构造性方法及其应用外文翻译外文翻译资料

构造性方法及其应用外文翻译

[1]James Ward Brown，Ruel V. Churchll 《Complex Variables and Applications》 McGraw-Hill Companies.

[2] Maurice D. Weir, Joel Hass, George B 《Thomas. Thomasrsquo;calculus : early transcendentals》 Pearson Education.

摘要：构造性方法，主要为构造图形方法在实数与定积分的应用。

关键词：构造图形； 具体； 实数；定积分

《复变函数及应用》

设和是对应于点，该点表示非零复数。令和，该数可以被写入以极坐标形式的点的坐标

如果，坐标是未定义的;因可以理解的是，当极坐标被使用时。

在复分析中，该实数不允许为负，并且表示半径矢量的长度;也就是说。实数表示的角度，以弧度为单位，为正实数时可以被解释为半径矢量（图6）。在计算中，具有无限数量的可能值，包括负数的可能，即相差。那些值的整数倍，可以从方程确定，其中对应的点是象限限定的。的每个值被称为的参数，和该组的所有这些值被表示为。的主值，记作，是唯一的值，使得.显然有，

此外，当为负实数，具有值，而不是。

图6

实例1复数，它位于第三象限，具有主值。也就是说，

。

必须强调的是，由于主要参数的范围是，是不正确的。

根据等式（2），

。

请注意，在公式（2）的右手侧的可以通过任何的特定值来代替，并且可以写作，例如，

。

符号，或者写作，由欧拉公式来定义

，

其中，要被用弧度测量。它使人们能够以指数形式写出极坐标形式（1）更紧凑

。

符号的选择将完全与Sec.29中相同。在Sec.7使用是一个自然的选择。

实例2，实例1中的复数有指数形式

。

我们已知，这也可以被写作。表达式（5），当然，只是一个数的指数形式的无限可能性；

。

注意如何表达式（4）中告诉我们的数字是中心在原点以为半径的圆，如图7所示。的值可以从图中得到，而不需要使用欧拉公式得到数值。这是，举例来说，通过几何得到数值是明显的

， ，和 。

也请注意，该方程写作

。

是圆的参数表示，中心在原点半径为。作为从该参数范围从增大到时，点从正实轴开始并沿逆时针方向横穿圆一次。一般地，圆，其中心是，半径为，具有参数表示

。

这可以通过注意点在逆时针方向横穿圆一旦对应于固定矢量的总和和长度的矢量的倾斜角，其从以不同矢量从到如（图8）所示。

《托马斯微积分：提前超越》

6定积分的应用

概述：第5章我们看到在闭区间上连续函数有一个定积分，这种函数的黎曼和没有极限。我们证明了我们可以用微积分基本定理对定积分。我们还发现曲线下的面积与区域之间的两个曲线可以计算为定积分。

在这一章中我们将定积分的应用找到曲面、平面曲线以及旋转表面的区域的长度和度量。我们还可以使用积分解决物理问题中涉及的力所做的功，流体力对平面壁和对象的中心位置的功。

6.1利用截面投影表示度量

在本节中我们定义了立体的体积利用截面面积。一个截面一个固体的平面区域相交的平面形成的（图6.1）。我们目前获得适当的截面发现三种不同的方法一个特定的固体的体积：纵向截面法，底面法，和垫圈法。如果我们想计算一个如图6.1中的固体的度量。我们开始通过扩展经典几何定义圆柱固体选取圆柱任意底面。如果有一个已知的圆柱体，已知底面积和高度，然后求圆柱形固体的体积有. 我们在该方程形式的基础上定义的体积的许多不规则固体的度量，如图6.1。如果固体在区间各点的横截面，是一个地区的面积，和一个是的连续函数，我们可以定义计算固体的体积为一个定积分。我们现在表明通过纵向截面法得到了该定积分。 图6.1

图6.1是一个截面的固体的相交形成的平面的垂直于x轴通过点在区间。

由平行平面截面

我们通过分隔点将分割成宽度（长度）为的子区间并切片的固体，如我们将面包的面包，由平面垂直于x轴

平面垂直于x轴的分区点，切成薄“长方体”（如薄片面包形状的）。一个典型的板面如图6.3所示。我们将近似的平面与之间的板面在看作柱形固体，由底面积和高度（图6.4）该柱状固体是的体积接近于，该固体的体积：

可近似为

的全部固体的体积可由这些分块小固体的体积的总和来计算，因此近似为：

这是函数在上的黎曼和。我们预计从近似这些总和，以改变作为该分区的使范数趋于零。将分区分为成n个子区间使得到：

所以我们定义黎曼和的极限定积分成为固体的体积。

定义 固体体积由截面区域积分得到将从到是从到的积分，

计算的固体体积的一般步骤：

1.画出固体和典型横截面。

2.使用适合的公式，对于一个典型的横截面的面积来说。

3.找到积分的极限。
4.对积分算出固体体积。

例1、金字塔高3米有一个侧面边长3米。横截面垂直于高度米从顶点向下金字塔底面是米。求金字塔的体积。

解法：此定义适用当是可积的，并且特别是当它是连续的。适用的定义来计算的固体体积，采取以下步骤：

1.绘制草图。我们绘制金字塔以其沿x轴的高度和其在顶点起源和包括一个典型的横截面（图6.5）。

图6.5

2.对应的表达式。横截面在处是边长为米的方形，所以其面积是

3.积分的限制。方形是平面从到

4.积分得到体积：

外文文献出处：

[1]James Ward Brown，Ruel V. Churchll. Complex Variables and Applications [M].New York: McGraw-Hill Companies, 2008:16-19.

[2] Maurice D. Weir, Joel Hass, George B. Thomas. Thomasrsquo;calculus : early transcendentals [M] Boston: Pearson Education, 2001:345-346.

附外文文献原文

《Complex Variables and Applications》^[1]

6.EXPONENTIAL FORM

Let and be coordinates of the point that corresponds to a nonzero complex number.Sinceand,the numbercan be written in polar form as

If,the coordinates is undefined ;and so it is understood that whenever polar coordinates are used.

In complex analysis ,the real number is not allowed to be negative and is the length of the radius vector for;that is, .The real number represents the angle, measured in radians ,that makes with the positive real axis when is interpreted as a radius vector(Fig.6).As in calculus , has an infinite number of possible values, including negative ones, that differ by integral multiples of.Those value can be determined from the equation ,where the quadrant containing the point corresponding to must be specified. Each value of is called an argument of ,and the set of all such values is denoted by .The principal value of ,denoted by,is that unique value such that .Evidently, then,

Also, when is a negative real number, has value ,not.

Figure6

Example 1. The complex number , which lies in the third quadrant, has principal argument . That is,

.

It must be emphasized that because of the restriction of the principal argument , it is not true that .

According to equation (2),

.

Note that the term on the right-hand side of equation (2) can be replaced by any particular value of and that one can write, for instance,

The symbol,or ,is defined by means of Eulerrsquo;s formula as

,

Where is to be measured in radians. It enables one to write the polar form(1) more compactly in exponential form as

.

The choice of the symbol will be fully motivated later on Sec.29. Its use in Sec.7 will, however, suggest that it is a natural choice.

Example 2.The number in Example 1 has exponential form

.

With the agreement that , this can also be written . Expression(5) is, of course, only one of an infinite number of possibilities for the exponential form of ;

.

Note how expression(4) with tells us that the numbers lie on the circle centered at the origin with radius unity, as shown in Fig.7. Values of are, then, immediate from that figure, without reference to Eulerrsquo;s formula. It is, for instance, geometrically obvious that

, , and .

Note, too, that the equation

.

is a parametric representation of the circle , centered at the origin with radius .As the parameter increases from to , the point starts from the positive real axis and traverses the circle once in the counterclockwise direction. More generally, the circle ,whose center is and whose radius is , has the parametric representation

.

This can be seen vectorially (Fig.8) by noting that a point traversing the circl

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构造性方法及其应用外文翻译原文

外文文献出处：

[1]James Ward Brown，Ruel V. Churchll. Complex Variables and Applications [M].New York: McGraw-Hill Companies, 2008.

[2] Maurice D. Weir, Joel Hass, George B. Thomas. Thomasrsquo;calculus : early transcendentals [M] Boston: Pearson Education, 2001.

外文文献原文

《Complex Variables and Applications》^[1]

6.EXPONENTIAL FORM

Let and be coordinates of the point that corresponds to a nonzero complex number.Sinceand,the numbercan be written in polar form as

If,the coordinates is undefined ;and so it is understood that whenever polar coordinates are used.

In complex analysis ,the real number is not allowed to be negative and is the length of the radius vector for;that is, .The real number represents the angle, measured in radians ,that makes with the positive real axis when is interpreted as a radius vector(Fig.6).As in calculus , has an infinite number of possible values, including negative ones, that differ by integral multiples of.Those value can be determined from the equation ,where the quadrant containing the point corresponding to must be specified. Each value of is called an argument of ,and the set of all such values is denoted by .The principal value of ,denoted by,is that unique value such that .Evidently, then,

Also, when is a negative real number, has value ,not.

Figure6

Example 1. The complex number , which lies in the third quadrant, has principal argument . That is,

.

It must be emphasized that because of the restriction of the principal argument , it is not true that .

According to equation (2),

.

Note that the term on the right-hand side of equation (2) can be replaced by any particular value of and that one can write, for instance,

The symbol,or ,is defined by means of Eulerrsquo;s formula as

,

Where is to be measured in radians. It enables one to write the polar form(1) more compactly in exponential form as

.

The choice of the symbol will be fully motivated later on Sec.29. Its use in Sec.7 will, however, suggest that it is a natural choice.

Example 2.The number in Example 1 has exponential form

.

With the agreement that , this can also be written . Expression(5) is, of course, only one of an infinite number of possibilities for the exponential form of ;

.

Note how expression(4) with tells us that the numbers lie on the circle centered at the origin with radius unity, as shown in Fig.7. Values of are, then, immediate from that figure, without reference to Eulerrsquo;s formula. It is, for instance, geometrically obvious that

, , and .

Note, too, that the equation

.

is a parametric representation of the circle , centered at the origin with radius .As the parameter increases from to , the point starts from the positive real axis and traverses the circle once in the counterclockwise direction. More generally, the circle ,whose center is and whose radius is , has the parametric representation

.

This can be seen vectorially (Fig.8) by noting that a point traversing the circle once in the counterclockwise direction corresponds to the sum of the fixed vector and a vector of length whose angle of inclination varies from to .

《Thomasrsquo;calculus : early transcendentals》^[2]

6 APPLICATIONS OF DEFINITE INTEGRALS

OVERVIEW In Chapter 5 we saw that a continuous function over a closed interval has a

definite integral, which is the limit of any Riemann sum for the function. We proved that

we could evaluate definite integrals using the Fundamental Theorem of Calculus. We also

found that the area under a curve and the area between two curves could be computed as

definite integrals.

In this chapter we extend the applications of definite integrals to finding volumes,

lengths of plane curves, and areas of surfaces of revolution. We also use integrals to

solve physical problems involving the work done by a force, the fluid force against a

planar wall, and the location of an objectrsquo;s center of mass.

6.1 Volumes Using Cross-Sections

In this section we define volumes of solids using the areas of their cross-sections. A crosssection

of a solid S is the plane region formed by intersecting S with a plane (Figure 6.1).

We present three different methods for obtaining the cross-sections appropriate to finding

the volume of a particular solid: the method of slicing, the disk method, and the washer

method.

Suppose we want to find the volume of a solid S like the one in Figure 6.1. We begin

by extending the definition of a cylinder from classical geometry to cylindrical solids with

arbitrary bases (Figure 6.2). If the cylindrical solid has a known base area A and height h,

then the volume of the cylindrical solid is

This equation forms the basis for defining the volumes of many solids that are not cylinders,

like the one in Figure 6.1. If the cross-section of the solidat each point in the interval

is a region of area, andis a continuous function of , we can define

and calculate the volume of the solid S as the definite integral of . We now show how

this integral is obtained by the method of slicing.

FIGURE 6.1 A cross-section of the

solidformed by intersectingwith a plane

perpendicular to the x-axis through the

pointin the interval .

FIGURE 6.1

Slicing by Parallel Planes

We partitioninto subintervals of width (length) and slice the solid, as we would a loaf of bread, by planes perpendicular to the x-axis at the partition points

The planesperpendicular to the x-axis at the partition points, sliceinto thin “sl

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