Multivariate Calculus - Bruce E. Shapiro

Lecture 22
Line Integrals
Suppose that we are traveling along a path from a point P towards a second point
Q, and that are position r(t) on this trajectory is described parametrically in terms
of a parameter t. We can associate two “directions” of travel along this path: one
from P to Q, and the second back in the other direction from Q to P. We can
visualize these directions by drawing arrows along the path.
Definition 22.1 Let C be a curve connecting two points P and Q that is parameterized
as
r(t), a ≤ t ≤ b
where r(a) = P and r(b) = Q.Then the curve is said to oriented if we associate a
direction of travel with it, and the path is said to be directed.
Furthermore, the path is said to be positively oriented if the direction of motion
corresponds to increasing t, and negatively oriented otherwise.
Figure 22.1: Examples of oriented curves, where a ≤ t ≤ b, P = r(a), and Q = r(b).
The curve on the left is positively oriented, because the motion follows the direction
of increasing t. The curve on the right is oriented but it is not positively oriented.
The arrows on the oriented curve really represent the direction of the tangent vectors;
we can approximate them by dividing the parameterization up into finite intervals
a = t 0 < t 1 < t 2 < · · · < t n = b (22.1)
175