surface integral

简明释义

面积分

英英释义

例句

1.In physics, the surface integral 表面积分 is crucial for determining the total electric field over a charged surface.

在物理学中，表面积分对于确定带电表面上的总电场至关重要。

2.The surface integral 表面积分 can be evaluated using parametrization of the surface.

可以通过对表面进行参数化来计算表面积分。

3.When dealing with fluid dynamics, the surface integral 表面积分 helps in calculating the flow rate through a surface.

在流体动力学中，表面积分有助于计算通过某个表面的流量。

4.In multivariable calculus, understanding the surface integral 表面积分 is essential for advanced applications.

在多变量微积分中，理解表面积分对于高级应用至关重要。

5.To calculate the flux of a vector field across a surface, we often use the concept of a surface integral.

为了计算一个向量场穿过一个表面的通量，我们通常使用表面积分的概念。

作文

In the field of mathematics, particularly in calculus, the concept of a surface integral plays a crucial role in understanding how functions behave over surfaces rather than just along curves. A surface integral is essentially an extension of the idea of a line integral to higher dimensions. It allows us to compute the integral of a function over a two-dimensional surface in three-dimensional space. This is particularly useful in physics and engineering, where we often need to calculate quantities like mass, charge, or fluid flow across a surface.To grasp the significance of a surface integral, let's consider an example involving the calculation of the total mass of a thin sheet of material that has a varying density across its surface. If we denote the density function as ρ(x, y), which varies with the coordinates on the surface, the total mass can be found by integrating this density function over the surface area of the sheet. The mathematical expression for this would be:M = ∬_S ρ(x, y) dS,where M represents the total mass, S is the surface over which we are integrating, and dS is an infinitesimal element of surface area. Here, the surface integral allows us to accumulate the mass contributions from every point on the surface.The computation of a surface integral involves parameterizing the surface, which means expressing the surface in terms of two parameters, typically denoted as u and v. For instance, if we have a surface defined by a function z = f(x, y), we can express it using parameters u and v, where x = u and y = v. The surface integral can then be evaluated using these parameters, transforming our original function into a form that is easier to integrate.One of the important applications of surface integrals is in physics, particularly in the study of electromagnetism. For example, when calculating the electric flux through a surface, we use a surface integral to integrate the electric field vector across the surface area. According to Gauss's law, the electric flux Φ through a closed surface is given by:Φ = ∬_S E · dS,where E is the electric field vector, and dS is the outward-pointing area vector of the surface. This application highlights how surface integrals bridge the gap between mathematical theory and practical physical phenomena.Moreover, surface integrals are also essential in fluid dynamics, where they are used to calculate the flow of fluids across surfaces. By integrating the velocity field of the fluid over a specified surface, we can determine the total flow rate across that surface, which is vital for engineering applications such as designing pipelines or analyzing airflow over wings.In conclusion, the surface integral is a powerful tool in both mathematics and applied sciences. It allows us to extend our understanding of integrals to two-dimensional surfaces, providing insights into various physical phenomena. Mastering the concept of surface integrals not only enhances our mathematical skills but also equips us with the necessary tools to tackle real-world problems in fields like physics and engineering. As we continue to explore the depths of calculus and its applications, the surface integral remains a fundamental concept that connects abstract mathematics with tangible realities.

在数学领域，特别是在微积分中，曲面积分的概念在理解函数在表面上的行为而不仅仅是沿着曲线的重要性上发挥着至关重要的作用。曲面积分本质上是将线积分的概念扩展到更高维度的一种方式。它使我们能够计算一个函数在三维空间中的二维表面上的积分。这在物理和工程中尤为有用，因为我们经常需要计算诸如质量、电荷或流体流动等穿过表面的量。为了理解曲面积分的重要性，让我们考虑一个涉及计算具有变化密度的薄材料片的总质量的例子。如果我们将密度函数表示为ρ(x, y)，它随表面上的坐标变化，总质量可以通过在表面积上对该密度函数进行积分来找到。这个数学表达式为：M = ∬_S ρ(x, y) dS，其中M代表总质量，S是我们正在积分的表面，dS是无穷小的表面积元素。在这里，曲面积分使我们能够累积来自表面每个点的质量贡献。计算曲面积分涉及对表面进行参数化，这意味着用两个参数（通常表示为u和v）来表示表面。例如，如果我们有一个由函数z = f(x, y)定义的表面，我们可以使用参数u和v来表达它，其中x = u和y = v。然后，可以使用这些参数来评估曲面积分，将原始函数转换为更易于积分的形式。曲面积分的一个重要应用是在物理学中，特别是在电磁学的研究中。例如，在计算通过表面的电通量时，我们使用曲面积分来将电场向量在表面积上进行积分。根据高斯定律，通过封闭表面的电通量Φ由以下公式给出：Φ = ∬_S E · dS，其中E是电场向量，dS是表面的外指面积向量。这个应用突显了曲面积分如何弥合数学理论与实际物理现象之间的差距。此外，曲面积分在流体动力学中也至关重要，它们被用来计算流体穿过表面的流动。通过对流体的速度场在指定表面上的积分，我们可以确定穿过该表面的总流量，这对于工程应用如设计管道或分析机翼上的气流至关重要。总之，曲面积分是数学和应用科学中的一种强大工具。它使我们能够将积分的理解扩展到二维表面，为各种物理现象提供了洞察。掌握曲面积分的概念不仅增强了我们的数学技能，还使我们具备了应对物理和工程等领域现实问题所需的工具。在我们继续探索微积分及其应用的深度时，曲面积分仍然是一个将抽象数学与具体现实连接起来的基本概念。