integrability

简明释义

英[ɪntɪɡrəˈbɪlɪti]美[ɪntɪɡrəˈbɪlɪti]

n. 可积（分）性

英英释义

单词用法

同义词

反义词

例句

1.With the concavity and integrability of sublinear terms near zero, the symmetry results for a class of sublinear elliptic equations are given by making use of the moving-plane method.

摘要本文利用次线性项在零点附近的凹性和可积性，用移动平面法给出了一类次线性椭圆方程正解的对称性。

2.A 2r Planar Robot Manipulator system is described, whose integrability is proved by the theory of Hamilton system.

描述了一类平面2r机械臂的模型，利用哈密顿系统理论证明了该系统的可积性。

3.This is usually a much stronger requirement than quadratic integrability.

这通常是一个比平方可积更强的要求。

4.Based on Darboux theory, this paper discussed the integrability of the Riemann Integral and provides a necessary and sufficient condition for integrability.

文章利用达布和理论，讨论了黎曼积分的可积性问题，给出了一个可积的充分必要条件。

5.A new integrable condition in a kind of special Abel equation is given in this paper. We can realize the special Abel equation integrability and accurate solution by computer.

给出一类特殊阿贝尔方程新的可积条件，实现了计算机对该类方程可积性及其精确解的自动判定。

6.With the concavity and integrability of sublinear terms near zero, the symmetry results for a class of sublinear elliptic equations are given by making use of the moving-plane method.

摘要本文利用次线性项在零点附近的凹性和可积性，用移动平面法给出了一类次线性椭圆方程正解的对称性。

7.These results can be used to study the integrability of conjugate harmonic functions and estimate the integrals for them.

这些结果能被用来研究共轭调和函数的可积性并且估计它们的积分。

8.The integrability 可积性 of a differential equation often dictates the methods we can use to solve it.

微分方程的integrability 可积性 通常决定了我们可以用来解决它的方法。

9.In physics, the integrability 可积性 of a system can reveal conserved quantities and symmetries.

在物理学中，系统的integrability 可积性 可以揭示守恒量和对称性。

10.The study of integrability 可积性 in complex systems is a growing field of research.

复杂系统中integrability 可积性 的研究是一个不断增长的研究领域。

11.Before applying numerical methods, we must check the integrability 可积性 of the function involved.

在应用数值方法之前，我们必须检查所涉及函数的integrability 可积性。

12.In mathematics, the concept of integrability 可积性 is crucial when determining whether a function can be integrated over a given interval.

在数学中，integrability 可积性 的概念在确定一个函数是否可以在给定区间内被积分时至关重要。

作文

In the realm of mathematics, particularly in calculus and analysis, the term integrability refers to the ability of a function to be integrated. This concept is fundamental as it determines whether we can find the area under a curve or the total accumulation of quantities represented by a function. Understanding integrability is essential for students and professionals alike, as it forms the basis for many applications in physics, engineering, and economics.To delve deeper, let's consider what it means for a function to be integrable. A function is said to be integrable if its integral exists over a specified interval. This can be understood through the lens of the Riemann integral, which is one of the most common methods for defining integrals. A function that is continuous on a closed interval is guaranteed to be Riemann integrable, which is a comforting notion for those studying calculus. However, the situation becomes more complex when we encounter functions that are not continuous or have discontinuities.For instance, the function f(x) = 1/x is not integrable over the interval [0, 1] because it approaches infinity as x approaches 0. This highlights an important aspect of integrability: it is not merely about having a function defined over an interval, but also about the behavior of the function within that interval. Thus, understanding the properties of functions—such as boundedness and continuity—is crucial in determining their integrability.Another interesting perspective on integrability comes from Lebesgue integration, which generalizes the concept beyond Riemann integration. Lebesgue's approach allows us to integrate a broader class of functions by focusing on measuring the 'size' of sets where the function takes certain values, rather than just partitioning the domain into intervals. This method shows that some functions can be integrable in the Lebesgue sense even if they are not Riemann integrable. The study of integrability thus opens up new avenues for analysis and provides deeper insights into the nature of functions.In practical terms, the concept of integrability is crucial in various fields. In physics, for example, the calculation of work done by a force involves integrating the force function over a distance. Similarly, in economics, the consumer surplus and producer surplus are determined using integrals that rely on the integrability of demand and supply functions. Therefore, a solid understanding of integrability equips individuals with the tools necessary to tackle real-world problems.In conclusion, integrability is a cornerstone of mathematical analysis that extends far beyond the classroom. It encapsulates the ability to integrate functions, providing a framework for understanding accumulation and area under curves. Whether through Riemann or Lebesgue integration, the concept of integrability offers profound insights into the behavior of functions and their applications across various disciplines. As such, mastering integrability is not only vital for academic success but also for practical problem-solving in numerous fields of study.

在数学领域，特别是在微积分和分析中，术语integrability指的是一个函数可以被积分的能力。这个概念是基础，因为它决定了我们是否能够找到曲线下的面积或表示某个函数的总累积量。理解integrability对于学生和专业人士来说至关重要，因为它构成了物理、工程和经济学中许多应用的基础。要深入了解，一个函数的可积性意味着什么？如果一个函数在指定区间上的积分存在，则该函数被称为可积的。通过黎曼积分的视角，我们可以理解这一点，这是定义积分的最常见方法之一。在封闭区间上连续的函数保证是黎曼可积的，这对学习微积分的人来说是一个令人安心的概念。然而，当我们遇到不连续或具有不连续性的函数时，情况变得更加复杂。例如，函数f(x) = 1/x在区间[0, 1]上是不可积的，因为当x趋近于0时，它趋向于无穷大。这突显了integrability的一个重要方面：这不仅仅是关于在一个区间内定义一个函数，还涉及到函数在该区间内的行为。因此，理解函数的性质——例如有界性和连续性——对于确定它们的integrability至关重要。另一个关于integrability的有趣视角来自勒贝格积分，它将概念推广到了超越黎曼积分的范围。勒贝格的方法允许我们通过关注函数取某些值的集合的“大小”来积分更广泛的函数类，而不仅仅是将定义域划分为区间。这种方法表明，某些函数即使在黎曼意义上不可积，也可以在勒贝格意义上是可积的。因此，研究integrability打开了新的分析途径，并提供了对函数本质的更深刻见解。在实际应用中，integrability的概念在各个领域都是至关重要的。例如，在物理学中，由力所做的功的计算涉及到在距离上对力函数进行积分。类似地，在经济学中，消费者剩余和生产者剩余是通过依赖于需求和供给函数的积分来确定的。因此，扎实理解integrability为个人提供了解决现实问题所需的工具。总之，integrability是数学分析的基石，远远超出了课堂的范畴。它概括了对函数进行积分的能力，为理解累积和曲线下的面积提供了框架。无论是通过黎曼还是勒贝格积分，integrability的概念都为我们提供了对函数行为及其在各个学科中的应用的深刻见解。因此，掌握integrability不仅对学业成功至关重要，而且对解决众多领域的实际问题也是如此。