does not really understand. It therefore is useful
to make sense of symbols such as “dx” in a rigorous way. Doing this will also help
prevent mistakes in imprecise manipulations of these symbols. The “casual user” of
differentials will gain nothing from this section and the next and probably should skip
them. At the end we shall simply have justified the classic notation and manipulation
