surface integral

简明释义

面积分

英英释义

例句

1.We learned how to compute a surface integral 表面积分 in our calculus class last week.

我们上周在微积分课上学习了如何计算表面积分.

2.In physics, the surface integral 表面积分 is often used to find the total charge over a charged surface.

在物理学中，表面积分常用于计算带电表面上的总电荷.

3.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评估表面积分.

4.The concept of surface integral 表面积分 is crucial in fluid dynamics for determining flow across surfaces.

在流体动力学中，表面积分的概念对于确定流体穿过表面是至关重要的.

5.To calculate the flux of a vector field across a surface, we need to evaluate the surface integral 表面积分.

为了计算一个向量场穿过一个表面的通量，我们需要评估这个表面积分.

作文

In the field of mathematics, particularly in calculus, the concept of a surface integral plays a crucial role in understanding multidimensional spaces. A surface integral allows us to extend the idea of an integral from one-dimensional curves to two-dimensional surfaces. This is particularly useful in physics and engineering, where we often deal with quantities that are distributed over surfaces rather than along lines. For instance, when calculating the total mass of a thin sheet of material, we can use a surface integral to sum up the mass density across the entire surface of the sheet.To understand how a surface integral works, let’s first revisit the concept of a regular integral. In one dimension, the integral of a function gives us the area under the curve of that function between two points. Similarly, in two dimensions, a surface integral measures the accumulation of a quantity over a surface in three-dimensional space. The mathematical formulation involves parameterizing the surface and then integrating a given function over this parameterization.For example, consider a simple case where we have a surface defined by a function z = f(x, y). To compute the surface integral of a scalar field over this surface, we would express the integral as follows: ∬_S f(x, y) dS, where S represents the surface and dS is the differential area element on the surface. The process typically involves calculating the Jacobian determinant to account for the stretching of area elements when mapping from the xy-plane to the surface.In physics, the application of surface integrals is extensive. For instance, in electromagnetism, Gauss's law relates the electric flux through a closed surface to the charge enclosed within that surface. This is expressed mathematically using a surface integral of the electric field over the surface. Here, the integral helps to quantify how much electric field lines pass through a given surface, which is fundamental in understanding electric fields and forces.Moreover, the concept of surface integrals is not limited to scalar fields; it also extends to vector fields. When dealing with vector fields, we often compute what is called a flux integral, which measures the flow of the vector field through a surface. This is particularly important in fluid dynamics and thermodynamics, where we need to analyze how fluids or heat transfer occurs through surfaces.In conclusion, the surface integral is a powerful mathematical tool that enables us to analyze and compute quantities over surfaces in multidimensional spaces. Its applications are widespread across various scientific disciplines, providing invaluable insights into the behavior of physical systems. By mastering the concept of surface integrals, students and professionals alike can enhance their understanding of complex phenomena in mathematics, physics, and engineering. As we continue to explore higher dimensions and intricate surfaces, the importance of surface integrals will undoubtedly grow, making it a fundamental topic in advanced studies of calculus and beyond.

在数学领域，特别是在微积分中，曲面积分的概念在理解多维空间方面起着至关重要的作用。曲面积分使我们能够将积分的概念从一维曲线扩展到二维表面。这在物理和工程中尤其有用，因为我们经常处理分布在表面上的量，而不是沿着线。例如，在计算薄材料片的总质量时，我们可以使用曲面积分来汇总整个表面上的质量密度。要理解曲面积分的工作原理，首先让我们回顾一下常规积分的概念。在一维中，函数的积分给我们提供了该函数在两个点之间的曲线下的面积。类似地，在二维中，曲面积分测量了三维空间中某个量在表面上的累积。其数学公式涉及对表面的参数化，然后在这个参数化上对给定函数进行积分。例如，考虑一个由函数z = f(x, y)定义的简单情况。为了计算标量场在该表面上的曲面积分，我们会将积分表示如下：∬_S f(x, y) dS，其中S代表表面，dS是表面上的微分面积元素。这个过程通常涉及计算雅可比行列式，以考虑在从xy平面映射到表面时面积元素的拉伸。在物理学中，曲面积分的应用非常广泛。例如，在电磁学中，高斯定律将通过封闭表面的电通量与该表面内的电荷联系起来。这通过对表面上的电场进行曲面积分来表达。在这里，积分有助于量化有多少电场线穿过给定表面，这对于理解电场和力是基础。此外，曲面积分的概念不仅限于标量场；它还扩展到矢量场。在处理矢量场时，我们通常计算称为通量积分的内容，这测量矢量场通过表面的流动。这在流体动力学和热力学中尤为重要，因为我们需要分析流体或热量如何通过表面转移。总之，曲面积分是一个强大的数学工具，使我们能够分析和计算多维空间中表面的量。其应用在各个科学领域广泛存在，为理解物理系统的行为提供了宝贵的见解。通过掌握曲面积分的概念，学生和专业人士都可以增强对数学、物理和工程中复杂现象的理解。随着我们继续探索更高维和复杂表面，曲面积分的重要性无疑会增长，使其成为微积分及其后续高级研究中的基本主题。