Mathematical Physics Vol 1

2.2 Integration

49

Figure 2.4: Curve orientation.

Consider two different points, T 1 and T 2 from the curve l , determined by the parameter values t 1 and t 2 ( t 1 < t 2 ) . If we assume that point T 1 is the preceding and T 2 the following, then an orientation is obtained (Fig. 2.4a). However, the opposite can also be assumed, namely that T 2 is the preceding and T 1 the following ( t 2 < t 1 ), which yields another orientation (Fig. 2.4b). Orientation of closed curve, where r ( t A )= r ( t B ) , which means that points A and B , corresponding to parameter values t A and t B , respectively, overlap, is determined by observing three points from the curve and selecting a curve orientation in one of the two ways shown in figure 2.5. The order ABC defines one orientation, while the order ACB defines another, opposite orientation.

Figure 2.5: Closed curve orientation.

Partition of oriented curve Partition of oriented curve ( l = AB ) division assumes that the points of partition

A = T 0 , T 1 ,..., T n = B

(2.66)

are numerated in order of succession (Fig. 2.6), namely that they correspond to the parameter values t 0 , t 1 ,..., t n , (2.67)

where, for the chosen orientation

t A = t 0 < t 1 < ··· < t n = t B .

(2.68)