数学专业英语2-5.ppt

New Words&Expressions:abscissa 横坐标横坐标 horizontal 水平的水平的analytic geometry 解析几何解析几何 hypotenuse 斜边斜边arbitrary 任意的任意的 integral 整数的整数的,积分的积分的,积分积分Cartesian 笛卡儿的笛卡儿的 intersect 相交相交Rene Descartes 笛卡儿笛卡儿 intertwine 融合，结合融合，结合circular 圆的，圆周的圆的，圆周的 leg 侧边，直角边侧边，直角边coordinate 坐标坐标 ordinate 纵坐标纵坐标 2.5 笛卡儿几何学的基本概念笛卡儿几何学的基本概念Basic Concepts of Cartesian GeometryNew Words&Expressions:the origin 坐标原点坐标原点 segment 线段线段parabolic 抛物线的抛物线的 three-dimensional 三维的三维的perpendicular 垂直的垂直的 triangle 三角形三角形polygonal 多边形的多边形的 the unit distance 单位长度单位长度quadrant 象限象限 vector 向量向量,矢量矢量reduce 归结，化简归结，化简 vertical 竖直的竖直的 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.5-A The coordinate system of Cartesian geometry就像前面提到的，积分的一个应用就是计算面积就像前面提到的，积分的一个应用就是计算面积.通常通常我们不讨论面积本身我们不讨论面积本身,相反相反,是讨论某物的面积是讨论某物的面积.This means that we have certain objects(polygonal regions,circular regions,parabolic segments etc.)whose areas we wish to measure.这这意意味味着着我我们们想想测测量量一一些些物物体体的的面面积积（多多边边形形区区域域，圆域，抛物弓形等。圆域，抛物弓形等。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.如如果果我我们们希希望望获获得得面面积积的的计计算算方方法法以以便便能能够够用用它它来来处处理理各各种种不不同同类类型型的的图图形形，我我们们就就必必须须首首先先找找出出表表述述这这些图形的有效方法。些图形的有效方法。The most primitive way of doing this is by drawing figures,as was done by the ancient Greeks.描描述述图图形形最最原原始始的的方方法法是是画画图图,就就像像古古希希腊腊人人做做的的那那样样.A much better way was suggested by Rene Descartes,who introduced the subject of analytic geometry(also known as Cartesian geometry).R.笛笛卡卡儿儿提提出出了了一一种种好好得得多多的的办办法法，并并建建立立了了解解析析几几何（也称为笛卡儿几何）这个学科。何（也称为笛卡儿几何）这个学科。Descartes idea was to represent geometric points by numbers.The procedure for points in a plane is this:笛卡儿的思想就是用数来表示几何点，在平面上找点笛卡儿的思想就是用数来表示几何点，在平面上找点的过程如下：的过程如下：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.选选两两条条互互相相垂垂直直的的参参考考线线(称称为为坐坐标标轴轴),其其中中一一条条是是水水平平的的(称称为为x轴轴),另另一一条条是是竖竖直直的的(称称为为y轴轴).它它们们的交点记为的交点记为O,称为原点称为原点.On the x-axis a convenient point is chosen to the right of O and its distance from O is called the unit distance.在在x轴轴上上，原原点点的的右右侧侧选选择择一一个个合合适适的的点点，该该点点与与原原点之间的距离称为单位长度。点之间的距离称为单位长度。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.沿沿着着y轴轴的的竖竖直直距距离离通通常常用用同同样样的的单单位位长长度度来来测测量量，不过不过有时采用不同的尺度有时采用不同的尺度(单位长度单位长度)较为方便。较为方便。现在现在xy平面上的每一个点都分配了一对数平面上的每一个点都分配了一对数,称为坐标称为坐标.这些数告诉我们如何定义一个点。这些数告诉我们如何定义一个点。The x-coordinate of a point is sometimes called its abscissa and the y-coordinate is called its ordinate.(P47第一段最后一句第一段最后一句)When we write a pair of numbers such as(a,b)to represent a point in a plane,we agree that the abscissa or x-coordinate a is written first.(P47第二段第一句第二段第一句)有时将一个点的有时将一个点的x坐标称为横坐标，坐标称为横坐标，y坐标称为纵坐标。坐标称为纵坐标。当用一对数当用一对数(a,b)来表示平面的点时，商定要把横坐来表示平面的点时，商定要把横坐标或者标或者x坐标写在第一个位置上。坐标写在第一个位置上。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.在在空空间间中中找找点点的的过过程程是是相相似似的的。在在空空间间中中选选取取相相交交于于一点的三条互相垂直的直线。一点的三条互相垂直的直线。这三条线确定了三个互相垂直的平面这三条线确定了三个互相垂直的平面,考虑用恰当的考虑用恰当的符号表示空间中的一个点到这些平面的距离，就可以符号表示空间中的一个点到这些平面的距离，就可以完整地描述这个点。完整地描述这个点。We shall discuss three-dimensional Cartesian geometry in more detail later on;for the present we confine our attention to plane analytic geometry.以以后后我我们们将将更更加加详详细细地地讨讨论论三三维维笛笛卡卡儿儿几几何何学学。目目前前将注意力集中于平面解析几何。将注意力集中于平面解析几何。A geometric figure,such as a curve in the plane,is a collection of points satisfying one or more special conditions.5-B Geometric figures一一个个几几何何图图形形是是满满足足一一个个或或多多个个特特殊殊条条件件的的点点集集，比比如平面上的曲线。如平面上的曲线。By translating these conditions into expressions,involving the coordinates x and y,we obtain one or more equations which characterize the figure in question.通通过过把把这这些些条条件件转转化化成成含含有有坐坐标标x和和y的的表表达达式式，我我们们就得到了一个或多个能刻画就得到了一个或多个能刻画该图形该图形特征的方程。特征的方程。Throughout their historical development,calculus and analytic geometry have been intimately intertwined.微微积积分分与与解解析析几几何何在在它它们们的的发发展展史史上上已已经经紧紧密密地地融融合合在一起了。在一起了。New discoveries in one subject led to improvements in the other.The development of calculus and analytic geometry in this book is similar to the historical development,in that the two subjects are treated together.一一门门学学科科的的新新发发现现会会导导致致另另一一门门学学科科的的进进步步。本本书书中中所所叙叙述述的的微微积积分分和和解解析析几几何何的的发发展展和和历历史史发发展展过过程程是是相似的，因为这两门学科是一起研究的。相似的，因为这两门学科是一起研究的。However,our primary purpose is to discuss calculus.Concepts from analytic geometry that are required for this purpose will be discussed as needed.不不过过，我我们们的的主主要要目目的的是是讨讨论论微微积积分分。为为此此，解解析析几几何中的概念将只是在必要时讨论。何中的概念将只是在必要时讨论。Actually,only a few very elementary concepts of plane analytic geometry are required to understand the rudiments of calculus.实实际际上上，仅仅仅仅在在微微积积分分入入门门阶阶段段需需要要用用到到平平面面解解析析几几何的几个基本概念。何的几个基本概念。A deeper study of analytic geometry is needed to extend the scope and applications of calculus,and this study will be carried out using vector method.如如果果想想拓拓展展微微积积分分的的范范围围与与应应用用，需需要要进进一一步步研研究究解解析几何，而这种研究需用到向量的方法来实现。析几何，而这种研究需用到向量的方法来实现。Until then,all that is required from analytic geometry is a little familiarity with drawing graph of function.而而这这之之前前，关关于于解解析析几几何何仅仅仅仅需需要要熟熟悉悉一一点点画画函函数数图图象的知识。象的知识。作业：P 51 2.汉译英(1),(4)3.英译汉(2)谢 谢！