题目
汉译英(1)常用英语字母和希腊字母来表示函数。Letters of the Englis. and Greek alphabets are often used to denote functions. (2)若 f 是一个给定的函数,x 是定义域里的一个元素,那么记号 f(x)用来表示由 f 确定的 对应于 x 的值。If f is a given function and if x is an object of its domain, the notation f(x) is used to designate that object in the range which is associated to x by the function f. (3)该射线将两个坐标轴的夹角分成两个相等的角。The ray makes.equal angles with the coordinates axes.(4)可以用许多方式给出函数思想的图解说明。The function idea may be illus.rated schematically in many ways.(5)容易证明,绝对值函数满足三角不等式。It is easy to proof that the absolute-value function satisfies the triangle inequality. (6)对于实数 x>0,函数 g(x)表示不超过 x 的素数的个数。For a given real number x.gt;0, the function g(x) is defined by the number of primes less than or equal to x.(7)函数是一种对应,它未必可以表示成一个简单的代数公式。A function is a.correspondence. It is not necessary to be expressed by a simple algebraic formula.(8)在函数的定义中,关于定义域和值域中的对象,没对其性质做出任何限制。The function idea places no restriction on the nature of the objects in the domain X and in the range Y.2.7 序列及其极限 序列及其极限(1)序列各项对 n 的相关性常利用下标来表示,写成如下形式: a n , x n 等。The depen.ence of every team of sequence on n is denoted by using subscript, and we write a n , x n and so on.(2)以正整数集为定义域的函数称为序列。A function whose domain is the se. of all positive integers is called an infinite sequence. (3)一个复值序列收敛当且仅当它的实部和虚部分别收敛。A complex-valued sequence converges if and only.if both the real part and the imaginary part converge separately.place汉译英(1)数学来源于人类的社会实践,包括工农业的劳动,商业、军事和科学技术研究等活动。Mathematices from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches.(2)如果没有运用数学,任何一个科学技术分支都不可能正常地发展。No modern s.ientific and technological branches could be regularly developed without the application of mathematics.(3)符号在数学中起着非常重要的作用,它常用于表示概念和命题。Notation. are a special and powerful tool of mathematics and are used to express conceptions and propositions very often.(4)17 世纪之前,人们局限于初等数学,即几何、三角和代数,那时只考虑常数。Before 17th century, man confine. himself to the elementary mathematics, i. e. , geometry, trigonometry and algebra, in which only the constants were considered. (5)方程与算数的等式不同在于它含有可以参加运算的未知量。Equation is.different from arithmetic identity in that it contains unknown quantity which can join operations.(4) 一个序列( a n )若满足: 对任意正数 ε , 存在另一个正数 N (N可能与ε 有关) 使得 a n - L < ε 对所有 n ? N 成立,就称( a n )收敛于 L。A sequence ( a n ) is said to have a limit L if, for every positive number ε , there isanother positive number N.(which may depend on ε ) such that In this case, we say the sequence ( a n ) converges to L. an ? L < ε for all n ? N.(5) 重要的是, 该集的每一个成员都用一个正整数标上记号。 这样一来, 就可以谈论第一项、 第二项和一般项,即第 n 项。The important thing is that each member of the set has been labeled with an integer so that we may speak of the first term. the second term and in general, the nth term. (6)若无另加申明,本章研究的序列都假定具有实的项或复的项。Unles. otherwise specified, all sequences in this chapter are assumed to have real or complex terms.(7)作为日常用语,sequence 和 series 是同义词;但作为数学术语,它们表示不同的概念。In everyday usage of the Englis. language, the words “sequence” and “series” are synonyms, but in mathematics these words have special technical meanings. (8)术语“收敛序列”指的是具有有限极限的序列,因此,极限为无限的序列不是收敛的, 而是发散的。The.phrase “convergent sequence. is used only for a sequence whose limit is finite. A sequence with an infinite limit is said to diverge not convergence.2.8 函数的导数和它的几何意义(1)差商表示函数 f 在连接 x 与 x+h 的区间上的平均变化率。Th. different quotient is referred to as the average rate of the change of f in the interval joining x to x+h.(2)速度等于位置函数的导数。Velocity is equal to the derivative of positing.(3)由定义导数的过程所提供的几何解释以一种自然的方式导出了关于曲线的切线思想。The.procedure used to define the derivative has a geometric interpretation which leads in a natural way to the idea of a tangent line to a curve.(4)差商表示直线 PQ 与水平线的夹角的正切。The difference quotient represents the trigonometric tangent of the angl. that PQ makes with the horizontal.(5)在直线运动中,速度的一阶导数称为加速度。For rectilinear motion. the first derivative of velocity is called acceleration. (6)我们约定 f(0)=f,即函数 f 的零阶导数就等于它本身。We make the convention that f(0)=f. that is the zeroth derivate is the function itself. (7)在运动的 9 秒钟内,物体的速度由 v (0) = -144 变成了 v (9) =144,也就是说,速度总共 增加了每秒 288 英尺。During the 9 second. of motion the velocity changes from v (0) = -144 to v (9) =144, that is, the total increase in velocity is 288 feet per second.(8)当 α 从 0 增加到π/2 时,tan α 所对应的直线趋于竖直位置。As α increasesfrom 0 to π/2 , tan.α approach a vertical position.2.9 微分方程简介(1)此时,微分方程就有无穷多个解,C的每个值对应一个解。The differential equation has infinitely many solutions, one for each value of C. (2)微分方程的阶指的是方程中最高阶导数的阶。By the order of an equation is meant the order of the highes. derivative which appears. (3)我们可以由已知的粒子运动速度或者加速度计算出粒子的位置。We could try to compute the position.of a moving particle from a knowledge of its velocity or acceleration.(4)如果一个微分方程的未知函数是多元函数,则称为偏微分方程。Ordinary and partial, depend on whether the unknown is.a function of just one variable or of two or more variables.(5)微分方程的研究直接受到力学、天文学和数学物理的推动。The s.udy of differential equations has been directly inspired by mechanics, astronomy, and mathematical physics.(6)许多应用问题要求我们从方程的解集中选出一个在某个点具有指定值的解。In many problems it.is necessary to select from the collection of all solutions one having a prescribed value at some point.(7)确定满足边界条件的解的问题称为边值问题。The problem.of determining such a solution that satisfies boundary condition is called a boundary-value problem.(8)人们设计许多高速运行的计算机来对各种积分做出近似估计。Automatic high-speeputing machines are often designed with this kind of problem in mind.2.10 线性空间中的相关与无关集(1)该式的两边同时关于t积分,我们就得到一个所需要的结论。Integrating bot. sid.s of this formula with respect to t. we can obtain a conclusion we need.(2)不难看出,这个命题仅仅建立在该空间是线性的这一事实上,与空间的其他性质无关。We clearly find that this proposition is based only on the fact that this space is a linearspace and not on any othe. special property of this space. (3)如果空间不存在有限基,就称该空间是无限维的。A s.ace is called infinite dimensional if it doesn’t have a finite basis.(4)假定这个结论对n-1个指数函数成立,我们将证明此结论对n个指数函数也成立。Assuming the conclusion is true for n-1 exponential functions, we will prove that it is truefor n.exponential function.(5)这两个定义在逻辑上是互相等价的。hese two definitions are logically equivalence. T(6)设X是线性空间V中k个元素组成的一个线性无关集合,L(X)是由X张成的子空间。那么,L(X)的每一个元素都可以表示成X的元素的线性组合。Let X be an independent set consisting of k elements in a linear space V and let L( )
汉译英(1)常用英语字母和希腊字母来表示函数。Letters of the Engli
s. and Greek alphabets are often used to denote functions. (2)若 f 是一个给定的函数,x 是定义域里的一个元素,那么记号 f(x)用来表示由 f 确定的 对应于 x 的值。If f is a given function and if x is an object of its domain, the notation f(x) is used to designate that object in the range which is associated to x by the function
f. (3)该射线将两个坐标轴的夹角分成两个相等的角。The ray make
s.equal angles with the coordinates axes.(4)可以用许多方式给出函数思想的图解说明。The function idea may be illu
s.rated schematically in many ways.(5)容易证明,绝对值函数满足三角不等式。It is easy to proof that the absolute-value function satisfies the triangle inequalit
y. (6)对于实数 x>0,函数 g(x)表示不超过 x 的素数的个数。For a given real number
x.gt;0, the function g(x) is defined by the number of primes less than or equal to x.(7)函数是一种对应,它未必可以表示成一个简单的代数公式。A function is
a.correspondenc
e. It is not necessary to be expressed by a simple algebraic formula.(8)在函数的定义中,关于定义域和值域中的对象,没对其性质做出任何限制。The function idea places no restriction on the nature of the objects in the domain X and in the range
Y.
2.7 序列及其极限 序列及其极限(1)序列各项对 n 的相关性常利用下标来表示,写成如下形式: a n , x n 等。The depe
n.ence of every team of sequence on n is denoted by using subscript, and we write a n , x n and so on.(2)以正整数集为定义域的函数称为序列。A function whose domain is the s
e. of all positive integers is called an infinite sequence. (3)一个复值序列收敛当且仅当它的实部和虚部分别收敛。A complex-valued sequence converges if and onl
y.if both the real part and the imaginary part converge separately.place汉译英(1)数学来源于人类的社会实践,包括工农业的劳动,商业、军事和科学技术研究等活动。Mathematic
es from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches.(2)如果没有运用数学,任何一个科学技术分支都不可能正常地发展。No modern
s.ientific and technological branches could be regularly developed without the application of mathematics.(3)符号在数学中起着非常重要的作用,它常用于表示概念和命题。Notatio
n. are a special and powerful tool of mathematics and are used to express conceptions and propositions very often.(4)17 世纪之前,人们局限于初等数学,即几何、三角和代数,那时只考虑常数。Before 17th century, man confin
e. himself to the elementary mathematics,
i. e. , geometry, trigonometry and algebra, in which only the constants were considere
d. (5)方程与算数的等式不同在于它含有可以参加运算的未知量。Equation i
s.different from arithmetic identity in that it contains unknown quantity which can join operations.(4) 一个序列{ a n }若满足: 对任意正数 ε , 存在另一个正数 N (N可能与ε 有关) 使得 a n - L < ε 对所有 n ? N 成立,就称{ a n }收敛于 L。A sequence { a n } is said to have a limit L if, for every positive number ε , there isanother positive number
N.(which may depend on ε ) such that In this case, we say the sequence { a n } converges to
L. an ? L < ε for all n ? N.(5) 重要的是, 该集的每一个成员都用一个正整数标上记号。 这样一来, 就可以谈论第一项、 第二项和一般项,即第 n 项。The important thing is that each member of the set has been labeled with an integer so that we may speak of the first ter
m. the second term and in general, the nth term. (6)若无另加申明,本章研究的序列都假定具有实的项或复的项。Unle
s. otherwise specified, all sequences in this chapter are assumed to have real or complex terms.(7)作为日常用语,sequence 和 series 是同义词;但作为数学术语,它们表示不同的概念。In everyday usage of the Engli
s. language, the words “sequence” and “series” are synonyms, but in mathematics these words have special technical meanings. (8)术语“收敛序列”指的是具有有限极限的序列,因此,极限为无限的序列不是收敛的, 而是发散的。Th
e.phrase “convergent sequenc
e. is used only for a sequence whose limit is finite. A sequence with an infinite limit is said to diverge not convergence.
2.8 函数的导数和它的几何意义(1)差商表示函数 f 在连接 x 与 x+h 的区间上的平均变化率。T
h. different quotient is referred to as the average rate of the change of f in the interval joining x to x+h.(2)速度等于位置函数的导数。Velocity is equal to the derivative of positin
g.(3)由定义导数的过程所提供的几何解释以一种自然的方式导出了关于曲线的切线思想。Th
e.procedure used to define the derivative has a geometric interpretation which leads in a natural way to the idea of a tangent line to a curve.(4)差商表示直线 PQ 与水平线的夹角的正切。The difference quotient represents the trigonometric tangent of the ang
l. that PQ makes with the horizontal.(5)在直线运动中,速度的一阶导数称为加速度。For rectilinear motio
n. the first derivative of velocity is called acceleration. (6)我们约定 f(0)=f,即函数 f 的零阶导数就等于它本身。We make the convention that f(0)=
f. that is the zeroth derivate is the function itself. (7)在运动的 9 秒钟内,物体的速度由 v (0) = -144 变成了 v (9) =144,也就是说,速度总共 增加了每秒 288 英尺。During the 9 secon
d. of motion the velocity changes from v (0) = -144 to v (9) =144, that is, the total increase in velocity is 288 feet per second.(8)当 α 从 0 增加到π/2 时,tan α 所对应的直线趋于竖直位置。As α increasesfrom 0 to π/2 , ta
n.α approach a vertical position.
2.9 微分方程简介(1)此时,微分方程就有无穷多个解,C的每个值对应一个解。The differential equation has infinitely many solutions, one for each value of
C. (2)微分方程的阶指的是方程中最高阶导数的阶。By the order of an equation is meant the order of the highe
s. derivative which appears. (3)我们可以由已知的粒子运动速度或者加速度计算出粒子的位置。We could try to compute the positio
n.of a moving particle from a knowledge of its velocity or acceleration.(4)如果一个微分方程的未知函数是多元函数,则称为偏微分方程。Ordinary and partial, depend on whether the unknown i
s.a function of just one variable or of two or more variables.(5)微分方程的研究直接受到力学、天文学和数学物理的推动。The
s.udy of differential equations has been directly inspired by mechanics, astronomy, and mathematical physics.(6)许多应用问题要求我们从方程的解集中选出一个在某个点具有指定值的解。In many problems i
t.is necessary to select from the collection of all solutions one having a prescribed value at some point.(7)确定满足边界条件的解的问题称为边值问题。The proble
m.of determining such a solution that satisfies boundary condition is called a boundary-value problem.(8)人们设计许多高速运行的计算机来对各种积分做出近似估计。Automatic high-spee
puting machines are often designed with this kind of problem in mind.
2.10 线性空间中的相关与无关集(1)该式的两边同时关于t积分,我们就得到一个所需要的结论。Integrating bo
t. si
d.s of this formula with respect to t. we can obtain a conclusion we need.(2)不难看出,这个命题仅仅建立在该空间是线性的这一事实上,与空间的其他性质无关。We clearly find that this proposition is based only on the fact that this space is a linearspace and not on any oth
e. special property of this space. (3)如果空间不存在有限基,就称该空间是无限维的。A
s.ace is called infinite dimensional if it doesn’t have a finite basis.(4)假定这个结论对n-1个指数函数成立,我们将证明此结论对n个指数函数也成立。Assuming the conclusion is true for n-1 exponential functions, we will prove that it is truefor
n.exponential function.(5)这两个定义在逻辑上是互相等价的。hese two definitions are logically equivalenc
e. T(6)设X是线性空间V中k个元素组成的一个线性无关集合,L(X)是由X张成的子空间。那么,L(X)的每一个元素都可以表示成X的元素的线性组合。Let X be an independent set consisting of k elements in a linear space V and let L( )
s. and Greek alphabets are often used to denote functions. (2)若 f 是一个给定的函数,x 是定义域里的一个元素,那么记号 f(x)用来表示由 f 确定的 对应于 x 的值。If f is a given function and if x is an object of its domain, the notation f(x) is used to designate that object in the range which is associated to x by the function
f. (3)该射线将两个坐标轴的夹角分成两个相等的角。The ray make
s.equal angles with the coordinates axes.(4)可以用许多方式给出函数思想的图解说明。The function idea may be illu
s.rated schematically in many ways.(5)容易证明,绝对值函数满足三角不等式。It is easy to proof that the absolute-value function satisfies the triangle inequalit
y. (6)对于实数 x>0,函数 g(x)表示不超过 x 的素数的个数。For a given real number
x.gt;0, the function g(x) is defined by the number of primes less than or equal to x.(7)函数是一种对应,它未必可以表示成一个简单的代数公式。A function is
a.correspondenc
e. It is not necessary to be expressed by a simple algebraic formula.(8)在函数的定义中,关于定义域和值域中的对象,没对其性质做出任何限制。The function idea places no restriction on the nature of the objects in the domain X and in the range
Y.
2.7 序列及其极限 序列及其极限(1)序列各项对 n 的相关性常利用下标来表示,写成如下形式: a n , x n 等。The depe
n.ence of every team of sequence on n is denoted by using subscript, and we write a n , x n and so on.(2)以正整数集为定义域的函数称为序列。A function whose domain is the s
e. of all positive integers is called an infinite sequence. (3)一个复值序列收敛当且仅当它的实部和虚部分别收敛。A complex-valued sequence converges if and onl
y.if both the real part and the imaginary part converge separately.place汉译英(1)数学来源于人类的社会实践,包括工农业的劳动,商业、军事和科学技术研究等活动。Mathematic
es from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches.(2)如果没有运用数学,任何一个科学技术分支都不可能正常地发展。No modern
s.ientific and technological branches could be regularly developed without the application of mathematics.(3)符号在数学中起着非常重要的作用,它常用于表示概念和命题。Notatio
n. are a special and powerful tool of mathematics and are used to express conceptions and propositions very often.(4)17 世纪之前,人们局限于初等数学,即几何、三角和代数,那时只考虑常数。Before 17th century, man confin
e. himself to the elementary mathematics,
i. e. , geometry, trigonometry and algebra, in which only the constants were considere
d. (5)方程与算数的等式不同在于它含有可以参加运算的未知量。Equation i
s.different from arithmetic identity in that it contains unknown quantity which can join operations.(4) 一个序列{ a n }若满足: 对任意正数 ε , 存在另一个正数 N (N可能与ε 有关) 使得 a n - L < ε 对所有 n ? N 成立,就称{ a n }收敛于 L。A sequence { a n } is said to have a limit L if, for every positive number ε , there isanother positive number
N.(which may depend on ε ) such that In this case, we say the sequence { a n } converges to
L. an ? L < ε for all n ? N.(5) 重要的是, 该集的每一个成员都用一个正整数标上记号。 这样一来, 就可以谈论第一项、 第二项和一般项,即第 n 项。The important thing is that each member of the set has been labeled with an integer so that we may speak of the first ter
m. the second term and in general, the nth term. (6)若无另加申明,本章研究的序列都假定具有实的项或复的项。Unle
s. otherwise specified, all sequences in this chapter are assumed to have real or complex terms.(7)作为日常用语,sequence 和 series 是同义词;但作为数学术语,它们表示不同的概念。In everyday usage of the Engli
s. language, the words “sequence” and “series” are synonyms, but in mathematics these words have special technical meanings. (8)术语“收敛序列”指的是具有有限极限的序列,因此,极限为无限的序列不是收敛的, 而是发散的。Th
e.phrase “convergent sequenc
e. is used only for a sequence whose limit is finite. A sequence with an infinite limit is said to diverge not convergence.
2.8 函数的导数和它的几何意义(1)差商表示函数 f 在连接 x 与 x+h 的区间上的平均变化率。T
h. different quotient is referred to as the average rate of the change of f in the interval joining x to x+h.(2)速度等于位置函数的导数。Velocity is equal to the derivative of positin
g.(3)由定义导数的过程所提供的几何解释以一种自然的方式导出了关于曲线的切线思想。Th
e.procedure used to define the derivative has a geometric interpretation which leads in a natural way to the idea of a tangent line to a curve.(4)差商表示直线 PQ 与水平线的夹角的正切。The difference quotient represents the trigonometric tangent of the ang
l. that PQ makes with the horizontal.(5)在直线运动中,速度的一阶导数称为加速度。For rectilinear motio
n. the first derivative of velocity is called acceleration. (6)我们约定 f(0)=f,即函数 f 的零阶导数就等于它本身。We make the convention that f(0)=
f. that is the zeroth derivate is the function itself. (7)在运动的 9 秒钟内,物体的速度由 v (0) = -144 变成了 v (9) =144,也就是说,速度总共 增加了每秒 288 英尺。During the 9 secon
d. of motion the velocity changes from v (0) = -144 to v (9) =144, that is, the total increase in velocity is 288 feet per second.(8)当 α 从 0 增加到π/2 时,tan α 所对应的直线趋于竖直位置。As α increasesfrom 0 to π/2 , ta
n.α approach a vertical position.
2.9 微分方程简介(1)此时,微分方程就有无穷多个解,C的每个值对应一个解。The differential equation has infinitely many solutions, one for each value of
C. (2)微分方程的阶指的是方程中最高阶导数的阶。By the order of an equation is meant the order of the highe
s. derivative which appears. (3)我们可以由已知的粒子运动速度或者加速度计算出粒子的位置。We could try to compute the positio
n.of a moving particle from a knowledge of its velocity or acceleration.(4)如果一个微分方程的未知函数是多元函数,则称为偏微分方程。Ordinary and partial, depend on whether the unknown i
s.a function of just one variable or of two or more variables.(5)微分方程的研究直接受到力学、天文学和数学物理的推动。The
s.udy of differential equations has been directly inspired by mechanics, astronomy, and mathematical physics.(6)许多应用问题要求我们从方程的解集中选出一个在某个点具有指定值的解。In many problems i
t.is necessary to select from the collection of all solutions one having a prescribed value at some point.(7)确定满足边界条件的解的问题称为边值问题。The proble
m.of determining such a solution that satisfies boundary condition is called a boundary-value problem.(8)人们设计许多高速运行的计算机来对各种积分做出近似估计。Automatic high-spee
puting machines are often designed with this kind of problem in mind.
2.10 线性空间中的相关与无关集(1)该式的两边同时关于t积分,我们就得到一个所需要的结论。Integrating bo
t. si
d.s of this formula with respect to t. we can obtain a conclusion we need.(2)不难看出,这个命题仅仅建立在该空间是线性的这一事实上,与空间的其他性质无关。We clearly find that this proposition is based only on the fact that this space is a linearspace and not on any oth
e. special property of this space. (3)如果空间不存在有限基,就称该空间是无限维的。A
s.ace is called infinite dimensional if it doesn’t have a finite basis.(4)假定这个结论对n-1个指数函数成立,我们将证明此结论对n个指数函数也成立。Assuming the conclusion is true for n-1 exponential functions, we will prove that it is truefor
n.exponential function.(5)这两个定义在逻辑上是互相等价的。hese two definitions are logically equivalenc
e. T(6)设X是线性空间V中k个元素组成的一个线性无关集合,L(X)是由X张成的子空间。那么,L(X)的每一个元素都可以表示成X的元素的线性组合。Let X be an independent set consisting of k elements in a linear space V and let L( )
题目解答
答案
错误
解析
本次题目主要是汉英翻译内容的展示，未明确提出问题，但根据提供的“这道题的答案是：错误”推测，可能是对展示内容中某部分翻译或表述的正确性判断。经检查，原文中“2.7 序列及其极限”部分第(4)点英文翻译里存在“In this case, we say the sequence { aₙ } converges to L. an ? L < ε for all n ? N”的表述错误，正确应为“In this case, we say the sequence { aₙ } converges to L: $|aₙ - L| < ε$ for all $n > N$”（原内容中“an ? L”应为“|aₙ - L|”，“n ? N”应为“n > N”），存在翻译错误，故答案为“错误”。