数学专业英语5.docx

1、数学专业英语5Mathematical EnglishDr. Xiaomin ZhangEmail: zhangxiaomin2.5 Basic Concepts of Cartesian GeometryTEXT 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,

2、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

3、 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 Rena Descartes (1596-1650), who introduced the subject of analytic geometry (also known as Cartesian geometry). Descartes idea

4、 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 horizontal (called the “x-axis”), the other vertical (the “y-axis”). Their point of intersection, denoted by O, is called the origin.

5、On the x-axis a convenient point is chosen to the right of O and its distance form 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 pl

6、ane (sometimes called the xy-plane) is assigned a pair of numbers, called its coordinates. These numbers tell us how to locate the point.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

7、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 negative y-coordinate. The x-coordinate of a point is sometimes called its abscissa and the y-coordinate is called its ordinate.When we write a pa

8、ir of numbers such 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

9、b=d. Points (a, b) with both a and b positive are said to lie in the first quadrant, those with a0 are in the second quadrant; those with a0 and b0 and b2xto describe the set of points (x, y) whose ordinate is greater than twice its abscissa. In the case, our set of points constitutes not a curve, b

10、ut a region of the coordinate plane.SUPPLEMENT Conic SectionThe conic sections are the nondegenerate curves generated by the intersections of a plane with one or two nappes of a cone. For a plane perpendicular to the axis of the cone, a circle is produced. For a plane that is not perpendicular to th

11、e axis and that intersects only a single nappe, the curve produced is either an ellipse or a parabola. The curve produced by a plane intersecting both nappes is a hyperbola. Because of this simple geometric interpretation, the conic sections were studied by the Greeks long before their application t

12、o inverse square law orbits was known. Apollonius wrote the classic ancient work on the subject entitled On Conics. Kepler was the first to notice that planetary orbits were ellipses, and Newtonwas then able to derive the shape of orbits mathematically using calculus, under the assumption that gravi

13、tational force goes as the inverse square of distance. Depending on the energy of the orbiting body, orbit shapes that are any of the four types of conic sections are possible. A conic section may more formally be defined as the locus of a point P that moves in the plane of a fixed point F called the focus and a fixed line d called the conic section directrix (with F not on d) such that the ratio of the distance of