数学专业英语中英文对照翻译2.5

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 geometry

As 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 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 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 procedure for points in a plane is this ：

Two perpendicular reference lines (called coordinate axes) are chosen, one horizont al (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 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 numbers, 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. Points 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 the 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 quadrant ,those with a<0 and b>0 are in the second quadrant ; and those with a<0 and b<0 are in the third quadrant ; and those with a>0 and b<0 are in the fourth

quadrant. Figure 2-5-1 shows one point in each quadrant.

The procedure for points in space is similar. We take three mutually perpendicular lines in space intersecting at a point (the origin) . These lines determine three mutually perpendicular planes ,and each point in space can be completely described by specifying , with appropriate regard for signs ,its distances from these planes. We shall discuee three-dimensional Cartesian geometry in more detail later on ; for the present we confine our attention to plane analytic geometry.

课文5-A：笛卡尔几何坐标系

正如前面所提到的，积分应用的一种是计算面积。通常我们不单独讨论面积，我们还讨论其它物体的面积。这意味着我们有特定的物体(多边形,圆域,抛物弓形等)希望能测量。如果我们希望获得面积的计算方法以便能用它来处理多种不同类型的图形，我们就必须首先找出描述这些图形的有效方法。

做这件事的最原始的方法就是描绘出图形，就如古希腊人所作的那样。笛卡尔(1596-1650)提出了一种好得多的方法并建立了解析几何(也称为笛卡尔几何)这个学科。笛卡尔的思想就是用数字来代表几何中的点，对应点的过程如下：

选取两条相互垂直的线(称为坐标轴)，一条水平的(称为x轴)，另一条是垂直(称为y轴)，它们的交点记为O,称为原点。在x轴上O

右边选定一个适当的点，并把它到O的距离称为单位长度。沿着y 轴的竖直距离通常也用相同的单位长度来测量，不过有时采用不同的尺度较为方便。现在平面(有时叫xy平面)上的每个点都对应一个数对，称它为坐标。这些数对告诉我们如何定为一个点。

图2-5-1说明了一些例子。坐标(3,2)的点位于y轴右边三个单位且在x轴上方两个单位长度的地方。3称为该点的x坐标，2称为该点的y坐标。y轴左边的点有负的x坐标，那些x轴下方的点有负的y坐标。点的x坐标有时称为横坐标，y坐标称为纵坐标。

当我们用一对数如(a,b)代表一个点时，我们商定横坐标，x坐标也就是a写在第一位。由于这个原因，数对(a,b)是一个有序数对。很明显，当且仅当a=c，b=d时，两个有序数对(a,b)和(c,d)代表同一个点。点(a,b)当a,b同为正时，该点位于第一象限，当a<0,b>0时位于第二象限，当a<0,b<0时位于第三象限，当a>0,b<0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 figures

A geometric figure, such 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 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,222

+=. This equation,

x y r

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 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 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 calculus . 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 with 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，x^2+y^2=r^2.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微积分的原理的。拓展微积分的范围和应用需要更深入的研究解析几何，这种学习将在之后的几章用和微积分方法一样的向量方法完成。直到那时，从解析几何的全部需要就是对绘图功能有些了解。

课文5-C Set of points in the plane

We have already showen that there is a one-to-one correspondence between points in a plane and paies of numbers (x,y) . Certain sets of points in the plane may be of special interest. For example , we may wish to examine the set of point comprising the circumference of a certain circle , or the set of points constituting the interior of a certain triangle. One may wonder if such sets of points may be succinctly described in acompact mathematical noyation.

We may write ()

{}

=

x y to describe the set of ordered pairs

y x

,|2

(x,y) , or corresponding points , such that the ordinate is equal to twice the abscissas. In effect ,then, the vertical line in (1) is read “such that” . By “the graph of the set of ordered pairs” is meant the set of all points of the plane corresponding to the set of ordered pairs. The student will readily infer that the set of points constituting the graph lies on a straight line.

Consistent the set (){}2,|y x =x y . Consistent with our previous interpretation , this symbol represents the set of ordered pairs (x,y) such that the ordinate is equal to the squqre of the abscissa. Here ,the total graph comprises a simple recognizable geometrical figure , a curve known as a parabola.

On the basis of these two example ,one may be tempted to believe that any abbitrarily drawn curve , which of course determines a set of points ordered pairs, could be described succinctly by a simple equation. Unfortunately ,this is not the case. For example , the broken line in figure 2-2-3 is one of such curves.

Consider now the set (){},|2y x >x y to describe the set of points (x,y) whose orsinate is greater than twice its abscissa. In this case ,our set of point constitutes not a curve , but a region of the coordinate piane.