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Theorem 5. If f x is continuous at x x0 and f x0 > 0 [or f x0 < 0], there exists an interval about x x0 in which f x > 0 [or f x < 0]. Theorem 6. If a function f x is continuous in an interval and either strictly increasing or strictly decreasing, the inverse function f 1 x is single-valued, continuous, and either strictly increasing or strictly decreasing.

A lightweight polyline is a 2-D line consisting of straight and arced segments. It s functionally similar to the legacy Polyline entity, but it s internally represented in a flat data structure as opposed to the tree-structured Polyline entity. This results in a more compact data size per entity and faster graphics regeneration; hence the lightweight name. You create a new LWPolyline object using the AddLightWeightPolyline method, as follows: Set LWPolylineObject = Object.AddLightWeightPolyline(Vertices)

2.1 THE COORDINATES OF A POINT We shall establish a correspondence between the points of a plane and pairs of real numbers. Choose two perpendicular lines in the plane of Fig. 2-1. Let us assume for the sake of simplicity that one of the lines is horizontal and the other vertical. The horizontal line will be called the x-axis and the vertical line will be called the y-axis.

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Thus, the value of the integral is obtained without reference to the curve joining P1 and P2 . This notion of the independence of path of line integrals of certain vector elds, important to theory and application, is characterized by the following three theorems: Theorem 1. A necessary and su cient condition that A dr be independent of path is that there C exists a scalar function such that A r .

Fig. 2-1 Next choose a coordinate system on the x-axis and one on the y-axis. The origin for both coordinate systems is taken to be the point O, where the axes intersect. The x-axis is directed from left to right, the y-axis from bottom to top. The part of the x-axis with positive coordinates is called the positive x-axis, and the part of the y-axis with positive coordinates the positive y-axis. Consider any point P in the plane. Take the vertical line through the point P, and let a be the coordinate of the point where the line intersects the x-axis. This number a is called the x-coordinate of P (or the abscissa of P). Now take the horizontal line through P, and let b be the coordinate of the point where the line intersects the y-axis.

CHAP. 10]

CHAP. 2]

Theorem 2. A necessary and su cient condition that the line integral, is that r A 0. A dr be independent of path

Table 8-6 provides a brief description of the Vertices argument. Table 8-6. The AddLightWeightPolyline Method Parameter

The number b is called the y-coordinate of P (or the ordinate of P). Every point has a unique pair (a, b) of coordinates associated with it. EXAMPLES In Fig. 2-2, the coordinates of several points have been indicated. We have limited ourselves to integer coordinates only for simplicity.

Theorem 3. If r A 0, then the line integral of A over an allowable closed path is 0, i.e., A dr 0.

Fig. 2-2 Conversely, every pair (a, b) of real numbers is associated with a unique point in the plane. EXAMPLES In the coordinate system of Fig. 2-3, to nd the point having coordinates (3, 2), start at the origin O, move three units to the right and then two units upward. To nd the point with coordinates ( 2, 4), start at the origin O, move two units to the left and then four units upward. To nd the point with coordinates ( 1, 3), start from the origin, move one unit to the left and then three units downward. Given a coordinate system, the entire plane except for the points on the coordinate axes can be divided into four equal parts, called quadrants. All points with both coordinates positive form the rst quadrant, quadrant I, in the upper right-hand corner (see Fig. 2-4). Quadrant II consists of all points with negative x-coordinate and positive y-coordinate; quadrants III and IV are also shown in Fig. 2-4.

If C is a plane curve, then Theorem 3 follows immediately from Green s theorem, since in the plane case r A reduces to @A1 @A2 @y @x

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