12.0.1 Arc Length And Orientations
The application of the integral considered here is the concept of the length of a
Definition 12.0.1 C is a smooth curve in ℝn if there exists an interval
⊆ ℝ and functions xi
→ ℝ such that the following conditions hold
- xi is continuous on .
- xi′ exists and is continuous and bounded on , with
xi′ defined as the derivative
from the right,
and xi′ defined similarly as the derivative from the left.
- For p
t → p is one to one on
1∕2≠0 for all t ∈.
- C = ∪.
With this definition of a smooth curve, it follows from the earlier material on the integral
and measure of a manifold that the one dimensional area (length) of this curve is given by
and that this definition is an intrinsic property of the curve itself being
independent of the parameterization.
Next it is necessary to consider the orientation of a curve.
Corollary 12.0.2 Let C be a smooth curve and let f :
→ C and g :
be two parameterizations satisfying (1) - (5). Then g−1 ∘ f is either strictly increasing
or strictly decreasing.
Proof: The function g−1 ∘ f is continuous and maps
. It is continuous
must be continuous. This follows because g
is one to one and continuous. Its
continuity requires it to map compact sets to compact sets. In other words, closed sets
to closed sets. Hence
= closed set. Let
g−1 ∘ f
f−1 ∘ g
Thus, by the same reasoning ϕ−1
is also continuous.
First it is shown that ϕ is either strictly increasing or strictly decreasing on
If ϕ is not strictly decreasing on
, then there exists
x1 < y1
, x1,y1 ∈
If for some other pair of points x2 < y2 with x2,y2 ∈
, the above inequality does not
hold, then since
is 1 −
Let xt ≡ tx1 +
and yt ≡ ty1
. It follows that xt < yt
for all t ∈
0 but by assumption, h
0 for any t ∈
This property of being either strictly increasing or strictly decreasing on
by the continuity of
. Therefore, ϕ−1
is also continuous by similar reasoning to the
In terms of the earlier material on orientable manifolds, a smooth curve is always
orientable. What the parameterization does is give a single chart in an atlas and obviously
= 1 which has positive determinant. To say that
gives the same orientation as
is to say that g−1 ∘ f
has positive derivative. This is the message of the next
Definition 12.0.3 If g−1 ∘ f is increasing, then f and g are said to be
equivalent parameterizations and this is written as f ∼ g. It is also said that the two
parameterizations give the same orientation for the curve when f ∼ g.
When the parameterizations are equivalent, they preserve the direction of motion along
the curve, and this also shows there are exactly two orientations of the curve since either
g−1 ∘ f is increasing or it is decreasing. This is not hard to believe. In simple language, the
message is that there are exactly two directions of motion along a curve. The difficulty is in
proving this is actually the case.
Lemma 12.0.4 The following hold for ∼.
Proof: Formula (12.1) is obvious because f−1 ∘ f
so it is clearly an increasing
function. If f ∼ g
then f−1 ∘g
is increasing. Now g−1 ∘f
must also be increasing because it is
the inverse of f−1 ∘ g.
This verifies (12.2
). To see (12.3
), f−1 ∘ h
so since both of these functions are increasing, it follows
f−1 ∘ h
is also increasing.
The symbol ∼ is called an equivalence relation. If C is such a smooth curve just described,
and if f :
is a parametrization of C
, consider g
, also a
. Now by Corollary 12.0.2
, if h
is a parametrization, then if f−1 ∘h
increasing, it must be the case that g−1 ∘h
is increasing. Consequently, either h ∼ g
or h ∼ f.
These parameterizations, h,
which satisfy h ∼ f
are called the equivalence class determined by
and those h ∼ g
are called the equivalence class determined by g.
classes are called orientations
. They give the direction of motion on C
see that going from f
corresponds to tracing out the curve in the opposite
Let C be a smooth curve contained in ℝp. A curve C is an “oriented curve” if the only
parameterizations considered are those which lie in exactly one of the two equivalence classes,
each of which is called an “orientation”. In simple language, orientation specifies a direction
over which motion along the curve is to take place. Thus, it specifies the order in which the
points of C are encountered. The pair of concepts consisting of the set of points making up
the curve along with a direction of motion along the curve is called an oriented curve.
Definition 12.0.5 Suppose F
∈ ℝp is given for each x ∈ C where C
is a smooth oriented curve and suppose x → F
is continuous. The mapping
x → F
is called a vector field. The line integral on an oriented curve C is given
where the function x is one of the allowed parameterizations of C in the given orientation of
C. In other words, there is an interval
and as t goes from a to b, x
moves in the
direction determined from the given orientation of the curve.
Theorem 12.0.6 The symbol ∫
CF⋅dR, is well defined in the sense that every
parametrization in the given orientation of C gives the same value for ∫
CF ⋅ dR.
Proof: Suppose g :
is another allowed parametrization. Thus g−1 ∘ f
increasing function ϕ
. Then since ϕ
is increasing, it follows from the change of variables
One can also string together smooth oriented curves in which the last point of one
coincides with the first point of the next and obtain a piecewise smooth curve. The
line integral over one of these is just the sum of the line integrals over the curves
which are joined together. This kind of fussiness is unpleasant but it is the usual
way of dealing with more general curves than smooth curves. It is much nicer to
consider curves of bounded variation from the beginning but this is not being done