chapter 25

 Gaussian Co-Ordinates

 

According to Gauss, this combined analytical¹ and geometrical mode of handling the problem can be arrived at in the following way. We imagine a system of arbitrary curves (see Fig². 4) drawn³ on the surface of the table. These we designate as u-curves, and we indicate each of them by means of a number. The curves u equals 1, u equals 2 and u equals 3 are drawn in the diagram. Between the curves u equals 1 and u equals 2 we must imagine an infinitely⁴ large number to be drawn, all of which correspond to real numbers lying between 1 and 2. We have then a system of u-curves, and this “infinitely dense” system covers the whole surface of the table. These u-curves must not intersect each other, and through each point of the surface one and only one curve must pass. Thus a perfectly⁵ definite value of u belongs to every point on the surface of the marble slab⁶. In like manner we imagine a system of v-curves drawn on the surface. These satisfy the same conditions as the u-curves, they are provided with numbers in a corresponding manner, and they may likewise be of arbitrary shape. It follows that a value of u and a value of v belong to every point on the surface of the table. We call these two numbers the co-ordinates of the surface of the table (Gaussian co-ordinates). For example, the point P in the diagram has the Gaussian co-ordinates u equals 3, v equals 1. Two neighbouring points P and upper P 1 on the surface then correspond to the co-ordinates

StartLayout 1st Row 1st Column upper P colon⁷ 2nd Column u comma v 2nd Row 1st Column upper P prime colon 2nd Column u plus d u comma v plus d v EndLayout

where du and dv signify very small numbers. In a similar manner we may indicate the distance (line-interval) between P and P′, as measured with a little rod, by means of the very small number ds. Then according to Gauss we have

d s squared equals g 11 d u squared plus 2 g 12 d u d v equals g 22 d v squared

where g 11 comma g 12 comma g 22, are magnitudes which depend in a perfectly definite way on u and v. The magnitudes g 11, g 12 and g 22, determine the behaviour of the rods relative to the u-curves and v-curves, and thus also relative to the surface of the table. For the case in which the points of the surface considered form a Euclidean continuum with reference to the measuring-rods, but only in this case, it is possible to draw the u-curves and v-curves and to attach numbers to them, in such a manner, that we simply have:

d s squared equals d u squared plus d v squared

Under these conditions, the u-curves and v-curves are straight lines in the sense of Euclidean geometry, and they are perpendicular⁸ to each other. Here the Gaussian coordinates¹⁰ are simply Cartesian ones. It is clear that Gauss co-ordinates are nothing more than an association of two sets of numbers with the points of the surface considered, of such a nature that numerical values differing very slightly from each other are associated with neighbouring points “in space.”

 

So far, these considerations hold for a continuum of two dimensions. But the Gaussian method can be applied¹¹ also to a continuum of three, four or more dimensions. If, for instance, a continuum of four dimensions be supposed available, we may represent it in the following way. With every point of the continuum, we associate arbitrarily four numbers, x 1 comma x 2 comma x 3 comma x 4, which are known as “co-ordinates.” Adjacent points correspond to adjacent values of the coordinates. If a distance ds is associated with the adjacent points P and upper P 1, this distance being measurable and well defined from a physical point of view, then the following formula holds:

d s squared equals g 11 d x 1 squared plus 2 g 12 d x 1 d x 2 ellipsis¹² g 44 d x 4 squared comma

where the magnitudes g 11, etc., have values which vary with the position in the continuum. Only when the continuum is a Euclidean one is it possible to associate the co-ordinates x 1 ellipsis x 4. with the points of the continuum so that we have simply

d s Baseline 2 equals d x 1 squared plus d x 2 squared plus d x 3 squared plus d x 4 squared period

In this case relations hold in the four-dimensional continuum which are analogous¹³ to those holding in our three-dimensional measurements.

 

However, the Gauss treatment for d s squared which we have given above is not always possible. It is only possible when sufficiently¹⁴ small regions of the continuum under consideration may be regarded as Euclidean continua. For example, this obviously holds in the case of the marble slab of the table and local variation of temperature. The temperature is practically constant for a small part of the slab, and thus the geometrical behaviour of the rods is almost as it ought to be according to the rules of Euclidean geometry. Hence the imperfections of the construction of squares in the previous section do not show themselves clearly until this construction is extended over a considerable portion of the surface of the table.

 

We can sum this up as follows: Gauss invented a method for the mathematical treatment of continua in general, in which “size-relations” (“distances” between neighbouring points) are defined. To every point of a continuum are assigned as many numbers (Gaussian coordinates) as the continuum has dimensions. This is done in such a way, that only one meaning can be attached to the assignment, and that numbers (Gaussian coordinates) which differ by an indefinitely small amount are assigned to adjacent points. The Gaussian coordinate⁹ system is a logical generalisation of the Cartesian co-ordinate system. It is also applicable to non-Euclidean continua, but only when, with respect to the defined “size” or “distance,” small parts of the continuum under consideration behave more nearly like a Euclidean system, the smaller the part of the continuum under our notice.

 

1 analytical   

adj.分析的；用分析法的

参考例句：

• I have an analytical approach to every survey.对每项调查我都采用分析方法。
• As a result,analytical data obtained by analysts were often in disagreement.结果各个分析家所得的分析数据常常不一致。

2 fig   

n.无花果(树)

参考例句：

• The doctor finished the fig he had been eating and selected another.这位医生吃完了嘴里的无花果，又挑了一个。
• You can't find a person who doesn't know fig in the United States.你找不到任何一个在美国的人不知道无花果的。

3 drawn   

v.拖，拉，拔出；adj.憔悴的，紧张的

参考例句：

• All the characters in the story are drawn from life.故事中的所有人物都取材于生活。
• Her gaze was drawn irresistibly to the scene outside.她的目光禁不住被外面的风景所吸引。

4 infinitely   

adv.无限地，无穷地

参考例句：

• There is an infinitely bright future ahead of us.我们有无限光明的前途。
• The universe is infinitely large.宇宙是无限大的。

5 perfectly   

adv.完美地，无可非议地，彻底地

参考例句：

• The witnesses were each perfectly certain of what they said.证人们个个对自己所说的话十分肯定。
• Everything that we're doing is all perfectly above board.我们做的每件事情都是光明正大的。

6 slab   

n.平板，厚的切片；v.切成厚板，以平板盖上

参考例句：

• This heavy slab of oak now stood between the bomb and Hitler.这时笨重的橡木厚板就横在炸弹和希特勒之间了。
• The monument consists of two vertical pillars supporting a horizontal slab.这座纪念碑由两根垂直的柱体构成，它们共同支撑着一块平板。

7 colon   

n.冒号，结肠，直肠

参考例句：

• Here,too,the colon must be followed by a dash.这里也是一样，应当在冒号后加破折号。
• The colon is the locus of a large concentration of bacteria.结肠是大浓度的细菌所在地。

8 perpendicular   

adj.垂直的，直立的；n.垂直线，垂直的位置

参考例句：

• The two lines of bones are set perpendicular to one another.这两排骨头相互垂直。
• The wall is out of the perpendicular.这墙有些倾斜。

9 coordinate   

adj.同等的，协调的；n.同等者；vt.协作，协调

参考例句：

• You must coordinate what you said with what you did.你必须使你的言行一致。
• Maybe we can coordinate the relation of them.或许我们可以调和他们之间的关系。

10 coordinates   

n.相配之衣物；坐标( coordinate的名词复数 )；（颜色协调的）配套服装；[复数]女套服；同等重要的人（或物）v.使协调，使调和( coordinate的第三人称单数 )；协调；协同；成为同等

参考例句：

• The town coordinates on this map are 695037. 该镇在这幅地图上的坐标是695037。 来自《简明英汉词典》
• The UN Office for the Coordination of Humanitarian Affairs, headed by the Emergency Relief Coordinator, coordinates all UN emergency relief. 联合国人道主义事务协调厅在紧急救济协调员领导下，负责协调联合国的所有紧急救济工作。 来自《简明英汉词典》

11 applied   

adj.应用的；v.应用，适用

参考例句：

• She plans to take a course in applied linguistics.她打算学习应用语言学课程。
• This cream is best applied to the face at night.这种乳霜最好晚上擦脸用。

12 ellipsis   

n.省略符号，省略(语法结构上的)

参考例句：

• In this textbook we will tolerate a certain amount of ellipsis.在这本书中我们允许一些简化。
• There is an ellipsis of "that" in that sentence.那个句子省略了"that"。

13 analogous   

adj.相似的；类似的

参考例句：

• The two situations are roughly analogous.两种情況大致相似。
• The company is in a position closely analogous to that of its main rival.该公司与主要竞争对手的处境极为相似。

14 sufficiently   

adv.足够地，充分地

参考例句：

• It turned out he had not insured the house sufficiently.原来他没有给房屋投足保险。
• The new policy was sufficiently elastic to accommodate both views.新政策充分灵活地适用两种观点。