球面坐标系中三重积分体积微元的分析

三重积分是高等数学教学中的一个重要知识点,特别是球面坐标系下的三重积分的计算.球面坐标系下三重积分的一个教学难点是如何清晰直观的理解体积微元.不同于坐标变换这种抽象的理论分析方式,从体积增量的角度出发,并通过MATLAB绘图,直观形象地给出球面坐标系下三重积分体积微元公式的分析过程.     

球坐标系中空间区域分类探讨与应用实例分析

针对球坐标系中三重积分计算的教学与学习过程中出现的有关积分区域确定的问题,定义了球坐标系中的空间区域的类型,分析了一类球坐标系中简单区域的图形特征,结合图形演示,给出了构建其不等式描述形式的过程与方法.针对不符合球坐标系中简单区域图形特征的立体区域,就如何通过有效分割,借助三重积分对积分区域的可加性完成计算过程进行了探讨和实例分析.     

一道竞赛题的三重积分的截面法

本文以一道三重积分的竞赛试题为例,对某些三重积分的计算采用截面法,计算简洁,方法灵活.     

三重积分的计算方法探析

本文通过两个例子说明如何选取合适的方法计算三重积分,帮助学生理解并掌握三重积分的计算,有效提高学生的数学思维能力.     

基于球坐标计算第一型曲面积分注记

给出了利用球坐标求解第一型曲面积分的方法及在球坐标系下第一型曲面积分转化为二重积分的公式,实例表明这种方法是可行的.此研究丰富了第一型曲面积分的计算方法,也可为在高校《高等数学》课程教学中对学生能力的培养提供素材.     

以二重积分为基础,学习三重积分

作三重积分的题，非但积分域不好画，而且，在画出积分域后，化作累次积分定限时还很容易作错。本文抓住三重积分与二重积分的联系，给读者介绍避免用立体图形作三重积分问题的方法，并进一步揭示三重积分在直角坐标系、往面坐标系和球面坐标系下计算方法的关系，我们从下面例1谈起：例1设f（x，y，z）连续，试将三次积分：化为先对此次对x、后对Z的三次积分。解：对于固定的XE（o，1），我们来研究二次积分：（见图1，这个二重积分的积分域是图中阴影部分所示的平面域风）。因此三次积分下一步，只要交换变量Z和Z的积分顺序，依旧采用二…     

轮换对称性在积分中的应用

在某些积分的计算过程中,若积分区域具备轮换对称性,则可以简化积分的计算过程.本文讨论了利用轮换对称性简化二重积分,三重积分,第一,二类曲线积分,第一,二类曲面积分的计算方法.(以下都在积分存在下予以讨论)     

由一道例题透析三重积分的解法

本文通过一道例题的多种解法透析三重积分计算中常用的截面法、投影法、利用柱坐标、球坐标和高斯公式求解三重积分.     

极坐标系中平面区域的分类与实例分析

针对极坐标系中二重积分计算教学与学习过程中出现的疑惑与问题,给出了极坐标系中积分区域分类的定义和图形展示,并就如何将复杂区域分割成极坐标系中的简单区域和构建相应简单区域的不等式描述形式的方法进行了分析,并借助具体实例对方法进行了验证.     

“截面法”在三重积分计算中的应用

众所周知,在三重积分的计算中,我们一般是将三重积分化为三次积分来计算.关于用“截面法”来计算三重积分,在一般高等数学教材中提及较少.实际上在某些情况下,利用“截面法”可将三重积分化为定积分来计算,使计算过程大大简化.     

一种新S曲线的建议

对于一元非线性回归计算,各种著作中均给出不同的线型及其相应的公式和线性化后的方程,供试验研究和工程、非工程计算中使用.S曲线的使用较为广泛,但它是一条不通过坐标原点的曲线,这与实际有出入,且尚欠合理.作者建议一条通过原点的S曲线,使之更符合实际.     

The Weierstrass—Enneper representation using hodographic coordinates on a minimal surface

In this paper we obtain the general solution to the minimal surface equation, namely its local Weierstrass-Enneper representation, using a system of hodographic coordinates. This is done by using the method of solving the Born-Infeld equations by Whitham. We directly compute conformal coordinates on the minimal surface which give the Weierstrass-Enneper representation. From this we derive the hodographic coordinate ρ∈ D ⊂ ℂ and σ its complex conjugate which enables us to write the Weierstrass-Enneper representation in a new way.     

Barycentric coordinates for convex sets

In this paper we provide an extension of barycentric coordinates from simplices to arbitrary convex sets. Barycentric coordinates over convex 2D polygons have found numerous applications in various fields as they allow smooth interpolation of data located on vertices. However, no explicit formulation valid for arbitrary convex polytopes has been proposed to extend this interpolation in higher dimensions. Moreover, there has been no attempt to extend these functions into the continuous domain, where barycentric coordinates are related to Green’s functions and construct functions that satisfy a boundary value problem. First, we review the properties and construction of barycentric coordinates in the discrete domain for convex polytopes. Next, we show how these concepts extend into the continuous domain to yield barycentric coordinates for continuous functions. We then provide a proof that our functions satisfy all the desirable properties of barycentric coordinates in arbitrary dimensions. Finally, we provide an example of constructing such barycentric functions over regions bounded by parametric curves and show how they can be used to perform freeform deformations.      

On some properties of coordinates in polynomial rings

In this paper, we study some properties of coordinates in the polynomial ring by introducing some concepts which are weaker than coordinates. In particular, in the polynomial ring in two variables over an algebraically closed field of characteristic zero, we show some relations between polynomials and their fibers on 𝔸².     

Node-pair finite volume/finite element schemes for the Euler equation in cylindrical and spherical coordinates

A numerical scheme is presented for the solution of the compressible Euler equations in both cylindrical and spherical coordinates. The unstructured grid solver is based on a mixed finite volume/finite element approach. Equivalence conditions linking the node-centered finite volume and the linear Lagrangian finite element scheme over unstructured grids are reported and used to devise a common framework for solving the discrete Euler equations in both the cylindrical and the spherical reference systems. Numerical simulations are presented for the explosion and implosion problems with spherical symmetry, which are solved in both the axial–radial cylindrical coordinates and the radial–azimuthal spherical coordinates. Numerical results are found to be in good agreement with one-dimensional simulations over a fine mesh.     

A general construction of barycentric coordinates over convex polygons

Barycentric coordinates are unique for triangles, but there are many possible generalizations to convex polygons. In this paper we derive sharp upper and lower bounds on all barycentric coordinates over convex polygons and use them to show that all such coordinates have the same continuous extension to the boundary. We then present a general approach for constructing such coordinates and use it to show that the Wachspress, mean value, and discrete harmonic coordinates all belong to a unifying one-parameter family of smooth three-point coordinates. We show that the only members of this family that are positive, and therefore barycentric, are the Wachspress and mean value ones. However, our general approach allows us to construct several sets of smooth five-point coordinates, which are positive and therefore barycentric. Dedicated to Charles A. Micchelli on his 60th Birthday Mathematics subject classifications (2000) 26C15, 65D05.     

关于型Ⅲ对称空间的热核构造

引进了第Ⅲ类典型城上的内切超圆坐标;计算出了矩阵极坐标下相应于不变度量的体积元素;利用积分变换构造出了不变度量的Ｌａｐｌａｃｅ－Ｂｅｌｔｒａｍｉ算子的热核．     

平行映射下保测地线的曲面

利用古典微分几何方法,讨论了在平行映射下测地线不变的曲面若干局部性质;研究了该类曲面的测地坐标系,并最终得出满足该条件的曲面只有三类.