exact differential

简明释义

恰当微分

英英释义

例句

1.In calculus, an exact differential is a differential that can be expressed as the derivative of a function.

在微积分中，exact differential是指可以表示为某个函数的导数的微分。

2.When solving differential equations, identifying an exact differential can simplify the process significantly.

在求解微分方程时，识别出exact differential可以显著简化过程。

3.The exact differential of a function indicates that the function has a well-defined integral.

一个函数的exact differential表明该函数具有良定义的积分。

4.To determine if a differential is exact differential, check if the mixed partial derivatives are equal.

要确定一个微分是否为exact differential，请检查混合偏导数是否相等。

5.The equation has an exact differential form, allowing us to find a potential function easily.

这个方程具有exact differential形式，使我们能够轻松找到一个势函数。

作文

In the field of mathematics, particularly in calculus, the concept of an exact differential plays a crucial role in understanding the behavior of functions and their integrals. An exact differential refers to a specific type of differential expression that indicates a function's total change over a certain path or region. To grasp the significance of this term, it is essential to explore its definition, applications, and implications in various mathematical contexts.An exact differential can be defined as a differential form that is the differential of some scalar function. This means that if we have a function f(x, y), then the differential df can be expressed in terms of its partial derivatives with respect to x and y. Specifically, for a function to have an exact differential, it must satisfy the condition that mixed partial derivatives are equal; that is, ∂²f/∂x∂y = ∂²f/∂y∂x. This property ensures that the integral of the differential is path-independent, meaning that the value of the integral between two points does not depend on the path taken.The importance of exact differentials extends beyond pure mathematics; it also finds applications in physics and engineering. For instance, in thermodynamics, the concept of an exact differential is used to describe state functions, such as internal energy and enthalpy. These state functions are dependent only on the current state of the system and not on how the system reached that state. In contrast, quantities like heat and work are not state functions and are associated with process paths, making them inexact differentials.Understanding exact differentials helps mathematicians and scientists identify conservative fields, which are vector fields where the line integral between two points is independent of the path taken. A classic example is the gravitational field near the Earth's surface. In such fields, the work done by gravity on an object moving from one height to another depends solely on the initial and final heights, exemplifying the concept of an exact differential.Moreover, the study of exact differentials leads to the development of important mathematical tools, such as potential functions and exact equations. When dealing with first-order differential equations, if the equation can be expressed in the form of an exact differential, it can often be solved more easily by finding the potential function whose differential equals the given expression. This approach simplifies the process of solving complex problems, making it a valuable technique in both theoretical and applied mathematics.In conclusion, the concept of an exact differential is fundamental in calculus and has far-reaching implications in various scientific disciplines. By recognizing the conditions under which a differential is considered exact, one can gain deeper insights into the nature of functions, integrals, and physical phenomena. The relationship between exact differentials and conservative fields highlights the interconnectedness of mathematics and the natural world, showcasing how abstract concepts can have practical applications. As students and professionals continue to explore these ideas, the significance of exact differentials will undoubtedly remain a cornerstone of mathematical education and research.

在数学领域，特别是微积分中，‘exact differential’的概念在理解函数及其积分的行为方面起着至关重要的作用。‘exact differential’指的是一种特定类型的微分表达式，表示某个函数在特定路径或区域上的总变化。要掌握这个术语的重要性，必须探讨其定义、应用和在各种数学背景下的意义。‘exact differential’可以定义为某个标量函数的微分形式。这意味着如果我们有一个函数f(x, y)，那么微分df可以用其对x和y的偏导数来表示。具体而言，为了使一个函数具有‘exact differential’，它必须满足混合偏导数相等的条件；即∂²f/∂x∂y = ∂²f/∂y∂x。这个性质确保了微分的积分与路径无关，这意味着两个点之间的积分值不依赖于所采取的路径。‘exact differentials’的重要性超越了纯数学，它在物理学和工程学中也找到了应用。例如，在热力学中，‘exact differential’的概念用于描述状态函数，如内能和焓。这些状态函数仅依赖于系统的当前状态，而与系统如何达到该状态无关。相比之下，像热量和功这样的量不是状态函数，与过程路径相关，因此被认为是非精确微分。理解‘exact differentials’帮助数学家和科学家识别保守场，即在这些向量场中，从一点到另一点的线积分与所采取的路径无关。一个经典的例子是地球表面附近的重力场。在这样的场中，重力对物体从一个高度移动到另一个高度所做的功仅依赖于初始和最终高度，体现了‘exact differential’的概念。此外，‘exact differentials’的研究导致了重要数学工具的发展，如势函数和精确方程。当处理一阶微分方程时，如果方程可以表示为‘exact differential’的形式，通常可以通过找到其微分等于给定表达式的势函数来更容易地解决。这种方法简化了复杂问题的解决过程，使其成为理论和应用数学中的一种宝贵技术。总之，‘exact differential’的概念在微积分中是基础性的，并在各种科学学科中具有深远的影响。通过认识到微分被认为是精确的条件，人们可以更深入地了解函数、积分和物理现象的本质。‘exact differentials’与保守场之间的关系突显了数学与自然界的相互联系，展示了抽象概念如何具有实际应用。随着学生和专业人士继续探索这些思想，‘exact differentials’的重要性无疑将继续成为数学教育和研究的基石。