高等数学视角下的弧微分公式推导及曲率公式适用条件

在利用弧微分公式推导曲率公式时,针对弧微分公式证明过程中“弧长和弦长的比值极限为1”假设不严谨的问题,本文在高等数学知识体系内,利用曲线直角坐标方程得到了弧长与有向弧段的值之间的关系,提出了一种证明弧微分公式的方法.之后,针对四种不同形式的曲线方程,推导并总结了不同方程下的曲率公式及其适用条件,并针对性地给出了两个典型例题.最后,给出了一个运用曲率公式求解实际工程问题的案例.     

椭圆弧长的级数表达式及其近似计算

导出了以椭圆参数方程离心角为变量的椭圆弧长级数表达式,可以计算任意离心角对应的椭圆弧长;在级数表达式基础上归纳了椭圆弧长近似计算公式,为解决实际应用问题提供了方便.     

飞越北极

本文对“飞机从北京出发、飞越北极直达底特律的所需时间 ,可比原航线节省多少时间”的问题进行讨论 ,并将航线选择归结为寻求曲面上的最短弧 .应用“曲面上最短弧为测地线”的事实进行了讨论 .模型 (一 )假设地球是球体 ,我们可通过单位向量的点乘与夹角的关系 ,加以解决 ;对于模型 (二 )设地球是旋转椭球体 ,我们利用微分几何学中测地线方程加以解决 ,并且把球面的纬度转化为旋转椭球面纬度 .对于 4组较特殊的点 ,纬度几乎相等或相近 ,或者两者之间的经度差过大时 ,用测地线计算比较困难 ,我们用椭圆弧 (长 )代替测地线长 ,结合数学软件 Mathematica的数值积分功能 ,可求得测地线长     

正交法在折纸飞机中的应用

纸折飞机是中小学生喜爱的自制玩具.多数纸飞机都是孩子们随意折叠的.由于随纸张及折叠方式的不同,飞机在飞行中平衡难以掌握,故飞行质量欠佳.为使纸飞机飞行平稳,飞行时间长,本人凭直观想象,选定四个因素,安排了这一定交试验. 一、指标:室内飞行时间 二、原料:居民身份证登记表(长26.4cm,宽19.1cm) 三、因素位级表 四、试验方案与结果. 利用L9(3～4)正交表折叠9架纸飞机,于无风的大厅中,每架举过头顶,轻轻一掷,同时以心中数数的方式计时,(由于缺乏秒表).时间越长越好. (上接第41页) 四、试验结果的分析 1.直接看:从试验结果看,5号飞机飞行…     

使用元素法必须满足的一个条件

设曲线弧由极坐标方程r＝r（θ）（a≤θ≤β）给出（如图）。其中r（θ）在[α，β]，用上具有连续导数。我们用如下方式来推导弧长公式。取θ为积分变量，其变化范围为[α，β]，相应于典型小区间「0，0＋doj的部分弧段BC的长用圆弧BD的长近似代替，得到弧长元素而课本（同济《高等数学》三版，以下说“课本”均指该书）上，在极坐标系下的曲线弧长公式为相比之下，可见（1）式肯定是错误的。门）式是仿照曲边扇形面积公式的推导过程用元素法推导出来的，而曲边扇形面积公式是完全正确的。同样使用元素法，为什么会一对一错呢？这就提出…     

基于第二类椭圆积分的椭圆弧长公式变换与应用

以第二类椭圆积分为理论基础,通过推导,将椭圆弧长公式变换为以椭圆离心角、极角等常用角度参数为自变量的第二类椭圆积分的标准形式,建立起椭圆弧长公式与第二类椭圆积分标准形式之间的关系,并分析了椭圆上的弧微分变化规律及椭圆周长与离心率的变化关系.公式反映了椭圆弧长的本质问题即为第二类椭圆积分问题.因此,各类涉及椭圆弧长计算的应用问题,均可化为第二类椭圆的计算问题,应用时直接调用各类编程软件的函数库中的第二类椭圆积分函数,无需复杂编程即可实现椭圆弧长的高精度计算.文章以GPS采用的WGS-84椭球子午线弧长为例进行计算分析,验证了给出的公式及相关分析的正确性及应用价值.     

一个二项展开式近似计算估值公式的修正

读郭长风、倪正藩两同志的《关于二项展开式近似公式的教学建议》一文(本刊81年第一期),很受启发.该文作者对应用二项展开式近似计算如何达到预定精确度提出了一般的判断方法.但文中提到“用相同的方法,可以算得”     

从等周不等式谈Green公式的一个应用

Green公式给出了平面图形面积与曲线积分的关系,利用这种关系给出了平面曲线等周不等式的一个简捷证明,并给出了平面图形面积的一个近似计算公式以及两个教学计算实例.     

一道高代考研题的充分条件

文献[1]中给出了华东师范大学1996年考研的一道高等代数题及其解法,利用二次函数求解公式给出另一种解法,根据这种解法又给出了该问题的一般形式可逆的充分条件.     

对“关于分期付款中的有关计算的探究”的探究

文 [1 ]对新教材关于数列的研究性课题 ,按“不计复利”的方式进行了探究 .文中给出的三种解法是由“复利”方式类比而来 .但这种类比是否可行 ?它究竟是不是符合“单利” ?我认为值得再探索 ;文中又讲到在银行还贷中 ,客户选定了某一种方法 ,与之对应的计算方法也就给定了 ,那么这三种解法背后对应着什么样的还款方式呢 ?先看解法 1 :“按课本提供的解法 ,有ｘ·( 1 + 1 .0 1 6 + 1 .0 32 + 1 .0 4 8+ 1 .0 6 4+ 1 .0 80 )=50 0 0 ( 1 + 0 .0 96 ) ( )6 .2 4ｘ =5480 ,ｘ 878.2 1元” .因为 878.2 1× 6 =52 6 9.2 6 >50 0 0元 ,故ｘ中除…     

The Neumann problem on ellipsoids

The Neumann problem on an ellipsoid in \(\mathbf {R}^n\) asks for a function harmonic inside the ellipsoid whose normal derivative is some specified function on the ellipsoid. We solve this problem when the specified function on the ellipsoid is a normalized polynomial (a polynomial divided by the norm of the normal vector arising from the definition of the ellipsoid). Specifically, we give a necessary and sufficient condition for a solution to exist, and we show that if a solution exists then it is a polynomial whose degree is at most the degree of the polynomial giving the specified function. Furthermore, we give an algorithm for computing this solution. We also solve the corresponding generalized Neumann problem and give an algorithm for computing its solution.     

A centro-projective inequality

We give a new integral formula for the centro-projective area of a convex body, which was first defined by Berck–Bernig–Vernicos. We then use the formula to prove that it is bounded from above by the centro-projective area of an ellipsoid and that equality occurs if and only if the convex set is an ellipsoid.     

Flow around an ellipsoid of revolution in a harmonic coaxial vector field

We give an integral equation defining a coaxial magnetic field near the surface of a superconductive axisymmetric body and the velocity of the liquid near the surface of an axisymmetric body situated coaxially to the flow of an ideal liquid. Using this equation in the case when the axisymmetric magnetic field before the placement of an ellipsoid of revolution coaxially to the field changed along the axis by a polynomial law, we analytically define the densities of the surface current and the force with which the magnetic field acts on the ellipsoid. Also the velocity of the liquid is determined near the surface of the ellipsoid of revolution and the force acting on the ellipsoid placed coaxially in the flow of an ideal liquid when the velocity of the liquid before the placement of the ellipsoid changed along the axis of symmetry by a polynomial law.     

An Ellipsoidal Billiard with a Quadratic Potential

There exists an infinite hierarchy of integrable generalizations of the geodesic flow on an n-dimensional ellipsoid. These generalizations describe the motion of a point in the force fields of certain polynomial potentials. In the limit as one of semiaxes of the ellipsoid tends to zero, one obtains integrable mappings corresponding to billiards with polynomial potentials inside an (n-1)-dimensional ellipsoid.In this paper, for the first time we give explicit expressions for the ellipsoidal billiard with a quadratic (Hooke) potential, its representation in Lax form, and a theta function solution. We also indicate the generating function of the restriction of the potential billiard map to a level set of an energy type integral. The method we use to obtain theta function solutions is different from those applied earlier and is based on the calculation of limit values of meromorphic functions on generalized Jacobians.     

Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid

We give a new proof of the complete integrability of the geodesic flow on the ellipsoid (in Euclidean, spherical or hyperbolic space). The proof is based on the construction of a metric on the ellipsoid whose non-parameterized geodesics coincide with those of the standard metric. This new metric is induced by the hyperbolic metric inside the ellipsoid (Klein's model).     

截面体与凸体的包含测度

本文建立了E^n中的凸体与其位似体的包含测度的相等关系，同时给出了凸体Minkowski和的包含测度的估计。最后证明了E^3中的关于原点对称的旋转椭球与其截面体相似的充要条件是K为以原点为心的球。     

关于格点覆盖的一个注记

对于一个给定椭球,本文给出了它的任一全等椭球都包含一个整点的一个充分必要条件.     

The sphere method and the robustness of the ellipsoid algorithm

We present a method for implementing the ellipsoid algorithm, whose basic iterative step is a linear row manipulation on the matrix of inequalities. This step is somewhat similar to a simplex iteration, and may give a clue to the relation between the two algorithms. Geometrically, the step amounts to performing affine transformations which map the ellipsoids onto a fixed sphere. The method was tried successfully on linear programs with up to 50 variables, some of which required more than 24 000 iterations. Geometrical properties of the iteration suggest that the ellipsoid algorithm is numerically robust, which is supported by our computational experience.     

Induced Circuits in Planar Graphs

In McDiarmid, B. Reed, A. Schrijver, and B. Shepherd (Univ. of Waterloo Tech. Rep., 1990) a polynomial-time algorithm is given for the problem of finding a minimum cost circuit without chords (induced circuit) traversing two given vertices of a planar graph. The algorithm is based on the ellipsoid method. Here we give an O(n²) combinatorial algorithm to determine whether two nodes in a planar graph lie on an induced circuit. We also give a min-max relation for the problem of finding a maximum number of paths connecting two given vertices in a planar graph so that each pair of these paths forms an induced circuit.     

John's Decomposition of the Identity in the Non-Convex Case

We prove an extension of the classical John's Theorem, that characterises the ellipsoid of maximal volume position inside a convex body by the existence of some kind of decomposition of the identity, obtaining some results for maximal volume position of a compact and connected set inside a convex set with nonempty interior. By using those results we give some estimates for the outer volume ratio of bodies not necessarily convex.