CHAPTER 1. 1 — FORMS §1. Differentials: Basic Rules Differentials ...

The above example tells is that the winding number of the loop going around the unit
circles n times is exactly n. In §3.1 we will see that W(γ, 0) is always an integer.
Forming pullbacks is a nice and easy operation. For example, we have
Rule PB1. (g ◦ f) ∗ ω = f ∗ (g ∗ ω).
Here, g ◦ f is the composite of g and f: if f sends x = (x1, . . . , xn) to y = (y1, . . . , ym)
and g sends y to z = (z1, . . . , zℓ), then g ◦ f sends x to z: (g ◦ f)(x) = z = g(y) = g(f(x)).
For a differential form ω in the z-space, g ∗ ω is a differential form in y-space with every
occurrence of zj (1 ≤ j ≤ ℓ) in ω replaced by zj = gj(y1, . . . , yn). Similarly, the pull
back f ∗ (g ∗ ω) of g ∗ ω is a differential form in x-space obtained by another substitution.
Certainly the result of consecutive substitutions by g followed by f is the same as the
single substitution by their composite g ◦ f. So Rule PB1 is clear. From this rule we can
deduce the identity
Indeed,
g◦γ
ω =
γ
g ∗ ω.
g◦γ ω = b
a (g ◦ γ)∗ ω = b
a γ∗ (g ∗ ω) =
γ g∗ ω.
A differential form ω is said to be exact if it is the differential of some function f,
that is ω = df. In that case, the pull back of ω = df = n k=1 (∂f/∂xk)dxk is given by
γ ∗ ω = n
k=1
∂f
∂xk
dxk(t) = d(f(x(t))) ≡
x(t)
d
f(x(t)) dt
dt
(1.7)
(the subscript x(t) indicates the point at which ∂f/∂xk is evaluated) and hence we have
γ ω = b
a γ∗ω = b d
a dtf(x(t)) dt = f(x(b)) − f(x(a)). We conclude:
γ
df = f(the terminal point of γ) − f(the initial point of γ). (1.8)
In particular, if γ is a loop, that is, x(b) = x(a), then
ω = 0. Thus the line integral γ γ ω
depends only on the end points but not on the path γ linking them. This may be called
the path-independence property of line integrals for exact forms. The differential form
xdy + ydz + zdx in Example 1.6 does not have this property, because its integrals along
two paths α and β with same end points are different. Notice that (1.7) actually shows
that γ ∗ df = dγ ∗ f, where γ ∗ f, as a function of t is f(x(t)). This is not surprising because
forming pull backs (or substitution) and taking differentials are independent actions like
“kicking and punching”. In general, we have
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