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Simple Closed Curve
A simple closed curve is a piecewise smooth curve whose initial and terminal points are equal and that does not cross or retrace its path. Examples of simple closed curves are the perimeters of a circle, a triangle, and a rectangle. The value of a line integral around a simple closed curve C depends on whether the length s is measured clockwise or counterclockwise, but does not depend on the initial point (Figure 13.2.11). The clockwise and counterclockwise line integrals of F around a simple closed curve C are denoted by
Figure 13.2.11 Integrals around Simple Closed Curves THEOREM 2 If C is a simple closed curve, then
and the values do not depend on the initial point of C. PROOF The equation in Theorem 2 holds because reversing the direction of the curve changes the sign of the line integral. Suppose C has the initial point A, and its direction is clockwise. Let A1 be any other point on C, and let Ct and C2 be as in Figure 13.2.12.
Figure 13.2.12 With the initial point A,
With the initial point A1,
These are equal as required.
Line integrals in space are developed in a similar way. Instead of an open rectangle we work in an open rectangular solid. A smooth curve C in space has three parametric equations with continuous derivatives, x = g(s), y = h(s), z = l(s), 0 ≤ s ≤ L. Given a continuous vector valued function F(x, y, z) = P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k and a smooth curve C in space, we define the line integral of F along C, in symbols,
as
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Last Update: 2006-11-20