The application of the integral considered here is the concept of the length of a curve.
The functions xi
Now consider what happens when the partition is refined by including more points. You can see from the following picture that the polygonal approximation would appear to be even better and that as more points are added in the partition, the sum of the lengths of the line segments seems to get close to something which deserves to be defined as the length of the curve C.
Thus the length of the curve is approximated by
Since the functions in the parametrization are differentiable, it is reasonable to expect this to be close to
which is seen to be a Riemannn sum for the integral ∫ ab
Definition 16.1.2 Let p
Would the same length be obtained if another parametrization were used? This is a very important question because the length of the curve should depend only on the curve itself and not on the method used to trace out the curve. The answer to this question is that the length of the curve does not depend on parametrization. The proof is somewhat technical so is given in the last section of this chapter.
Does the definition of length given above correspond to the usual definition of length in the case when the curve is a line segment? It is easy to see that it does so by considering two points in ℝn p and q. A parametrization for the line segment joining these two points is
Using the definition of length of a smooth curve just given, the length according to this definition is
Thus this new definition which is valid for smooth curves which may not be straight line segments gives the usual length for straight line segments.
The proof that curve length is well defined for a smooth curve contains a result which deserves to be stated as a corollary. It is proved in Lemma 16.2.6 on Page 1004 but the proof is mathematically fairly difficult so it is presented later.
Definition 16.1.4 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. The symbol ∼ is for the word “similar”.
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.
Proof: Formula 16.6 is obvious because f−1 ∘ f
The symbol ∼ is called an equivalence relation. If C is such a smooth curve just described, and if f :
Sometimes people wonder why it is required, in the definition of a smooth curve that p′
In this case, t = x1∕3 and so y = x2∕3. Thus the graph of this curve looks like the picture below. Note the pointy place. Such a curve should not be considered smooth.
So what is the thing to remember from all this? First, there are certain conditions which must be satisfied for a curve to be smooth. These are listed above. Next, if you have any curve, there are two directions you can move over this curve, each called an orientation. This is illustrated in the following picture.
Either you move from p to q or you move from q to p.
Definition 16.1.7 A curve C is piecewise smooth if there exist points on this curve p0,p1,
Note that Example 16.1.6 is an example of a piecewise smooth curve although it is not smooth.