The application of the integral considered here is the concept of the length of acurve.
Definition 12.0.1C is a smooth curvein ℝ^{n}if there exists an interval
[a,b]
⊆ ℝ and functions x_{i} :
[a,b]
→ ℝ such that the following conditions hold
x_{i} is continuous on
[a,b]
.
x_{i}^{′} exists and is continuous and bounded on
[a,b]
, with x_{i}^{′}
(a)
defined as the derivative
from the right,
lim xi(a+-h)-− xi(a),
h→0+ h
and x_{i}^{′}
(b)
defined similarly as the derivative from the left.
For p
(t)
≡
(x (t),⋅⋅⋅,x (t))
1 n
, t → p
(t)
is one to one on
(a,b)
.
|p′(t)|
≡
( ∑ )
ni=1 |x′i(t)|2
^{1∕2}≠0 for all t ∈
[a,b]
.
C = ∪
{(x1 (t),⋅⋅⋅,xn (t)) : t ∈ [a,b]}
.
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
∫_{a}^{b}
|p ′(t)|
dt 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 12.0.2Let C be a smooth curve and let f :
[a,b]
→ C and g :
[c,d]
→ Cbe two parameterizations satisfying (1) - (5). Then g^{−1}∘ f is either strictly increasingor strictly decreasing.
Proof:The function g^{−1}∘ f is continuous and maps
[a,b]
to
[c,d]
. It is continuous
because g^{−1} 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
( −1)
g
^{−1}
(closed set)
= closed set. Let ϕ
(x)
= g^{−1}∘ f
(x)
,
ϕ^{−1}
(y)
= f^{−1}∘ g
(y)
. Thus, by the same reasoning ϕ^{−1} is also continuous.
First it is shown that ϕ is either strictly increasing or strictly decreasing on
(a,b)
.
If ϕ is not strictly decreasing on
(a,b)
, then there exists x_{1}< y_{1}, x_{1},y_{1}∈
(a,b)
such
that
(ϕ(y1)− ϕ (x1))(y1 − x1) > 0.
If for some other pair of points x_{2}< y_{2} with x_{2},y_{2}∈
(a,b)
, the above inequality does not
hold, then since ϕ is 1 − 1,
(ϕ(y2)− ϕ (x2))(y2 − x2) < 0.
Let x_{t}≡ tx_{1} +
(1− t)
x_{2} and y_{t}≡ ty_{1} +
(1 − t)
y_{2}. It follows that x_{t}< y_{t} for all t ∈
[0,1]
.
Now define
h (t) ≡ (ϕ(yt)− ϕ(xt))(yt − xt).
Then h
(0)
< 0,h
(1)
> 0 but by assumption, h
(t)
≠0 for any t ∈
(0,1)
, a contradiction.
This property of being either strictly increasing or strictly decreasing on
(a,b)
carries over
to
[a,b]
by the continuity of ϕ. Therefore, ϕ^{−1} is also continuous by similar reasoning to the
above. ■
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
D
(f−1 ∘ f)
= 1 which has positive determinant. To say that g gives the same orientation as
f is to say that g^{−1}∘ f has positive derivative. This is the message of the next
definition.
Definition 12.0.3If g^{−1}∘ f is increasing, then f and g are said to beequivalent parameterizations and this is written asf ∼ g.It is also said that the twoparameterizations give the same orientation for the curve whenf ∼ 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.
= t 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 =
( )
f−1 ∘g
∘
( )
g−1 ∘h
and
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 :
[a,b]
→ C is a parametrization of C, consider g
(t)
≡ f
((a+ b)− t)
, also a
parametrization of C. Now by Corollary 12.0.2, if h is a parametrization, then if f^{−1}∘h is not
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
f and those h ∼ g are called the equivalence class determined by g. These two
classes are called orientations of C. They give the direction of motion on C. You
see that going from f to g corresponds to tracing out the curve in the opposite
direction.
Let C be a smooth curve contained in ℝ^{p}. 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 12.0.5Suppose F
(x)
∈ ℝ^{p}is given for each x ∈ C where Cis a smooth oriented curve and supposex → F
(x )
is continuous. The mappingx → F
(x)
is called avector field. The line integral on an oriented curve C is givenby
∫ ∫
b ′
C F ⋅dR ≡ a F (x(t))⋅x (t)dt
where the function x is one of the allowed parameterizations of C in the given orientation ofC. In other words, there is an interval
[a,b]
and as t goes from a to b, x
(t)
moves in thedirection determined from the given orientation of the curve.
Theorem 12.0.6The symbol∫_{C}F⋅dR,is well defined in the sense that everyparametrization in the given orientation of C gives the same value for∫_{C}F ⋅ dR.
Proof:Suppose g :
[c,d]
→ C is another allowed parametrization. Thus g^{−1}∘ f is an
increasing function ϕ. Then since ϕ is increasing, it follows from the change of variables
formula that
∫ ∫
d ′ b ′ ′
c F (g (s))⋅g (s)ds = a F (g(ϕ(t))) ⋅g (ϕ (t))ϕ (t)dt
∫ ∫
b d-( ( −1 )) b ′
= a F(f (t)) ⋅dt g g ∘f (t) dt = a F(f (t))⋅f(t)dt.■
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
here.