格林公式的一个应用

运用格林公式给出平面上任意多边形的面积公式和重心坐标公式。     

反函数的原函数公式及其应用

在本文第一部分给出了反函数原函数公式一个简单的证明，第二部分列举了几个用反函的原函数公式求积的例子，最后用此公式给出了Ｙｏｕｎｇ不等式一个证明，此证明不必借助于定积分的几何意义和图形面积的几何直观，从而是数学意义上更为严格的证明。     

计算旋转体体积的一般积分公式

0引言本文首先讨论了平面曲线在直线上的投影长函数 ,平面曲线 (图形 )绕一共面直线旋转所得旋转体的体积函数 ,给出了它们的积分表示式 ,进而得出计算旋转体体积的一般积分公式。关于旋转体体积的计算问题 ,一般标准分析教材 [1,2 ] 中只讨论了平面图形绕坐标轴旋转所得旋转体的体积的积分公式 ,为了应用上的便利本文将其推广 ,给出平面图形绕任一共面直线旋转所得旋转体体积计算的一般积分公式。一般认为平面曲线是 (开 )直线段到平面内的一一的 ,双方连续的 ,在上映射的象[3] .在直线段a≤ t≤ b上引入坐标 t,在平面上引入笛卡尔直角坐标…     

类比梯形推导棱台体积公式

类比,在数学学习中起到至关重要的作用,不仅一些结论可以通过类比得到,而且在方法上也可以通过类比．在推导棱台体积公式时,通过降维变成平面图形——梯形,先给出梯形面积公式的两种证法,而后将这两种方法类比应用到棱台上求体积,实现问题的圆满解决．     

用祖暅原理求平面图形面积的探索

我们知道：能用初等方法求出面积的平面图形有圆与多边形（可分割成三角形来求）,除此以外就屈指可数了．本文与同学们一起探索用祖暅 原理求平面图形的面积,直到推导出椭圆的面积公式,一起来吧,其乐无穷。     

平面图形的形心在旋转体体积计算中的应用

本文讨论平面图形绕平面内的直线旋转的旋转体体积与被旋转的平面图形的形心的关系.当被旋转的平面图形的内部与旋转轴没有交点时,得到了用被旋转的平面图形的面积以及被旋转的平面图形的形心到旋转轴的距离表示的旋转体体积公式.     

用面积法高效解题

用面积法解题是根据题目给出的条件,利用等积变换原理和有关面积计算的公式、定理或图形的面积关系进行解题的方法.所谓高效解题是"走解题的直线距离",说白了,就是将转化的环节减少一些,少走弯路.     

格林公式的几何意义

我们知道格林公式指出了二重积分和曲线积分的联系。本文的目的是要讨论格林公式的几何意义。格林公式:设P(x,y),Q(x,y)以及偏导数(?)P/(?)y和(?)Q/(?)x是闭单连通区域(D)上的连续函数,(D)的边界(L)是光滑的简单曲线,那末有从格林公式的证明知道,不需要增加条件,下式亦成立:和以下就(1)式来讨论几何意义,对于(2)式是类似的。我们假定读者已经知道什么是平面图形的面积。现在给出一个判别平面图形是否有面积的方法:     

射影公式的一种推广

由于用几何方法解决一般立体图形的夹角（除异面直线所成的角外）与距离等问题射影公式起着关键作用．但是用射影公式必须知道射影的位置,对于一般图形,直接用射影公式难以定位,本文给出射影公式的一个推论,即可解决立体图形的各种夹角与距离等问题．     

一个基本图形，两个二面角公式

一个基本图形，两个二面角公式３５１１００福建省莆田中山中学陈天雄《立体几何》（必修）Ｐ．２７图１—３２，亦即Ｐ１１７题３的图形（如图１），是一个具有广泛应用的重要的基本图形，前者揭示斜线与平面所成的角的最小性，后者给出了线线角、线面角的关系式：（其中...     

Cubature Formula for Spherical Basis Function Networks

Some mathematical models in geophysics and graphic processing need to compute integrals with scattered data on the sphere. Thus cubature formula plays an important role in computing these spherical integrals. This paper is devoted to establishing an exact positive cubature formula for spherical basis function networks. The authors give an existence proof of the exact positive cubature formula for spherical basis function networks, and prove that the cubature points needed in the cubature formula are not larger than the number of the scattered data.     

Handling the difficulties of technical school students in the construction and interpretation of graphic representations

In this work, an attempt is made to evaluate the errors that have to do with the interpretation and construction of graphic representations. Although the students are studying in the second year of technical high school (secondary education), i.e. in schools with an emphasis in technical subjects (post junior secondary), it is observed that they find difficult to handle the graphic representations. This is an attempt to overcome the difficulties that exist by putting the students to work on electrical machines. They read, construct or verify the graphic representations of the electrical machines using electrical instruments with which they carry out measurements and take values. It is evaluated after the end of each activity and also after a suitable interval of time that there is an improvement in the students’ abilities (interpretation and construction) in graphic representations.     

n元椭球面的判定及所围椭球体的体积

给出n元二次曲面是椭球面的充要条件和所对应的n维椭球体的体积计算公式,并且条件和体积的计算中只用到曲面中的系数行列式,使判定和体积计算较为方便.     

基于谱密度评价随机性

为了评价一个平稳过程的随机性, 我们基于谱密度提出了一个图方法. 当图中的散点呈现线性关系的时候, 我们可以判定这个序列是随机的. 为了说明这个思想, 我们用模拟的办法来检验伪随机数的随机性. 另外, 我们也用了一个实际数据来考察数据的相关性. 这两个例子都说明了我们的图方法是非常有效的.     

Systematic approaches to experimentation: The case of Pick's theorem

In this paper two 10th graders having an accumulated experience on problem-solving ancillary to the concept of area confronted the task to find Pick's formula for a lattice polygon's area. The formula was omitted from the theorem in order for the students to read the theorem as a problem to be solved. Their working is examined and emphasis is given to highlighting the students’ range of systematic approaches to experimentation in the context of problem solving and aspects of control that are reflected in these approaches.     

An analysis of single-index model with monotonic link function

The single-index model with monotonic link function is investigated. Firstly, it is showed that the link function h（.） can be viewed by a graphic method. That is, the plot with the fitted response y on the horizontal axis and the observed y on the vertical axis can be used to visualize the link function. It is pointed out that this graphic approach is also applicable even when the link function is not monotonic. Note that many existing nonparametric smoothers can also be used to assess h（.）. Therefore, the I-spline approximation of the link function via maximizing the covariance function with a penalty function is investigated in the present work. The consistency of the criterion is constructed. A small simulation is carried out to evidence the efficiency of the approach proposed in the paper.     

The role of the graphic calculator in mediating graphing activity

The PIGMI (Portable Information Technologies for supporting Graphical Mathematics Investigations) Project ¹ investigated the role of portable technologies in facilitating development of students' graphing skills and concepts. This paper examines the impact of a recent shift towards calculating and computing tools as increasingly accessible, everyday technologies on the nature of learning in a traditionally difficult curriculum area. The paper focuses on the use of graphic calculators by undergraduates taking an innovative new mathematics course at the Open University. A questionnaire survey of both students and tutors was employed to investigate perceptions of the graphic calculator and the features which facilitate graphing and linking between representations. Key features included visualization of functions, immediate feedback and rapid graph plotting. A follow-up observational case study of a pair of students illustrated how the calculator can shape mathematical activity, serving a catalytic, facilitating and checking role. The features of technology-based activities which can structure and support collaborative problem solving were also examined. In sum, the graphic calculator technology acted as a critical mediator in both the students' collaboration and in their problem solving. The pedagogic implications of using portables are considered, including the tension between using and over-using portables to support mathematical activity.     

A mathematical model for the calculation of ceramic batch formulas

In the ceramic field the process of forming a ceramic body consists of several steps that must be executed efficiently and controlled closely to achieve the desired product. A major step in the process is the selection of raw materials, which not only will provide the necessary oxides, but also will melt and fine well to form a high-quality product. This process, if carried out on a trial-and-error manner, may lead to an uneconomical production cost due to wrong materials being selected. In addition, some important physical batch characteristics may be ignored owing to the increase in computational requirements if more constraints are added to the process. In this paper we propose the mathematical modelling approach, which has gained very little attention from researchers in this field, as an alternative to the trial-and-error approach. The model presented in this paper may be used to calculate the batch formula of a designed product without first knowing its empirical formula. An illustrated example of the use of this model is also provided.     

Minimal Graphic Functions on Manifolds of Nonnegative Ricci Curvature

We study minimal graphic functions on complete Riemannian manifolds ∑ with nonnegative Ricci curvature, euclidean volume growth, and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville‐type theorem with such growth via splitting for tangent cones of ∑ at infinity. When, in contrast, we do not impose any growth restrictions for minimal graphic functions, we also obtain a Liouville‐type theorem under a certain nonradial Ricci curvature decay condition on ∑. In particular, the borderline for the Ricci curvature decay is sharp by our example in the last section. © 2015 Wiley Periodicals, Inc.     

A Generalization of Intersection Formulae of Integral Geometry

We establish extensions of the Crofton formula and, under some restrictions, of the principal kinematic formula of integral geometry from curvature measures to generalized curvature measures of convex bodies. We also treat versions for finite unions of convex bodies. As a consequence, we get a new intuitive interpretation of the area measures of Aleksandrov and Fenchel–Jessen.