L1calculus on Euclidean space

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微积分calculus英文单词

微积分英语单词Absolute convergence ：绝对收敛Absolute extreme values ：绝对极值Absolute maximum and minimum ：绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative ：反导数Approximate integration ：近似积分Approximation ：逼近法Arc length ：弧长Area ：面积Asymptote ：渐近线Average speed ：平均速率Average velocity ：平均速度Axes, coordinate ：坐标轴Axes of ellipse ：椭圆之轴at a point ：在一点处之连续性as the slope of a tangent ：导数看成切线之斜率by differentials ：用微分逼近between curves ：曲线间之面积Binomial series ：二项级数Cartesian coordinates ：笛卡儿坐标一般指直角坐标Cartesian coordinates system ：笛卡儿坐标系Cauch’s Mean Value Theorem ：柯西均值定理Chain Rule ：连锁律Change of variables ：变数变换Circle ：圆Circular cylinder ：圆柱Closed interval ：封闭区间Coefficient ：系数Composition of function ：函数之合成Compound interest ：复利Concavity ：凹性Conchoid ：蚌线Cone ：圆锥Constant function ：常数函数Constant of integration ：积分常数Continuity ：连续性Continuous function ：连续函数Convergence ：收敛Coordinate ：s ：坐标Cartesian ：笛卡儿坐标cylindrical ：柱面坐标Coordinate axes ：坐标轴Coordinate planes ：坐标平面Cosine function ：余弦函数Critical point ：临界点Cubic function ：三次函数Curve ：曲线Cylinder ：圆柱Cylindrical Coordinates ：圆柱坐标Distance ：距离Divergence ：发散Domain ：定义域Dot product ：点积Double integral ：二重积分Decreasing function ：递减函数Decreasing sequence ：递减数列Definite integral ：定积分Degree of a polynomial ：多项式之次数Density ：密度Derivative ：导数Determinant ：行列式Differentiable function ：可导函数Differential ：微分Differential equation ：微分方程Differentiation ：求导法Directional derivatives ：方向导数Discontinuity ：不连续性Disk method ：圆盘法domain of ：导数之定义域differential ：微分学Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation ：方程式Even function ：偶函数Expected Valued ：期望值Exponential Function ：指数函数Exponents , laws of ：指数率Extreme value ：极值Extreme Value Theorem ：极值定理Factorial ：阶乘First Derivative Test ：一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions ：分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理from the left ：左连续from the right ：右连续Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式Half-angle formulas ：半角公式Harmonic series ：调和级数Helix ：螺旋线Higher Derivative ：高阶导数Horizontal asymptote ：水平渐近线Horizontal line ：水平线Hyperbola ：双曲线Hyperboloid ：双曲面horizontal ：水平渐近线Implicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：瑕积分Increasing/Decreasing Test ：递增或递减试验法Increment ：增量Increasing Function ：增函数Indefinite integral ：不定积分Independent variable ：自变数Indeterminate from ：不定型Inequality ：不等式Infinite point ：无穷极限Infinite series ：无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积分式Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem ：中间值定理Interval ：区间Inverse function ：反函数Inverse trigonometric function ：反三角函数Iterated integral ：逐次积分integral ：积分学implicit ：隐求导法Laplace transform ：Leplace 变换Law of Cosines ：余弦定理Least upper bound ：最小上界Left-hand derivative ：左导数Left-hand limit ：左极限Lemniscate ：双钮线Length ：长度Level curve ：等高线L'Hospital's rule ：洛必达法则Limacon ：蚶线Limit ：极限Linear approximation：线性近似Linear equation ：线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Line in the plane ：平面上之直线Line in space ：空间之直线Lobachevski geometry ：罗巴切夫斯基几何Local extremum ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数linear ：线性逼近法Maximum and minimum values ：极大与极小值Mean Value Theorem ：均值定理Multiple integrals ：重积分Multiplier ：乘子Natural exponential function ：自然指数函数Natural logarithm function ：自然对数函数Natural number ：自然数Normal line ：法线Normal vector ：法向量Number ：数of a function ：函数之连续性on an interval ：在区间之连续性Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题Order ：阶Ordinary differential equation ：常微分方程 Origin ：原点Orthogonal ：正交的Parabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallelepiped ：平行六面体Parallel lines ：并行线Parameter ：参数Partial derivative ：偏导数Partial differential equation ：偏微分方程 Partial fractions ：部分分式Partial integration ：部分积分Partiton ：分割Period ：周期Periodic function ：周期函数Perpendicular lines ：垂直线Piecewise defined function ：分段定义函数 Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Point-slope form ：点斜式Power function ：幂函数Product ：积polar ：极坐标partial ：偏导数partial ：偏微分方程partial ：偏微分法Quadrant ：象限Quotient Law of limit ：极限的商定律Quotient Rule ：商定律rectangular ：直角坐标Radius of convergence ：收敛半径Range of a function ：函数的值域Rate of change ：变化率Rational function ：有理函数Rationalizing substitution ：有理代换法Rationalizing substitution ：有理代换法Rational number ：有理数Real number ：实数Rectangular coordinates ：直角坐标Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution, solid of ：旋转体Revolution, surface of ：旋转曲面Riemann Sum ：黎曼和Riemannian geometry ：黎曼几何Right-hand derivative ：右导数Right-hand limit ：右极限Root ：根Saddle point ：鞍点Scalar ：纯量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative ：二阶偏导数Sector ：扇形Sequence ：数列Series ：级数Set ：集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant asymptote ：斜渐近线Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface ：平滑曲面Solid of revolution ：旋转体Space ：空间Speed ：速率Spherical coordinates ：球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing ：严格递减Strictly increasing ：严格递增Sum ：和Surface ：曲面Surface integral ：面积分Surface of revolution ：旋转曲面Symmetry ：对称slant ：斜渐近线spherical ：球面坐标Tangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector ：切向量Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分term by term ：逐项求导法under a curve ：曲线下方之面积vertical ：垂直渐近线Value of function ：函数值Variable ：变数Vector ：向量Velocity ：速度Vertical asymptote ：垂直渐近线Volume ：体积X-axis ：x 轴x-coordinate ：x 坐标x-intercept ：x 截距Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点。

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6 Diﬀerential Calculus of Vectors in Orthogonal Curvilinear 6.1 The Del Operator . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Gradient of a Scalar Function . . . . . . . . . . . . . . . 6.3 Coordinate Surface Normals . . . . . . . . . . . . . . . . . . 6.3.1 What are the Coordinate Surface Normals? . . . . . 6.3.2 Cross Products of Coordinate Surface Normals . . . . 6.3.3 Unit Tangent Vector Relations . . . . . . . . . . . . . 6.4 Divergence of a Vector Field . . . . . . . . . . . . . . . . . . 6.5 Curl of a Vector Field . . . . . . . . . . . . . . . . . . . . . 6.6 Laplacian of a Scalar Function . . . . . . . . . . . . . . . . .

Quantum Deformations of Multi-Instanton Solutions in the Twistor Space

a r X i v :q -a l g/95724v121J u l1995QUANTUM DEFORMATIONS OF MULTI-INSTANTON SOLUTIONS IN THE TWISTOR SPACE B.M.Zupnik Bogoliubov Laboratory of Theoretical Physics ,JINR,Dubna,Moscow Region,141980,E-mail:zupnik@thsun1.jinr.dubna.su To be published in ”Pis’ma v ZhETP”v.62,n.4We consider the quantum-group self-duality equation in the framework of the gauge theory on a deformed twistor space.Quantum deformations of the Atiyah-Drinfel’d-Hitchin-Manin and t’Hooft multi-instanton solutions are constructed.The quantum-group gauge theory was considered in the framework of the algebra of local diﬀerential complexes [1]-[3]or as a noncommutative generalization of the ﬁbre bundles over the classical or quantum basic spaces [4,5].We prefer to use local constructions of the noncommutative connection forms or gauge ﬁelds as a deformed analogue of the local gauge ﬁelds.In particular,the quantum-group self-duality equation (QGSDE)has been considered in the deformed 4-dimensional Euclidean space,and an explicit formula for the corresponding one-instanton solution has been constructed [3].This solution can be treated as q -deformation of the BPST-instanton [6].We shall discuss here quantum deformations of the general multi-instanton solutions [7].The conformal covariant description of the classical ADHM solution was considered in Ref[8].We shall study the quantum deformation of this version of the twistor formalism.It is convenient to discuss ﬁrstly the deformations of the complex conformal group GL (4,C ),complex twistors and the complex linear gauge groups.Let R ab cd ,(a,b,c,d...=1...4)be the solution of the 4D Yang-Baxter equation satis-fying also the Hecke relation R R ′R =R ′R R ′(1)R 2=I +(q −q −1)R (2)where q is a complexparameter.Note that the standard notation for these R -matrices isR =ˆR 12,R ′=ˆR 23[9].Consider also the SL q (2,C )R -matrixR αβµν=qδαµδβν+εαβ(q )εµν(q )(3)where ε(q )is the deformed antisymmetric symbol.Noncommutative twistors were considered in Ref[10].We shall use the R -matrix ap-proach to deﬁne the diﬀerential calculus on the deformed twistor space.Let z αa and dz αa be the components of the q -twistor and their diﬀerentialsR αβµνz µa z νb =z αc z βd R dc ba (4)z αa dz βb =R αβµνdz µc z νd R dc ba (5)dz αa dz βb =−R αβµνdz µc dz νd R dc ba (6)1One can deﬁne also the algebra of partial derivatives∂aαR abcd ∂cα∂dβ=∂aµ∂bνRνµβα(7)∂aαzβb=δa bδβα+RβµανR da cb zνd∂cµ(8) Consider the4D deformedεq-symbolR bafe εefcdq=−1Let us consider the quantum deformation of the GL(2)t’Hooft solution[8]=q−3dzαa(∂aµΦ)Φ−1εσµ(q)εσβ(q)(18)AαβΦ= i(X i)−1,X i=(y,b i)=εabcd q y ab b i cd(19) where b i cd are the noncommutative isotropic6D vectorsdb i cd=0,(b i,b i)=0(20)[y ab,X i]=[b i cd,X i]=0(21) The central elements X i of the(B,z)-algebra do not commute with dz=q2dzαa X i(22)X i dzαaStress that Aαβsatisﬁes Eq(16)and its quantum trace is a U(1)-gaugeﬁeld with the zeroﬁeld-strengthTr q A=−q−3dΦΦ−1,Tr q dA=0(23) The QGSDE for Aαβis equivalent to theﬁnite-diﬀerence Laplace equation for the functionΦon the q-twistor space∆baΦ(X i)= i∆ba1εαβ(q)∂bβ∂aαΦ=(∂ba+11+q2where g(z)is the nondegenerate(k×k)matrix with the central elementsqg AB(z)=This curvature contains the self-dual2-form(14)only.It should be stressed that all R-matrices of our deformation scheme satisfy the Hecke relation with the common parameter q.The other possible parameters of diﬀerent R-matrices are independent.The case q=1corresponds to the unitary deformations(R2= I)of the twistor space and the gauge groups.It is evident that the trivial deformation of the z-twistors is consistent with the nontrivial unitary deformation of the gauge sector and vice versa.The Euclidean conformal q-twistors are a representation of the U∗(4)×SU q(2)group. The antiinvolution for these twistors has the following form:(zαa)∗=εαβ(q)zβb C b a(47) where C is the charge conjugation matrix for U∗(4).We can use the gauge group U q(N) in the framework of our approach.An analogous construction can be considered for the real twistors and the gauge group GL q(N,R).The author would like to thank A.T.Filippov,E.A.Ivanov,A.P.Isaev and V.I.Ogievet-sky for helpful discussions and interest in this work.I am grateful to administration of JINR and Laboratory of Theoretical Physics for hospitality.This work was supported in part by ISF-grant RUA000,INTAS-grant93-127 and the contract No.40of Uzbek Foundation of Fundamental Research.[1]A.P.Isaev,Z.Popowicz,Phys.Lett.B281,271(1992);Phys.Lett.B307,353(1993)[2]A.P.Isaev,J.Math.Phys.35,6784(1995)[3]B.M.Zupnik,Pis’ma ZhETF,61,434(1995);Preprints JINR E2-94-449,hep-th/9411186;E2-94-487,q-alg/9412010[4]T.Brzezinski,Sh.Majid,Comm.Math.Phys.157,591(1993)[5]M.J.Pﬂaum,Comm.Math.Phys.166,279(1994)[6]A.A.Belavin,A.M.Polyakov,A.S.Schwartz and Yu.S.Tyupkin,Phys.Lett.B59,85(1975)[7]M.F.Atiyah,V.G.Drinfel’d,N.J.Hitchin and Yu.I.Manin,Phys.Lett.A65,185(1978)[8]W.Siegel,Preprint ITP-SB-94-66,Stony Brook,1994;hep-th/9412011[9]N.Yu.Reshetikhin,L.A.Takhtadjan and L.D.Faddeev,Algeb.Anal.1,178(1989)[10]S.A.Merkulov,Z.Phys.C52,583(1991)5。

Differential geometry of GL$_h(1 1)$

E-mail address: sacelik@.tr
I. INTRODUCTION In the last few years, the theory of quantum (super) groups like GL(2), GL(1|1), etc., were generalized in two ways. Both of the generalizations are based on the deformation of the algebra of functions on the matrix (super) groups generated by coordinate functions Tji which normally commute. These deformations of Lie (super) groups are algebraic structures depending on one (or more) continuous parameter. We have a standard Lie (super) group for particular values of the deformation parameters. Quantum (super) groups1−3 present the examples of (graded) Hopf algebras. They have found application in diverse areas of physics and mathematics4 . The q -deformation of Lie (super) groups can be realized on a quantum (super) space in which coordinates are noncommuting2 . Recently the diﬀerential calculus on noncommutative (super) space has been intensively studied both by mathematicians and mathematical physicists. There is much activity in diﬀerential geometry on quantum groups. Throughout the recent development of diﬀerential calculus on the quantum groups two principal concepts are readily seen. First of them, formulated by Woronowicz5 , is known as bicovariant diﬀerential calculus on the quantum groups. Another concept, introduced by Woronowicz6 and Schirmacher et al7 proceeds from the requirement of a calculus only. There are many papers in this ﬁeld8 . We shall consider the second concept. Another type of deformation, the so called h-deformation, which is a new class of quantum deformations of Lie groups and Lie algebras has recently been intensively studied9 . This deformation may be obtained as a contraction of the q -deformation10 . There is much interest in studies relating to various aspects of the h-deformed algebra. The diﬀerential geometry of SLh (2) was given in11 . In this work, we introduce a right-invariant diﬀerential calculus on the quantum supergroup GLh (1|1). This quantum supergroup was obtained in ref. 12 using a contraction procedure given in Ref. 10. Let us brieﬂy discuss the content of the paper. In the second section, the basic notations of the Hopf algebra structure on the quantum supergroup GLh (1|1) are introduced. In the third section we shall obtain the commutation relations for the group parameters (the matrix elements) and their diﬀerentials so we have a diﬀerential algebra. This diﬀerential algebra (extended algebra) has a Hopf algebra structure. Later, we shall construct the Cartan-Maurer one-forms and obtain the needed commutation relations. Using these commutation relations, we shall describe the quantum superalgebra for the vector ﬁelds (superalgebra generators) for GLh (1|1) and derive the commutation relations between the group parameters and the algebra generators. We shall 2

剑桥学习科学手册(第一、二部分)

四、研究目标
（1）为了更好地理解认知过程和社会化过程以产生最有效的学习；（2）为了用学习科学的知识来重新设计我们的课堂和其他学习环境。
五、概念界定
研究多种情境下的学习（正式学习与非正式学习）；多学科、跨学科的新科学；为未来学校绘制发展蓝图。
本书目的：通过展示不同的人设计学习环境和课堂的方式，来建立新的学习科学。
了解对话的过程，即知识产生的过程；能够批判地检查论据的逻辑性
将陈述性知识和程序性知识视为静态知识，认为这些知识只来自于权威著作
要求学习者反思其理解及学习的过程
仅仅记忆知识，没有对目的和学习策略等进行反思
学习科学的基础
建构主义
认知科学
教育技术学
社会文化研究
学科知识研究
科学知识是情境性的、实践性的、通过协作产生的。
在认知分析中运用交互
群体活动分析可能会涉及个体活动中不明显的重要活动
可以将群体分解为单独个体的方式研究解释群体
从认知到交互
将认知原则从个人延伸到群体活动是相当有价值的
情景化视角
情境化方法的明确特征是不再关注个体学习者，分析的主要焦点在于活动系统：包含学习者、教师、课程材料、软件工具以及自然环境在内的复杂的社会组织。
监控问题解决活动获得产生式规则的过程
表征学生的多种策略和典型迷失概念
知识跟踪
模型跟踪
认知模型
工作机制
设计原则
以产生式规则集的形式展现学生的能力在问题——解决的情境中提供指导在问题解决之前传达目标结构促进对问题解决知识的正确、总体的理解将学习外部的工作记忆负荷减弱到最低程度对与预期绩效模型相关的错误提供即时反馈
内隐学习与大脑
对理解学习的个体差异提供帮助

斯宾诺莎几何学方法

斯宾诺莎几何学方法Spinoza's geometric method in his "Ethics" has been a subject of much debate and discussion among scholars and philosophers.斯宾诺莎在他的《伦理学》中所采用的几何学方法一直是学者和哲学家们争论和讨论的课题。

One perspective on Spinoza's geometric method is that it represents an attempt to provide a rigorous and systematic account of his philosophical ideas. In using the geometric method, Spinoza aimedto present his arguments in a clear and logical manner, much like the propositions and proofs found in geometry.斯宾诺莎的几何学方法代表了他试图对自己的哲学思想进行严谨系统的解释。

在使用几何学方法时，斯宾诺莎的目标是以清晰逻辑的方式呈现他的论证，就像在几何学中发现的命题和证明一样。

Another perspective is that the use of the geometric method reflects Spinoza's aim to ground his philosophy in reason and to demonstrate the coherence and interconnectedness of his ideas. Bypresenting his philosophical ideas in a geometric format, Spinoza sought to show the necessary connections between different aspects of his thought, as well as the logical implications of his central claims.另一个观点是，使用几何学方法反映了斯宾诺莎的目标，即将他的哲学根植于理性，并展示他的思想的连贯性和相关性。

Cartan Calculus on the Quantum Space ${cal R}_q^{3}$

$Cartan Calculus on the Quantum Space ${cal R}_q^{3}$$

a r X i v :m a t h /0607383v 2 [m a t h .Q A ] 11 D e c 2006YTUMB 2006-02,July 2006CARTAN CALCULUS ON THE QUANTUMSPACE R 3qSalih C ¸elik 1,2,E.Mehmet ¨Ozkan 1and Erg¨u n Ya¸s ar 11Yildiz Technical University,Department of Mathematics,34210DAVUTPASA-Esenler,Istanbul,TURKEY.2E-mail:sacelik@.trABSTRACTTo give a Cartan calculus on the extended quantum 3d space,the noncommutativediﬀerential calculus on the extended quantum 3d space is extended by introducing inner derivations and Lie derivatives.1.INTRODUCTIONThe noncommutative diﬀerential geometry of quantum groups was introduced by Woronowicz[11,12].In this approach the diﬀerential calculus on the group is deduced from the properties of the group and it involves functions on the group, diﬀerentials,diﬀerential forms and derivatives.The other approach,initiated by Wess and Zumino[10],followed Manin’s emphasis[5]on the quantum spaces as the primary objects.Diﬀerential forms are deﬁned in terms of noncommuting coordinates,and the diﬀerential and algebraic properties of quantum groups acting on these spaces are obtained from the properties of the spaces.The diﬀerential calculus on the quantum3d space similarly involves functions on the3d space,diﬀerentials,diﬀerential forms and derivatives.The exterior derivative is a linear operator d acting on k-forms and producing(k+1)-forms, such that for scalar functions(0-forms)f and g we haved(1)=0,d(fg)=(d f)g+(−1)deg(f)f(d g)where deg(f)=0for even variables and deg(f)=1for odd variables,and for a k-formω1and any formω2∧ω2)=(dω1)∧ω2+(−1)kω1∧(dω2).d(ω1A fundamental property of the exterior derivative d isd∧d=:d2=0.There is a relationship of the exterior derivative with the Lie derivative and to describe this relation,we introduce a new operator:the inner derivation.Hence the diﬀerential calculus on the quantum3d space can be extended into a large calculus.We call this new calculus the Cartan calculus.The connection of the inner derivation denoted by i a and the Lie derivative denoted by L a is given by the Cartan formula:L a=i a◦d+d◦i a.This and other formulae are explaned in Ref.6-8.We now shall give a brief overview without much discussion.Let us begin with some information about the inner derivations.Generally,for a smooth vectorﬁeld X on a manifold the inner derivation,denoted by i X,is a linear operator which maps k-forms to(k−1)-forms.If we deﬁne the inner derivation i X on the set of all diﬀerential forms on a manifold,we know that i X is an antiderivation of degree−1:(α∧β)=(i Xα)∧β+(−1)kα∧(i Xβ)iXwhereαandβare both diﬀerential forms.The inner derivation i X acts on0-and 1-forms as follows:(f)=0,iX(d f)=X(f).iXWe know,from the classical diﬀerential geometry,that the Lie derivative L can be deﬁned as a linear map from the exterior algebra into itself which takes k-forms to k-forms.For a0-form,that is,an ordinary function f,the Lie derivative is just the contraction of the exterior derivative with the vectorﬁeld X:L X f=i X d f.For a general diﬀerential form,the Lie derivative is likewise a contraction,taking into account the variation in X:L Xα=i X dα+d(i Xα).The Lie derivative has the following properties.If F(M)is the algebra of functions deﬁned on the manifold M thenL X:F(M)−→F(M)is a derivation on the algebra F(M):L X(af+bg)=a(L X f)+b(L X g),L X(fg)=(L X f)g+f(L X g),where a and b real numbers.The Lie derivative is a derivation on F(M)×V(M)where V(M)is the set of vectorﬁelds on M:L X1(fX2)=(L X1f)X2+f(L X1X2).The Lie derivative also has an important property when acting on diﬀerential forms.Ifαandβare two diﬀerential forms on M thenL X(α∧β)=(L Xα)∧β+(−1)kα∧(L Xβ)whereαis a k-form.The extended calculus on the quantum plane was introduced in Ref.3using the approach of Ref.6.In this work we explicitly set up the Cartan calculus on the quantum3d space using approach of Ref1.2.REVIEW OF SOME STRUCTURES ON R3qIn this section we give some information on the Hopf algebra structures of the quantum3d space and its diﬀerential calculus[2]which we shall use in order to establish our notions.2.1The algebra of polynomials on the quantum3d spaceThe quantum three dimensional space is deﬁned as an associative algebra gener-ated by three noncommuting coordinates x,y and z with three quadratic relationsxy=qyx,yz=qzy,xz=qzx,where q is a non-zero complex number.This associative algebra over the complex number,C,is known as the algebra of polynomials over the quantum three di-mensional space and we shall denote it by R3q.In the limit q−→1,this algebra is commutative and can be considered as the algebra of polynomials C[x,y,z]overthe usual three dimensional space,where x,y and z are the three coordinate func-tions.We denote the unital extension of R3q by A,i.e.it is obtained by adding a unit element.2.2The Hopf algebra structure on AOne extends the algebra A by including inverse of x which obeysxx−1=1=x−1x.The deﬁnitions of a coproduct,a counit and a coinverse on the algebra A as follows [2]:(1)The C-algebra homomorphism(coproduct)∆A:A−→A⊗A is deﬁned by∆A(x)=x⊗x,∆A(y)=x⊗y+y⊗x,∆A(z)=z⊗1+1⊗z,which is coassociative:(∆A⊗id)◦∆A=(id⊗∆A)◦∆Awhere id denotes the identity map on A.(2)The C-algebra homomorphism(counit)ǫA:A−→C is given byǫA(x)=1,ǫA(y)=0,ǫA(z)=0.The counitǫA has the propertyµ◦(ǫA⊗id)◦∆A=µ′◦(id⊗ǫA)◦∆Awhereµ:C⊗A−→A andµ′:A⊗C−→A are the canonical isomorphisms, deﬁned byµ(k⊗u)=ku=µ′(u⊗k),∀u∈A,∀k∈C.(3)The C-algebra antihomomorphism(coinverse)S A:A−→A is deﬁned byS A(x)=x−1,S A(y)=−x−1yx−1,S A(z)=−z.The coinverse S satisﬁesm◦(S A⊗id)◦∆A=ǫA=m◦(id⊗S A)◦∆Awhere m stands for the algebra product A⊗A−→A.The coproduct,counit and coinverse which are speciﬁed above supply the algebra A with a Hopf algebra structure.2.3Diﬀerential algebraWeﬁrst note that the properties of the exterior diﬀerential d.The exterior diﬀer-ential d is an operator which gives the mapping from the generators of A to the diﬀerentials:d:u−→d u,u∈{x,y,z}.We demand that the exterior diﬀerential d has to satisfy two properties:the nilpotencyd2=0and the Leibniz ruled(fg)=(d f)g+(−1)deg(f)f(d g).A deformed diﬀerential calculus on the quantum3d space is as follows:the commutation relations with the coordinates of diﬀerentialsx d x=d x x,x d y=q d y x,x d z=q d z x,y d x=q−1d x y,y d y=d y y,y d z=q d z y,z d x=q−1d x z,z d y=q−1d y z,z d z=d z z.This algebra is denoted byΓ1.The commutation relations between the diﬀerentialsd x∧d x=0,d y∧d y=0,d z∧d z=0.d x∧d y=−q d y∧d x,d y∧d z=−q d z∧d y,d x∧d z=−q d z∧d x.This algebra is denoted byΓ2.A diﬀerential algebra on an associative algebra A is a graded associative algebra Γequipped with an operator d that has the above properties.Furthermore,the algebraΓhas to be generated byΓ0∪Γ1∪Γ2,whereΓ0is isomorphic to A.LetΓbe the quoitent algebra of the free associative algebra on the set{x,y,z,d x,d y,d z} modulo the ideal J that is generated by the relations of R3q,Γ1andΓ2.To proceed,one can obtain the relations of the coordinates with their partial derivatives using the expression+d y∂y+d z∂z)f.d f=(d x∂xConsequently one has∂x x=1+x∂x,∂x y=q−1y∂x,∂x z=q−1z∂x,∂y x=qx∂y,∂y y=1+y∂y,∂y z=q−1z∂y,∂z x=qx∂z,∂z y=qy∂z,∂z z=1+z∂z.Using the fact that d2=0,oneﬁnds∂x∂y=q∂y∂x,∂x∂z=q∂z∂x,∂y∂z=q∂z∂y.The relations between partial derivatives and diﬀerentials are found as∂x d x=d x∂x,∂x d y=q−1d y∂x,∂x d z=q−1d z∂x,∂y d x=q d x∂y,∂y d y=d y∂y,∂y d z=q−1d z∂y,∂z d x=q d x∂z,∂z d y=q d y∂z,∂z d z=d z∂z.We can deﬁne three one-forms using the generators of A.If we call themωx,ωy andωz then one can deﬁne them as follows:ωx=d x x−1,ωy=d y x−1−d x x−1yx−1,ωz=d z.We denote the algebra of forms generated by three elementsωx,ωy andωz by Ω.The generators of the algebraΩwith the generators of A satisfy the following rulesxωx=ωx x,xωy=qωy x,xωz=qωz x,yωx=ωx y,yωy=qωy y,yωz=qωz y,zωx=ωx z,zωy=ωy z,zωz=ωz z.The commutation rules of the generators ofΩareωx∧ωx=0,ωy∧ωy=0,ωz∧ωz=0,ωx∧ωy=−ωy∧ωx,ωy∧ωz=−ωz∧ωy,ωx∧ωz=−ωz∧ωx.The algebraΩis a graded Hopf algebra[2].2.4Lie algebraThe commutation relations of Cartan-Maurer forms allow us to construct the algebra of the generators.In order to obtain the quantum Lie algebra of the algebra generators weﬁrst write the Cartan-Maurer forms asx,d x=ωxy+ωy x,d y=ωxd z=ω.zThe diﬀerantial d can then the expressed in the formd f=(ωT x+ωy T y+ωz T z)f.xHere T x,T y and T z are the quantum Lie algebra generators.Considering an arbitrary function f of the coordinates of the quantum3d space and using that d2=0,weﬁnd the following commutation relations for the(undeformed)Lie algebra[2]:[T x,T y]=0,[T x,T z]=0,[T y,T z]=0.The commutation relations between the generators of algebra and the coordinates areT x x=x+x T x,T x y=y+y T x,T x z=z T x,T y x=qx T y,T y y=x+qy T y,T y z=z T y,T z x=qx T z,T z y=qy T z,T z z=1+z T z.The(quantum)Lie algebra generators can be expressed in terms of the generators of the quantum3d space and partial diﬀerentials:T x≡x∂x+y∂y,T y≡x∂y,T z≡∂z.The commutation relations of the Lie algebra generators T x,T y and T z with the diﬀerentials are followingT x d x=d x T x,T x d y=d y T x,T x d z=d z T x,T y d x=q d x T y,T y d y=q d y T y,T y d z=d z T y,T z d x=q d x T z,T z d y=q d yT z,T z d z=d z T z.The commutation rules of the Lie algebra generators with one-forms as followsT xωx=ωx T x−ωx,T xωy=ωy T x−ωy,T xωz=ωz T x,T yωx=ωx T y,T yωy=ωy T y−ωx,T yωz=ωz T y,T zωx=ωx T z,T zωy=ωy T z,T zωz=ωz T z.The Hopf algebra structure of the Lie algebra generators is given by∆(T x)=T x⊗1+1⊗T x,∆(T y)=T y⊗1+q T x⊗T y,∆(T z)=T z⊗1+q T x⊗T z,ǫ(T x)=0,ǫ(T y)=0,ǫ(T z)=0,S(T x)=−T x,S(T y)=−q−T x T y,S(T z)=−q−T x T z.2.5The dual of the Hopf algebra AIn this section,in order to obtain the dual of the Hopf algebra A deﬁned in section 2,we shall use the method of Refs.4and9.A pairing between two vector spaces U and A is a bilinear mapping<,>:U x A−→C,(u,a)→<u,a>.We say that the pairing is non-degenerate if<u,a>=0(∀a∈A)=⇒u=0and<u,a>=0(∀u∈U)=⇒a=0.Such a pairing can be extended to a pairing of U⊗U and A⊗A by<u⊗v,a⊗b>=<u,a><v,b>.Given bialgebras U and A and a non-degenerate pairing<,>:U x A−→C(u,a)→<u,a>∀u∈U∀a∈Awe say that the bilinear form realizes a duality between U and A,or that the bialgebras U and A are in duality,if we have<uv,a>=<u⊗v,∆A(a)>,<u,ab>=<∆U(u),a⊗b>,<1U,a>=ǫA(a),and<u,1A>=ǫU(u)for all u,v∈U and a,b∈A.If,in addition,U and A are Hopf algebras with coinverseκ,then they are said to be in duality if the underlying bialgebras are in duality and if,moreover,we have<S U(u),a>=<u,S A(a)>∀u∈U a∈A.It is enough to deﬁne the pairing between the generating elements of the two algebras.Pairing for any other elements of U and A follows from above relations and the bilinear form inherited by the tensor product.For example,foru′k⊗u′′k,∆U(u)=kwe have<u′k,a><u′′k,b><u,ab>=<∆U(u),a⊗b>=kAs a Hopf algebra A is generated by the elements x,y and z,and a basis is given by all monomials of the formf=x k y l z mwhere k,l,m∈Z+.Let us denote the dual algebra by U q and its generating elements by A and B.The pairing is deﬁned through the tangent vectors as follows<X,f>=kδl,0δm,0,<Y,f>=δl,1δm,0,<Z,f>=δl,0δm,1.We also have<1U,f>=ǫA(f)=δk,0.Using the deﬁning relations one gets<XY,f>=δl,1δm,0and<Y X,f>=δl,1δm,0where diﬀerentiation is from the right as this is most suitable for diﬀerentiation in this basis.Thus one obtains one of the commutation relations in the algebra U q dual to A as:XY=Y X.Similarly,one hasXZ=ZX,Y Z=ZY.The Hopf algebra structure of this algebra can be deduced by using the duality. The coproduct of the elements of the dual algebra is given by∆U(X)=X⊗1U+1U⊗X,∆U(Y)=Y⊗q−X+1U⊗Y,∆U(Z)=Z⊗q−X+1U⊗Z.The counity is given byǫU(X)=0,ǫU(Y)=0,ǫU(Z)=0.The coinverse is given asS U(X)=−X,S U(Y)=−Y q X,S U(Z)=−Zq X.We can now transform this algebra to the form obtained in section5by making the following identities:T x≡X,T y≡q X/2Y q X/2,T z≡q X/2Zq X/2which are consistent with the commutation relation and the Hopf structures.3.EXTENDED CALCULUS ON THE QUANTUM3D SPACEA Lie derivative is a derivation on the algebra of tensorﬁelds over a manifold. The Lie derivative should be deﬁned three ways:on scalar functions,vectorﬁelds and tensors.The Lie derivative can also be deﬁned on diﬀerential forms.In this case,it is closely related to the exterior derivative.The exterior derivative and the Lie derivative are set to cover the idea of a derivative in diﬀerent ways.These diﬀer-ences can be hasped together by introducing the idea of an antiderivation which is called an inner derivation.3.1Inner derivationsIn order to obtain the commutation rules of the coordinates with inner derivations, we shall use the approach of Ref. 1.Similarly other relations can also obtain. Consequently,we have the following commutation relations:•the commutation relations of the inner derivations with x,y and zx=x i x,i x y=q−1y i x,i x z=q−1z i x,ixix=q x i y,i y y=y i y,i y z=q−1z i y,yx=q x i z,i z y=q y i z,i z z=z i z.iz•the relations of the inner derivations with the partial derivatives∂x,∂y and ∂z∂x=∂x i x,i x∂y=q∂y i x,i x∂z=q∂z i x,ixi∂x=q−1∂x i y,i y∂y=∂y i y,i y∂z=q∂z i y,y∂x=q−1∂x i z,i z∂y=q−1∂y i z,i z∂z=∂z i z.iz•the commutation relations between the diﬀerentials and the inner derivations ∧d x=1−d x∧i x,i x∧d y=−q−1d y∧i x,ixi∧d x=−q d x∧i y,i y∧d y=1−d y∧i y,yi∧d x=−q d x∧i z,i z∧d y=−q d y∧i z,zi∧d z=−q−1d z∧i x,i y∧d z=−q−1d z∧i y,x∧d z=1−d z∧i z.iz3.2Lie derivationsIn this section weﬁnd the commutation rules of the Lie derivatives with functions,i.e.the elements of the algebra A,their diﬀerentials,etc.,using the approach of[1]as follows:•the relations between the Lie derivatives and the elements of AL x x=1+x L x,L x y=q−1y L x,L x z=q−1z L x,L y x=q x L y,L y y=1+y L y,L y z=q−1z L y,L z x=q x L z,L z y=q y L z,L z z=1+z L z.•The relations of the Lie derivatives with the diﬀerentialsL x d x=d x L x,L x d y=q−1d y L x,L x d z=q−1d z L x,L y d x=q d x L y,L y d y=d y L y,L y d z=q−1d z L y,L z d x=q d x L z,L z d y=q d y L z,L z d z=d z L z.Other commutation relations can be similarly obtained.To complete the descrip-tion of the above scheme,we get below the remaining commutation relations as follows:•the Lie derivatives and partial derivativesL x∂x=∂x L x,L x∂y=q∂y L x,L x∂z=q∂z L x,L y∂x=q−1∂x L y,L y∂y=∂y L y,L y∂z=q∂z L y,L z∂x=q−1∂x L z,L z∂y=q−1∂y L z,L z∂z=∂z L z.•the inner derivations∧i y=−q i y∧i x,ix∧i z=−q i z∧i x,ix∧i z=−q i z∧i y.iy•the Lie derivatives and the inner derivationsL x i x=i x L x,L x i y=q i y L x,L x i z=q i z L x,L y i x=q−1i x L y,L y i y=i y L y,L y i z=q i z L y,L z i x=q−1i x L z,L z i y=q−1i y L z,L z i z=i z L z.•the Lie derivativesL x L y=q L y L x,L x L z=q L z L x,L y L z=q L z L y.Note that the Lie derivatives can be written as follows:L x=x−1T x−x−1yx−1T y,L y=x−1T y,L z=T z.ACKNOWLEDGMENTThis work was supported in part by TBTAK the Turkish Scientiﬁc and Technical Research Council.REFERENCES1.Celik,Salih:J.Math.Phys.47(8):Art.No:0835012.Celik,Sultan A.and Yasar,E.:Czech.J.Phys.56(2006),229.3.Chryssomalakos,C.,Schupp P.and Zumino,B.:”Induced extended calculuson the quantum plane”,hep-th/9401141.4.Dobrev,V.K.:J.Math.Phys.33(1992),3419.5.Manin,Yu I.:”Quantum groups and noncommutative geometry”,(MontrealUniv.Preprint,1988).6.Schupp,P.,Watts,P.,Zumino,B.:Lett.Math.Phys.25(1992),139.7.Schupp,P.,Watts P.,Zumino,B.:”Cartan calculus on quantum Lie alge-bras”,hep-th/9312073.8.Schupp,P.:”Cartan calculus:Diﬀerential geometry for quantum groups”,hep-th/9408170.9.A.Sudbery,A.:Proc.Workshop on Quantum Groups,Argogne(1990)eds.T.Curtright,D.Fairlie and C.Zachos,pp.33-51.10.Wess,J.and Zumino,B.:Nucl.Phys.(Proc.Suppl.)18B(1990),302.11.Woronowicz,S.L.:Commun.Math.Phys.111(1987),613.12.Woronowicz,mun.Math.Phys.122(1989),125.。

calculus

calculusCalculus: An Introduction to the Mathematical Study of ChangeIntroductionCalculus is a branch of mathematics that deals with the concept of change. It provides a powerful framework for understanding and analyzing a wide range of phenomena, from the motion of objects to the behavior of complex systems. In this document, we will explore the fundamentals of calculus, its applications, and some key concepts that underpin this fascinating field of study.History of CalculusThe origins of calculus can be traced back to ancient civilizations such as Babylon and Egypt, where rudimentary techniques for solving geometric problems and calculating areas and volumes were developed. However, it was in the 17th century that calculus took significant leaps forward with the contributions of two great mathematicians, Isaac Newton and Gottfried Wilhelm Leibniz.Newton and Leibniz independently developed a framework that allowed for the systematic treatment of rates of change and accumulation. Newton's approach, known as the method of fluxions, focused on the concept of instantaneous rates of change. Leibniz, on the other hand, introduced the notation we still use today, including the integral (∫) and the derivative (d/dd).Calculus ConceptsThe two main branches of calculus are differential calculus and integral calculus. Differential calculus focuses on the study of rates of change and slopes of curves, while integral calculus deals with the accumulation of quantities and the calculation of areas and volumes.Derivatives: The derivative of a function represents the rate at which the function is changing at any given point. It allows us to determine slopes of curves, maximal and minimal values, and even the behavior of functions on a larger scale. The derivative of a function is denoted by d′(d) or dd/dd.Integrals: Integrals provide a way to calculate the accumulation of quantities over intervals. They allow us to compute areas, volumes, and the total change of a function over a given interval. The integral of a function is denoted by ∫d(d)dd.Applications of CalculusCalculus has a wide range of applications in various fields, including physics, engineering, economics, and biology. Here are a few examples of how calculus is used in these disciplines:Physics: Calculus helps describe and predict the motion of objects, whether they are falling bodies, projectiles, or astronomical bodies. It also plays a crucial role in understanding concepts like acceleration, velocity, and force.Engineering: Engineers use calculus to analyze and design structures, control systems, and electrical circuits. Calculus is instrumental in optimizing systems by determining the optimal values of various parameters.Economics: Calculus is used in economics to model and analyze the behavior of markets, consumers, and producers. It provides a framework for understanding concepts such as supply and demand, marginal utility, and optimization of profit functions.Biology: Calculus is essential in mathematical modeling of biological processes, such as population growth, the spread of diseases, and the behavior of ecosystems. It enables researchers to make predictions and understand complex interactions within these systems.ConclusionCalculus is a fundamental branch of mathematics that revolutionized the way we understand and analyze change. Its concepts and principles have wide-ranging applications across many disciplines. From describing the motion of objects to modeling economic behavior and biological processes, calculus provides deep insights into the underlying mathematics governing these phenomena. As you continue your study of calculus, you will uncover even more fascinating applications and develop a deeper appreciation for its significance in the world around us.。

微积分英文版课件

Applications of Derivatives
Local Extrema
Discover how derivatives help identify local maximums and minimums of functions.
Mean Value Theorem
Explore the mean value theorem and its applications in calculus.
Gradients and Directional Derivatives
2
derivatives and their applications in multivariable calculus.
Learn about gradients and
directional derivatives for
Derivatives
1
Definition of a Derivative
Uncover the definition and
Differentiability and Continuity
2
fundamental properties of derivatives.
Understand the relationship
Discover the conditions for a function to be continuous and its implications.
Explore the different types of discontinuities and their characteristics.
Conclusion
Review of Key Concepts

数学分析ch11-1euclid空间上的基本的定理

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详细描述
勾股定理表述为“直角三角形斜边的平方等于两直角边的平方和”，这个定理的证明需要深入的数学知识，包括代数、三角学和解析几何等。在欧几里得几何中，勾股定理的证明通常依赖于其他的基本假设和定理，而这些假设和定理本身也需要复杂的证明。
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THANKS
感谢观看
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REPORTING
在光学中，欧几里得空间也被用来描述光线的传播路径和方向等物理量，使得光学的研究更加深入和系统化。
在经典力学中，欧几里得空间被用来描述物体的运动轨迹和速度等物理量，使得经典力学的研究更加严谨和精确。
在工程学中的应用
欧几里得空间在工程学中也有着广泛的应用，如建筑设计、机械
设计、电子工பைடு நூலகம்等。
在建筑设计领域，欧几里得空间被用来描述建筑物的形状、大小和位置等参数，使得建筑设计更
四边形。
黎曼几何
黎曼几何是以德国数学家黎曼命名的几何体系，它以流形为研究对象，用度量张量来描述空间中的距离和角度。
黎曼几何是非欧几里得几何的一种推广，它允许空间中的距离和角度在不同点处有不同的值。
黎曼几何在广义相对论中是描述引力场的数学工具，也是微分几何的重要分支。
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PART 05
欧几里得空间中的问题与挑战
加精确和可靠。
在机械设计领域，欧几里得空间被用来描述机器零件的位置和运动轨迹等参数，使得机械设计更
加严谨和精确。
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PART 04
欧几里得空间的扩展
REPORTING
射影几何
射影几何是研究图形在射影变换下不变性质的几何分支，主要研究点、直线和平面之间的相互关系。

calculus on manifolds 解答手册

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Integration on Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.1 Algebraic Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
t u
I Exercise 2 (1-2). When does equality hold in Theorem 1-1 (3) kx C y k 6 kx k C ky k ? ˇ ˇ Proof. We reprove that ˇhx ; y iˇ 6 kx k ky k for every x ; y 2 Rn . Obviously, if x D 0 or y D 0, then hx ; y i D kx k ky k D 0. So we assume that x ¤ 0 and y ¤ 0. We ﬁrst ﬁnd some w 2 Rn and ˛ 2 R such that hw; ˛ y i D 0. Write w D x ˛ y . Then 0 D hw; ˛ y i D hx ˛ y ; ˛ y i D ˛ hx ; y i ˛ 2 ky k2
ˇ ˇ 0 11=2 0 11=2 ˇX ˇ X X ˇ ˇ ˇ @ f .xi / g .xi / xi ˇ f .xi /2 xi A @ g .xi /2 xi A : ˇ ˇ6 ˇ i ˇ i i

Fuzzy spaces, the M(atrix) model and the quantum Hall effect

IC/2004/37 CCNY-HEP-04/5
Fuzzy spaces, the M(atrix) model and the quantum Hall eﬀect 1
arXiv:hep-th/0407007v2 17 Jul Department of Physics and Astronomy Lehman College of the CUNY Bronx, NY 10468 E-mail: karabali@ V. P. NAIR Physics Department City College of the CUNY New York, NY 10031 E-mail: vpn@ S. RANDJBAR-DAEMI Abdus Salam International Centre for Theoretical Physics Trieste, Italy E-mail: seif@ictp.trieste.it
2
A natural family of symplectic manifolds of ﬁnite volume are given by the co-adjoint orbits of a compact semisimple Lie group G. (In this case, there is no real distinction between co-adjoint and adjoint orbits. For quantization of co-adjoint orbits, see [7, 8].) One can quantize such spaces, at least when a Dirac-type quantization condition is satisﬁed, and the resulting Hilbert space corresponds to a unitary irreducible representation of the group G. In this way, we can construct fuzzy analogs of spaces which are the co-adjoint orbits. In the following, we will work through this strategy for the case of CPk = SU (k + 1)/U (k). In this review, we will focus on fuzzy spaces, how they may appear as solutions to M(atrix) theory and their connection to generalizations of the quantum Hall eﬀect. There is a considerable amount of interesting work on noncommutative spaces, particularly ﬂat spaces, in which case one has inﬁnite-dimensional matrices, and the properties of ﬁeld theories on them. Such spaces can also arise in special limits of string theory. We will not discuss them here, since there are excellent reviews on the subject [9].

罗氏欧几里得立方体

罗氏欧几里得立方体全文共四篇示例，供读者参考第一篇示例：罗氏欧几里得立方体欧几里得（Euclid）是古希腊数学家、几何学家，被认为是欧几里得几何学的奠基人，他的代表作《几何原本》对西方数学的发展产生了深远的影响。

而在罗氏(Roche)体中，这位伟大的数学家的名字也得到了永恒的记忆。

罗氏欧几里得立方体是一种几何学中的立体图形，名字来源于欧几里得，因为这种立方体的特点与欧几里得的几何学原理密切相关。

罗氏欧几里得立方体具有独特的形状和性质，被广泛应用于数学、工程学和建筑领域等方面。

罗氏欧几里得立方体的结构十分坚固，具有良好的稳定性和强度，常被用于建筑中的支撑结构或者作为装饰物。

这种立方体的形状规整，每个面都是一个正方形，侧面是一个梯形，上下底面则是正方形。

罗氏欧几里得立方体的对角线相等，底面和侧面的边长也是相等的，整体看起来非常稳重和美观。

除了在建筑领域中得到广泛应用外，罗氏欧几里得立方体还被广泛运用于数学和几何学的研究当中。

它的特殊形状使其成为研究几何学性质和定理的理想实例，尤其是在立体几何学和立体图形的研究中。

研究罗氏欧几里得立方体可以帮助人们更好地理解不同立体图形之间的关系和相互作用，从而推动几何学理论的发展。

在工程学领域，罗氏欧几里得立方体的稳定性和强度使其成为设计结构的理想选择。

工程师们可以利用这种立方体的形状和性质来设计各种复杂的支撑结构和建筑物，保证其稳固性和安全性。

罗氏欧几里得立方体在工程学中的应用范围非常广泛，涵盖了建筑、桥梁、隧道等各个领域。

罗氏欧几里得立方体是一种具有独特形状和性质的立体图形，广泛应用于数学、工程学和建筑领域。

它的稳定性和强度使其在设计结构和建筑物时发挥着重要作用，同时也通过研究几何学性质和定理来促进数学理论的发展。

未来，随着科学技术的不断进步，罗氏欧几里得立方体的应用范围将会更加广泛，为人类的发展和进步做出新的贡献。

第二篇示例：罗氏欧几里得立方体是英国数学家罗夫·罗奇在欧几里德几何学的基础上所建立的一种新的几何学，在立方体世界里展现了不可思议的魔幻世界。

微积分的名称ppt课件市公开课金奖市赛课一等奖课件

第6页
窮竭法
當矩形數目愈來愈多，它們面積之和會愈來愈迫近曲邊形面積。
第7页
不可分元法
對一個平面片而言，其“不可分元”是指它一條弦(chord)
對一個立體而言，其“不可分元”是指它一個平面截面
第8页
不可分元法
第9页
Democritus(德謨克利特 460~370BC)
根據不可分元想法，推出稜錐(或圓錐)體積是含有同樣底和高稜柱(或圓柱)體積三分之一。
26 歲出任駐巴黎大使，結識 Huygens，並開
始鑽研數學，研讀 Descartes 及 Pascal 的著作
27 歲出使倫敦
30 歲任 Elector of Hanover 的顧問及圖書館
長
1700 年服務於 Elector of Brandenburg 宮廷
1716 年去世
萊布尼茲十分博學，對法律、歷史、神學、
但書寫得非常艱澀。 1703 年任英國皇家學會主席 62 歲放棄教授職位，轉任倫敦大英造幣廠，同年被封為爵士死時年 85 歲
第25页
第26页
萊布尼茲(Leibnitz 1646~1716)
第27页
萊布尼茲(Leibnitz 1646~1716)貢獻
1684年發表《一種求極大極小和切線新办法，適用於分式和無窮量，以及這種新办法奇妙類型計算》
《莊子天下篇》曰：「一尺之棰，日取其半，萬世不竭。」
劉徽創割圓術，謂「割之彌細，所失彌少。割之又割，以至於不可割，則與圓合體而無所失矣。」
祖氏父子：『夫疊棋成立積，緣冪勢既同，則積不容異。』(注：“冪”指截面面積，“勢”指高度)
第15页
問題引路
第一類問題：已知距離表為時間函數，求速度和加速度。反過來，已知加速度表為時間函數，求距離和速度。 (例：Galileo曾探討此類問題)

大一上册微积分课件《CalculusI》

• But it does not tell us how to find these extreme values.
• We start by looking for local extreme values.
Fermat's Theorem If f has a local maximum or minmum at c , and if f (c) exists , then f (c) =0
minimum m can’t be taken on at the endpoints
at the same time. So ,without loss of generality,
we suppose that M is taken on at some point c
in (a,b), so f(c) =M is a local maximum and f(x)
is differentiable at x=c, by the Fermat’s
theorem, we know that
f (c) 0
c
Caution• :The conditions cannot be weakened.
Y
Example
f (x) x x [1,1]
-1
0 1X
Example
• 1) Find the values of f at the critical numbers of f
in (a,b).
• 2) Find the values of f at the endpoints of the interval.
• 3) The largest of the values from step 1) and step 2) is the absolute maximum value; the smallest of these values is the absolute

continental calculus league 题目

continental calculus league 题目全文共四篇示例，供读者参考第一篇示例：continental calculus league 是一个国际性的数学竞赛联赛，旨在促进世界各国学生对微积分的学习和掌握。

该联赛每年举办一次，参赛学生需要通过各种难度不同的数学题目来展示自己的数学能力。

这不仅是一个比赛，更是一个学习的过程和一种挑战。

在这个比赛中，学生们可以在全球的舞台上展示自己的才华，并与来自世界各地的同龄人一较高下。

continental calculus league 的题目涵盖了微积分的各个方面，包括定积分、不定积分、微分方程等。

这些题目要求学生具备扎实的数学基础和逻辑思维能力，能够独立分析和解决问题。

在参加这个比赛之前，学生们需要通过多门预备课程的学习和考试，为参赛做好准备。

continental calculus league 是一个非常具有挑战性和意义的数学竞赛联赛，可以帮助学生们提高自己的数学水平和解题能力，并为他们的未来发展打下良好的基础。

希望越来越多的学生能够参加这个比赛，享受数学带来的乐趣和挑战。

【文章结束】第二篇示例：大家好，今天我要和大家分享的是关于Continental Calculus League（大陆微积分联赛）的题目。

微积分是数学中一个非常重要的分支，它被广泛应用在自然科学、工程等各个领域。

而Continental Calculus League则是一个专门为热爱微积分的学生举办的比赛，旨在培养学生的数学思维和解决问题的能力。

在Continental Calculus League的比赛中，参赛选手会面对一系列各种难度的微积分题目，涵盖了微积分的各个方面，如极限、导数、积分等。

这些题目既考验了参赛选手的理论基础，又考察了他们的实际运用能力。

通过比赛，参赛选手们可以锻炼自己的数学思维和解决问题的能力，同时还可以结交志同道合的伙伴，共同进步。

下面，让我们来看一些来自Continental Calculus League的经典题目：1. 求函数f(x) = x^2 - 3x + 2在区间[0,1]上的最大值和最小值。

欧氏平面快速k-瓶颈斯坦纳树近似算法

欧氏平面快速k-瓶颈斯坦纳树近似算法
李晓丹
【期刊名称】《电脑编程技巧与维护》
【年(卷),期】2016(000)006
【摘要】瓶颈k-Steiner树问题描述如下：给定n个点和一个正整数k，寻找一棵Steiner树用至多k个Steiner点将n个点连接起来，使得此Steiner树的最长边最短。

L. Wang和D.-Z. Du证明了适用于欧几里得平面瓶颈斯坦纳树算法的近似性能比为2，并且给出了一个适用于该问题时间复杂度为(nlogn+kn)的算法，在欧几里得平面上和近似性能比为2的前提下，通过引入最大堆和斐波那契堆分别对该算法进行优化，优化后算法的时间复杂度分别达到(nlogn+klogn)和(nlogn+k)，优化后的算法在现实中可以更好地应用。

【总页数】4页(P31-34)
【作者】李晓丹
【作者单位】中南民族大学计算机科学学院，武汉430074
【正文语种】中文
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Consider vector operations in the same space.
2.3 Definition: A vector field V on R3 is a function that assigns to each point p of R3 a tangent vector V(p) to R3 at p.
U1 (p) (1,0,0) p U 2 ( p) (0,1,0) p U 3 (p) (0,0,1) p
For each point p of R3, We call U1, U2,U3-collectively-the natural frame field on R3.
2.5 Lemma: If V is a vector on R3 , there are three uniquely determined real-valued functions v1,v2,v3 on R3 such that
idean Space
1.1 Definition:
Euclidean 3-Space R3 is the set of all ordered triples of real numbers. Such a triple p=(p1,p2,p3) is called a point of R3.
V ( p) v1U1 v2U2 v3U3 viUi
Question1: Let V=xU1+yU2 and W=2x2U2-U3, compute the vector field WxV, and find its value at the point p=(-1,0,2).
Course information

Expectations
Keep up to date Look over textbooks or lecture notes as soon as possible after class, and definitely before the next lecture. Do the Problem This course is not a spectator sport. Doing problem is even more important than learning theory. Ask Questions Confused? Ask!
vp
p + v = (3,4,5)
p = (1,1,3)
vp
vq
Vp // Vq
Tangent Vectors
2.2 Definition: Let p be a point of R3. The set Tp(R3) consisting of all tangent vectors that have p as point of application is called tangent space of R3 at p.
(V W )(p) V (p) W (p) ( fV )(p) f (p)V (p)
Point-wise principle
Natural frame field
2.4 Definition: Let U1, U2,U3 be the vector fields on R3 such that
1.2 Definition: A real-valued function f on R3 is differentiable (or infinitely differentiable, or smooth, or of class ) provided that all partial derivatives of f, of all orders, exist and are continuous.
Differential Geometry Lecture 1
Informal overview, Preliminary knowledge
Ming Chen Nov. 1, 2011
1
What is Differential Geometry
Applications
a) In Physics, three uses will be mentioned: Differential geometry is the language in which Einstein’s general theory of relativity is expressed. The study of electromagnetism. Differential geometry has applications to both Lagrangian mechanics and Hamiltonian mechanics. b) In economics, differential geometry has applications to the field of econometrics. c) Geometric modeling (including computer graphics) and computer-aided geometric design draw on ideas from differential geometry. d) In engineering, differential geometry can be applied to solve problems in digital signal processing.o e) In probability, statistics, and information theory, one can interpret various structures as Riemannian manifolds, which yields the field of information geometry, particularly via the Fisher information metric. f) In structural geology, differential geometry is used to analyze and describe geologic structures. g) In computer vision, differential geometry is used to analyze shapes. h) In image processing, differential geometry is used to process and analyse data on non-flat surfaces.
Rolled up
Course information

Required Texts
Differential Geometry of Curves and Surfaces, by Manfredo do Carmo. Elementary Differential Geometry, by Barrett O’Neill
Directional derivatives
3.2 Lemma: If vp=(v1,v2,v3)p is a tangent vector to R3. Then
Vp [ f ] vi
f ( p) x i
Proof. Let p=(p1,p2,p3); then p+t v= (p1+t v1,p2+t v2,p3+t v3) Use the chain rule to compute the derivative at t=0 of the function f(p+t v)=f(p1+t v1,p2+t v2,p3+t v3) Since
d ( f ( p tv )) |t 0 dt
is called the derivative of f with respect to vp
Example 1: Compute vp[f] for the function f = x2yz, with p=(1,1,0) and v=(1,0,-3). p+tv=(1,1,0)+t(1,0,-3)=(1+t,1,-3t) f(p+tv)=(1+t)2.1.(-3t)=-3t-6t2-3t3. d(f(p+tv))/dt = -3-12t-9t2, hence at t=0, we get vp[f]=-3.
d(pi+t vi)/dt=vi
Proved.
Example 1: Compute vp[f] for the function f = x2yz, with p=(1,1,0) and v=(1,0,-3).
f f f 2 xyz, x 2 z, x 2 y, x y z

Assessment
Written exam 90% (closed book). To qualify for the written exam, you have to attend the course regularly and participate actively. Performance in class 10%. Give correct answers for questions or ask a good question in the class. Bonus score. 10 points. Using Visual C++ or Matlab or Maple or any other math packages to solve one assignment. (No cheating is allowed, or you will get “F” directly in the final exam . )