第二类曲面积分的计算方法

利用两类曲面积分的联系、分面投影法、合一投影法和高斯公式解答一个第二类曲面积分的题目。     

Stokes公式的一个注记

利用Stokes公式证明了一个对满足散度为零的向量场的第二型曲面积分可化为其边界封闭曲线的第二型曲线积分来计算的定理.该定理对于满足上述条件向量场的曲面积分,给出了具体转化为曲线积分进行计算的公式,最后利用该公式计算了一个例子.     

第二类曲线(面)积分的向量学习法再探

以通量概念引入第二类曲面积分、以环流量概念引入第二类曲线积分,并用向量形式表达高斯公式、斯托克斯公式等关系,以期达到第二类曲线(面)积分部分的知识点符号表达简明、计算和公式容易记忆的目的.     

用向量函数证明积分换元公式

运用二元向量函数和三元向量函数证明二重积分和三重积分的换元公式,推导曲面的表面积公式,并计算第二类曲面积分.     

第一类曲线积分的计算方法探讨

对第一类曲线积分的计算技巧及方法进行探讨,指出计算时可以把积分曲线方程代入到被积函数中,也可以利用对称性及轮换对称性简化计算,并提出可以利用公式法及均匀曲线型构件的质心公式来计算.     

第一类曲面积分的新方法

从一个具体的物理问题入手,探讨了如何将第一类曲面积分转化为两个第一类曲线积分的累次形式,从而给出了这两类积分之间的关系,并通过举例说明该公式可以用来直观简便地计算第一类曲面积分.     

巧用两类曲面积分之间的联系计算第二类曲面积分

第二类曲面积分是数学分析课程中的重点,也是难点.本文主要介绍利用两类曲面积分之间的联系计算第二类曲面积分,为初学者求解这类问题提供一种思路和方法.     

一类曲面积分的简单计算与推广

第一类曲面积分的积分表达式具有如下特点时:(1)积分曲面是可求曲面面积的曲面;(2)被积函数是单变量函数或可化为单变量函数的函数,利用积分元素法,能将其直接化为定积分计算,这种简单的算法还可以推广到计算具有类似特征的三重积分.     

关于对坐标的曲面积分的一种算法

在高等数学中,计算对坐标的曲面积分时,要把积分曲面投影到坐标面上.由于积分曲面是有向的,且被积函数要在积分曲面上取值,所以计算起来比较困难.特别,对坐标y、z;z、x;x、y的三种曲面积分都在积分表达式中时,运算将会更困难些,下面介绍一个计算公式,利用此公式,有时可以使计算简便.     

第二类曲面积分的计算方法

通过一道例题的讲解,介绍了第二类曲面积分的四种计算方法,使学生对第二类曲面积分的计算有更深的理解和掌握.     

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.     

Three-dimensional analog of the Cauchy type integral

On a smooth closed surface, we consider integrals of the Cauchy type with kernel depending on the difference of arguments. They cover both double-layer potentials for second-order elliptic equations and generalized integrals of the Cauchy type for first-order elliptic systems. For the functions described by such integrals, we find sufficient conditions providing their continuity up to the boundary surface. We obtain the corresponding formulas for their limit values.     

On some integrals of Glauert’s type

Some principal value integrals of Glauert’s type are calculated analytically. These integrals appear when dealing with the hydrodynamic impact problem of a two-dimensional wedge onto a liquid free surface in the potential theory.     

第一类曲面积分的一种解法

运用第一类曲线积分方法解决一类特殊的第一类曲面积分问题,并举例说明此方法的简便性.     

第一类曲面积分的一类特殊解法

对一些特殊曲面(如柱面、旋转曲面)上的第一类曲面积分,转化为二重积分的基本方法可能非常复杂,用类似计算三重积分的平行截面法能更简洁地解决此类问题.     

Semi-analytic treatment of the three-dimensional Poisson equation via a Galerkin BIE method

A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach.     

Semi-analytic treatment of the three-dimensional Poisson equation via a Galerkin BIE method

A systematic treatment of the three-dimensional Poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a Galerkin approximation. Developed to conveniently deal with domain integrals without a volume-fitted mesh, the proposed method initially converts domain integrals featuring the Newton potential and its gradient into equivalent surface integrals. Then, the resulting boundary integrals are evaluated by means of well-established cubature methods. In this transformation, weakly-singular domain integrals, defined over simply- or multiply-connected domains with Lipschitz boundaries, are rigorously converted into weakly-singular surface integrals. Combined with the semi-analytic integration approach developed for potential problems to accurately calculate singular and hypersingular Galerkin surface integrals, this technique can be employed to effectively deal with mixed boundary-value problems without the need to partition the underlying domain into volume cells. Sample problems are included to validate the proposed approach.     

A Two-dimensional Contact Problem with Given Region of Adhesion

The two-dimensional indentation of an elastic half-plane bya rigid punch under normal and tangential load is considered.The contact area is divided into an inner region with adhesion,the dimension of which is known beforehand, surrounded by tworegions in which inward slip takes place. The problem is formulatedin terms of a coupled pair of Cauchy type integrals for thenormal and shear stresses at the surface of the half-plane.In the case of friction-free slip these integrals are combinedto an inhomogeneous Fredholm equation which is solved by a methodof successive approximations. In the case when the inward slipis governed by Coulomb friction, the problem is solved by anumerical method.     

Indefinite integrals of products of special functions

A method is given for deriving indefinite integrals involving squares and other products of functions which are solutions of second-order linear differential equations. Several variations of the method are presented, which applies directly to functions which obey homogeneous differential equations. However, functions which obey inhomogeneous equations can be incorporated into the products and examples are given of integrals involving products of Bessel functions combined with Lommel, Anger and Weber functions. Many new integrals are derived for a selection of special functions, including Bessel functions, associated Legendre functions, and elliptic integrals. A number of integrals of products of Gauss hypergeometric functions are also presented, which seem to be the first integrals of this type. All results presented have been numerically checked with Mathematica.     

一类中值问题的另证

通过考察微分中值定理中的中值点的性质,利用积分中值定理,得到一类有关定积分的中值问题的一种新证明．