数学专业英语

举一个简单的例子，当常微分方程满足f‟(x) = f(x) 的关系时，最特殊的情况是指数函数 f(x) = ex
We shall see presently that every solution of (9.1) f‟(x) = f(x) must be of the form f(x) = Cex , where C may be any constant.
The study of differential equations is one part of mathematics that, perhaps more than any other , has been directly inspired by mechanics, astronomy, and mathematical physics .Its history began in the 17th century when Newton, Leibniz, and the Bernoullis solved some simple differential equations arising from problems in geometry and mechanics .
displacement Bernoulli
n.位移
n. (人名) 伯努利
The Bernoullis 伯努利(家族) mathematical physics 数学物理
A large variety of scientific problems arise in which one tries to determine something from its rate of change.
这些方程叫做微分方程，他们的研究方式 是最难的数学分ations are classified under two main headings: ordinary and partial, depending on whether the unknown is a function of just one variable or of two or more variables. classified 分类 variable 变量
polynomial [,pɑlɪ'nomɪəl] n.多项式
logarithm ['lɔgə,rɪðəm] n.对数
logarithm function 对数函数
exponential [,ɛkspo'nɛnʃəl] adj.指数的 exponential function 指数函数 nature n.性质，自然 initial [ɪ'nɪʃəl] -value problem 初值问题 initial condition 初始条件
另一方面，像 就是偏微分方程的一个例 子。这种称为拉普拉斯的特殊公式常应用在电磁原理及流体 力学等领域。它有多种形式，比 如： ， , and
The second partial derivative of the function f with respect to x plus the second partial derivative of the function f with respect to y equal zero the function f equal x plus two times y exponential function of x multiply cos y logarithm function of the second power of x plus the second power of y
例如，我们可以试着通过已知的速度或 加速度来计算一个移动质点的位置。
Or a radioactive substance may be disintegrating at a known rate and we may be required to determine the amount of material present after a given time.
微分方程根据未知量是单变量函数或多变量函 数分成两个主题：常微分方程和偏微分方程。
A simple example of an ordinary differential equation is the relation (9.1) f‟(x) = f(x) which is satisfied, in particular by the exponential function, f(x) = ex.
我们可以看出每一个f‟(x) = f(x)的结果都是 f(x) = Cex 的形式，C为任意常数。
On the other hand, an equation like is an example of a partial differential equation . This particular one , called Laplace‟s equation , appears in the theory of electricity and magnetism, fluid mechanics, and elsewhere . It has many different kinds of solutions , among which are , and
微分方程研究作为数学的一部分，正广泛的被机械学、天文学 以及数学物理学等领域的专家学者所使用。它的历史可以追溯 到17世纪，牛顿、莱布尼茨和伯努利在一次处理几何、机械学 的问题时发现了微分方程。
These early discoveries, beginning about 1690, gradually led to the development of a lot of “special tricks” for solving certain special kinds of differential equations .Although these special tricks are applicable in relatively few cases, they do enable us to solve many differential equations that arise in mechanics and geometry, so their study is of practical importance . Some of these special methods and some of the problems which they help us solve are discussed near the end of this chapter . 自1960年起，这些早期的发现逐渐转变成对于特定种类微分方程 解决办法的技巧探究。尽管这些技巧只能应用在与其相关的一小部 方程中，但它们已经使我们能够处理很多机械学和天文学中出现的 微分方程问题了，所以他们的研究成果是具有实用意义的。我们将 会在本章末继续讨论这些技巧以及其帮助我们所解决的问题。
近似估计
integral [„ɪntəgrəl] n.积分 ； adj.整数的
integrate integration v.对…积分 n.积分 n.力学
mechanics [mə'kænɪks] astronomy n.天文学
disintegrate [dɪs'ɪntəgret] v.解体，衰变 terminology [,tɝmə'nɑlədʒɪ] n.术语
present 留存的 disintegrate 分裂 放射性物质 radioactive substance
又如，某种放射性物质可能正在以已知的速 度进行衰变，需要我们确定在给定的时间后 遗留物质的总量。
In examples like these, we are trying to determine an unknown function from prescribed information expressed in the form of an equation involving at least one of the derivatives of the unknown function. prescribe 规定 express 表示
Experience has shown that it is difficult to obtain mathematical theories of much generality about solutions of differential equations, except for a few types .Among these are the so-called linear differential equations which occur in a great variety of scientific problems .The simplest types of linear differential equations and some of their applications are also discussed in this introductory chapter .A more thorough study of linear equations is carried out in Volume Ⅱ. 经验表明，对于微分方程解决办法的概述，除却一小部分外，我 们很难得出数学原理。这其中就包括了经常出现在各种各样的科 学的问题中的，所谓的线性微分方程。关于线性微分方程的简单 种类及其应用也将会在这部分介绍章节中继续探讨。第二册中将 对于线性微分方程进行更深入的研究。
in the form of an equation 以方程的形式 在类似的例子中，我们力求由方程的形式表述 的信息来确定未知函数，而这种方程至少包含 了未知函数的一个导数。
These equations are called differential equations , and their study forms one of the most challenging branches of mathematics.
这里which引起的从句修饰problems
大量的科学问题需要人们根据事物的 变化率去决定该事物的量
For example, we could try to compute the position of a moving particle from a knowledge of its velocity or acceleration. compute 计算 particle 质点 position 位置，方位 a knowledge of 已知
differential equation [ɪ'kweʃən] 微分方程
ordinary differential equation 常微分方程
partial ['pɑrʃəl] differential equation 偏微分方程 rational ['ræʃən!] function 有理函数 trigonometric [trɪgənə'mɛtrɪk] function 三角函数 inverse trigonometric function 反三角函数 approximate [ə'prɑksəmɪt] evaluation [ɪ,vælju'eʃən]