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1、2.5 笛卡尔几何学的基本概念（basic concepts of Cartesian geometry）课文 5-A the coordinate system of Cartesian geometryAs mentioned earlier, one of the applications of the integral is the calculation of area. Ordinarily , we do not talk about area by itself ,instead, we talk about the area of something . This means
2、 that we have certain objects (polygonal regions, circular regions, parabolic segments etc.) whose areas we wish to measure. If we hope to arrive at a treatment of area that will enable us to deal with many different kinds of objects, we must first find an effective way to describe these objects.The
3、 most primitive way of doing this is by drawing figures, as was done by the ancient Greeks. A much better way was suggested by Rene Descartes, who introduced the subject of analytic geometry (also known as Cartesian geometry). Descartes idea was to represent geometric points by numbers. The procedur
4、e for points in a plane is this ：Two perpendicular reference lines (called coordinate axes) are chosen, one horizontal (called the “x-axis”), the other vertical (the “y-axis”). Their point of intersection denoted by O, is called the origin. On the x-axis a convenient point is chosen to the right of
5、O and its distance from O is called the unit distance. Vertical distances along the Y-axis are usually measured with the same unit distance ,although sometimes it is convenient to use a different scale on the y-axis. Now each point in the plane (sometimes called the xy-plane) is assigned a pair of n
6、umbers, called its coordinates. These numbers tell us how to locate the points.Figure 2-5-1 illustrates some examples.The point with coordinates (3,2) lies three units to the right of the y-axis and two units above the x-axis.The number 3 is called the x-coordinate of the point,2 its y-coordinate. P
7、oints to the left of the y-axis have a negative x-coordinate; those below the x-axis have a negtive y-coordinate. The x-coordinateof a point is sometimes called its abscissa and the y-coordinateis called its ordinate.When we write a pair of numberssuch as (a,b) to represent a point, we agree that th
8、e abscissa or x-coordinate,a is written first. For this reason, the pair(a,b) is often referred to as an ordered pair. It is clear that two ordered pairs (a,b) and (c,d) represent the same point if and only if we have a=c and b=d. Points (a,b) with both a and b positiveare said to lie in the first q
9、uadrant ,those with a0 are in the second quadrant ; and those with a0 and b0 时位于第二象限，当 a0,b0 时位于第四象限。图 2-5-1 画出了每个象限的一个点。在空间中点的表示方法是相似的。我们取空间中交于一点(原点) 的三条相互垂直的线。这些线决定了三个相互垂直的平面，且空间中的每一个点通过它到三个平面的距离选取合适的记号，都能完全具体的指定出来。我们之后应该更细节的讨论三维笛卡尔几何，目前我们限制于关注平面解析几何。课文 5-B Geometric figuresA geometric figure, suc
10、h as a curve in the plane , is a collection of points satisfying one or more special conditions. By translating these conditions into expressions, involving the coordinates x and y, we obtain one or more equations which characterize the figure in question , for example, consider a circle of radius r
11、 with its center at the origin, as show in figure 2-5-2. let P be an arbitrary point on this circle, and suppose P has coordinates (x, y). Then the line segment OP is the hypotenuse of a right triangle whose legs have lengths |x| and |y| and hence, by the theorem of Pythagoras, . This 22xyrequation,
12、 called a Cartesian equation of the circle , is satisfied by all points (x,y) on the circle and by no others , so the equation completely characterizes the circle. This example illustrates how analytic geometry is used to reduce geometrical statements about points to analytical statements about real
13、 numbers.Throughout their historical development, calculus and analytic geometry have been intimately intertwined. New discoveries in one subject led to improvements in the other. The development of calculus and analytic geometry in this book is similarto the historical development, in that the two
14、subjects are treated together . However our primary purpose is to discuss calculus . Concepts from analytic geometry that are required for this purpose will be discussed as needed . Actually, only a few very elementary concepts of plane analytic geometry are required to understand the rudiments of c
15、alculus . A deeper study of analytic geometry is needed to extend the scope and applications of calculus , and this study will be carried out in later chapters using vector methods as well as the methods of calculus. Until then, all that is required from analytic geometry is a little familiarity wit
16、h drawing graph of function.课文 5-B：几何图形一个几何图形，比如平面上的一条曲线，是满足一个或多个特殊条件点的集合。通过把这些条件转化成含有坐标 x 和 y 的表达式，我们就得到了一个或多个能刻画该图形特征的方程。例如，考虑一个中心在原点半径为 r 的圆，如图 2-5-2.让 P 是这个圆上的任意一点，并且假设 P 的坐标为(x,y)。然后线段 OP 是一个边长为|x| 和|y|的直角三角形的斜边，因此由毕达哥拉斯定理，x2+y2=r2.这个等式叫做圆的笛卡尔等式，仅圆上所有点(x,y)满足它，所以这个等式完全描绘了圆。这个例子说明解析几何如何被用来把点的几何特征归纳为真实数据的解析特征。微积分与解析几何在它们的发展史上已经互相融合在一起了。一个领域的新的发现导致另一个领域的提高。在这本书微积分和解析几何的发展与历史发展是相似的，因此这两个学科被放在一起看待的。然而，我们初始的目的是讨论微积分。为了这个目的，来自解析几何的概念需要被讨论。实际上，仅仅一些很基本的平面解析几何的概念是需要熟知
