A polynomial approach to the constructive induction of structural knowledge

1. Introduction. Mathematical models of many physical, biological, and economic processes involve systems of linear, constant coefficient ordinary differential equations x ˙ (t) = Ax(t). Here A is a given, fixed, real or complex n-by-n matrix. A solution vector x(t) is sought which satisfies an initial condition x(0) = x0 . In control theory, A is known as the state companion matrix and x(t) is the system response. In principle, the solution is given by x(t) = etA x0 , where etA can be formally defined by the convergent power series etA = I + tA + t 2 AE MOLER AND CHARLES VAN LOAN
We will primarily be concerned with matrices whose order n is less than a few hundred, so that all the elements can be stored in the main memory of a contemporary computer. Our discussion will be less germane to the type of large, sparse matrices which occur in the method of lines for partial differential equations. Dozens of methods for computing etA can be obtained from more or less classical results in analysis, approximation theory, and matrix theory. Some of the methods have been proposed as specific algorithms, while others are based on less constructive characterizations. Our bibliography concentrates on recent papers with strong algorithmic content, although we have included a fair number of references which possess historical or theoretical interest. In this survey we try to describe all the methods that appear to be practical, classify them into five broad categories, and assess their relative effectiveness. Actually, each of the “methods” when completely implemented might lead to many different computer programs which differ in various details. Moreover, these details might have more influence on the actual performance than our gross assessment indicates. Thus, our comments may not directly apply to particular subroutines. In assessing the effectiveness of various algorithms we will be concerned with the following attributes, listed in decreasing order of importance: generality, reliability, stability, accuracy, efficiency, storage requirements, ease of use, and simplicity. We would consider an algorithm completely satisfactory if it could be used as the basis for a general purpose subroutine which meets the standards of quality software now available for linear algebraic equations, matrix eigenvalues, and initial value problems for nonlinear ordinary differential equations. By these standards, none of the algorithms we know of are completely satisfactory, although some are much better than others. Generality means that the method is applicable to wide classes of matrices. For example, a method which works only on matrices with distinct eigenvalues will not be highly regarded. When defining terms like reliability, stability, and accuracy, it is important to distinguish between the inherent sensitivity of the underlying problem and the error properties of a particular algorithm for solving that problem. Trying to find the inverse of a nearly singular matrix, for example, is an inherently sensitive problem. Such problems are said to be poorly posed or badly conditioned. No algorithm working with finite precision arithmetic can be expected to obtain a computed inverse that is not contaminated by large errors. An algorithm is said to be reliable if it gives some warning whenever it introduces excessive errors. For example, Gaussian elimination without some form of pivoting is an unreliable algorithm for inverting a matrix. Roundoff errors can be magnified by large multipliers to the point where they can make the computed result completely erroneous, but there is no indication of the difficulty. An algorithm is stable if it does not introduce any more sensitivity to perturbation than is inherent in the underlying problem. A stable algorithm produces an answer which is exact for a problem close to the given one. A method can be stable and still not produce accurate results if small changes in the data cause large changes in the answer. A method can be unstable and still be reliable if the instability can be detected. For example, Gaussian elimination with either partial or complete pivoting must be regarded as a mildly unstable algorithm because there is a possibility that the matrix elements will grow during the elimination and the resulting roundoff errors will not be small when compared with the original data. In practice, however, such growth is rare and can be detected.

Lesson 1Mathematics as a Language of Science assert vt. 断言；坚持主张；维护表明qualitative adj. 性质的；定性的 quantitative adj. 量的；数量的；定量的；与数量有关的 astronomy n. 天文学 postulate n. 假定, 基本条件, 基本原理 vt. 要求, 假定 vi.要求 hypothetical adj. 假设的, 假定的,爱猜想的Lesson 2 deduction n. 减除, 扣除, 减除额, 推论, 演绎 induction n. 归纳；归纳法；归纳所得之结论 verification n. 验证；证实 correlate vt. 使相互关联 vi. 和...相关discard vt. 丢弃, 抛弃 v. 放弃 discredit n. 不信任；失信 consistent adj. 一致的, 调和的, 坚固的, [数、统]相容的 inadequacy n. 不充分 ,不适当,不适合,不足额 conic, conical adj 圆锥的；圆锥形的 ellipse n. 椭圆, 椭圆形 ellipt (n.) hyperbolic adj.双曲线的 hyperbola (n.) parabolic adj. 用寓言表达的: 抛物线的，像抛物线的parabola (n.) algebraic adj. 代数的, 关于代数学的 mineralogy n. 矿物学 refraction n. 折光, 折射 stimulus n. 刺激物, 促进因素, 刺激, 刺激 impetus n. 冲力推动力；刺激 Lesson 3 Axioms, definitions and Theorems axiom n. [数]公理 definition n. 阐明；确定定义；界说 extravagant adj. 奢侈的, 浪费的, 过分的, 放纵的 collinear adj. 在同一直线上的, 同线的 convex adj. 凸出的；凸面的 segment n. 部分；片段；节, 弓形；圆缺；弧形, 线段 conswquently adv. 从而, 因此 in terms of adv. 根据, 按照, 用...的话, 在...方面 pretense n. 主张, 要求, 伪称, 借口, 自称 Lesson 4 Geometry and Geometrical terms term n. 学期, 期限, 期间, 条款, 条件, 术语 triangle n. [数]三角形, 三人一组, 三角关系 parallelogram n. 平行四边形 straight angle n. [数]平角right angle n. 直角 acute angle n. 锐角 obtuse angle n. 钝角 reflex angle n. 优角 rectilinear adj 直线的；由直线组成的；循直线进行的 isosceles triangle n. 等腰三角形 equilateral triangle n. 等边三角形 right triangle n. 直角三角形 obtuse triangle n. 钝角三角形 acute triangle n. 锐角三角形 equiangular triangle n. 正三角形,等角三角形 hypotenuse n. （直角三角形的）斜边 circle 圆 center 中心；中央；圆心 diameter n. 直径 radius n. 半径, 范围, 辐射光线, 有效航程, 范围, 界限circumference n. 圆周, 周围 Lesson 5 The Method of Limits limit n. 限度,极限,极点 infinite adj. 无限的；无穷的 infinitesimal adj. 无穷小的, 极小的, 无限小的calculus n. 微积分学, 结石 exemplify vt. 例证, 例示, 作为...例子 inscribe v. 记下 polygon n. [数]多角形, 多边形 diminish v. (使)减少, (使)变小 curvilinear adj 曲线的, 由曲线组成的 intuition n. 直觉, 直觉的知识 integral n. [数学] 积分, 完整, 部分 defective adj. 有缺陷的, (智商或行为有)欠缺的 differential coefficient 微分系数 arithmetical adj. 算术的, 算术上的 convergence n. 集中, 收敛 criterion n. (批评判断的)标准, 准据, 规范 sequence n. 次序, 顺序, 序列 irrational numbers n.[数]无理数 domain ，定义域 contradiction 矛盾 reversal n. 颠倒, 反转, 反向, 逆转, 撤销 Lesson 6 Function continuous variable 连续变量;［连续变数］ variation 变分, 变化 interval 区间 independent variable 自变量 dependent variable 应变量rectangular coordinate 直角坐标 abscissa n. 〈数〉横坐标 ordinate n. [数]纵线, 纵座标 gradient adj. 倾斜的 n. 梯度, 倾斜度, 坡度 slope n. 斜坡, 斜面, 倾斜 v. (使)顺斜 Lesson 7 Differential and Integral calculus differential adj. 微分的 n. 微分 (differentiation) Integral n. [数学] 积分, 完整, 部分 (integration) calculus n. 微积分学, 结石 interrelation n. 相互关系 trigonometry n. 三角法 exponential adj. 指数的, 幂数的 logarithm n. [数] 对数 derivative n. 导数；微商 tangent n. 切线, [数]正切 counterclockwise adj. 反时针方向的 adv. 反时针方向 (clockwise) definite integral 定积分 approximation n. 接近, 走近, [数]近似值 culminate v. 达到顶点mean n. 平均数, 中间, 中庸 differential equation 微分方程 extreme value n. 极值multiple integral 多重积分 double integral line integral functional analysis 泛函分析 Lesson 8 The Concept of Cardinal Number (I) cardinal number n. 基数（如：1, 2, 3, ... 有别于序数） denumerable adj. 可数的 aggregate n. 合计, 总计, 集合体 adj. 合计的, 集合的, 聚合的 v. 聚集, 集合, 合计 purport n. 主旨 v. 声称fancier n. 空想家, 培育动物(或植物)的行家, 爱好者 sniff v. 用力吸, 嗅, 闻到, 发觉, 轻视, 用力吸气 n. 吸, 闻, 吸气声, 嗤之以鼻 scheme n. 安排, 配置, 计划, 阴谋, 方案, 图解, 摘要 v. 计划, 设计, 图谋, 策划, * n.（计算数学）方法，格式 superior n. 长者, 高手, 上级 adj. 较高的, 上级的, 上好的, 出众的, 高傲的 cumbersome adj.讨厌的, 麻烦的, 笨重的 instruction n. 指示, 用法说明(书), 教育, 指导, 指令drastically adv. 激烈地, 彻底地 conservation 守衡律 quadrature n. 求积, 求积分interpolation n. 插值 extrapolation n. [数]外推法, 推断 internal point 内点identical adj. 同一的, 同样的 generalized solution 广义解 functional 泛函hydrodynamics 流体力学，水动力学 divergence 发散（性），梯度，发散量 play an important (fundamental ... ) role 起着重要的（...）作用 integro-interpolation method 积分插值法 Variational method 变分方法 comparatively adv. 比较地, 相当地deficiency n. 缺乏, 不足 fictive adj. 虚构的, 想象上的, 虚伪的 self-adjoint (nonself-adjoint) 自治的，自伴的，自共轭的 finite element method 有限元法 spline approximation 样条逼近 Particles-in-the-Cell 网格质点法 herald n. 使者, 传令官, 通报者, 先驱, 预兆 vt. 预报, 宣布, 传达, 欢呼 advection n. 水平对流phenomenological adj. 现象学的, 现象的 fluctuation n. 波动, 起伏 optimism n. 乐观, 乐观主义 pessimism n. 悲观, 悲观主义 unjustified adj 未被证明其正确的mean-square 均方 dispersion n. [数] 离差, 差量 Polynomial n adj. [数]多项式的interpolation 插值 arithmetic n. 算术, 算法 rounding errors 舍入误差 multiple n. 倍数, 若干 subjective adj. 主观的, 个人的 objective adj. 客观的, outcome n. 结果,成果 pattern n. 样品 toss v. 投, 掷 exhaust vt. 用尽, 耗尽, 抽完, 使精疲力尽divisible adj. 可分的 dice, die n. 骰子 assign vt. 分配, 指派 attach vt. 缚上, 系上, 贴上 v. 配属, 隶属于 pitfall n. 缺陷 chairperson 主席 mechanics n. (用作单数)机械学、力学, (用作复数)技巧, 结构 statics n. [物]静力学 dynamics n. 动力学adequately adv. 充分地 celestial adj. 天上的 macroscopic adj. 肉眼可见的, 巨观的classical field theory 经典场理论 rigit adj. 刚硬的, 刚性的, 严格的 elastic adj.弹性的 plastic n. 可塑的,塑性的,塑料的 quantum n. 量, 额, [物] 量子, 量子论inception n. 起初, 获得学位 pertain v. 适合, 属于 gravitation n. 地心吸力, 引力作用 tide n. 潮, 潮汐, 潮流, 趋势 monumental adj. 纪念碑的, 纪念物的, 不朽的, 非常的 encompass v. 包围, 环绕, 包含或包括某事物 ingredient n. 成分, 因素acquainted adj. 有知识的, 知晓的 synonymous adj. 同义的 configuration n. 构造, 结构, 配置, 外形 reference n. 提及, 涉及, 参考, 参考书目 inertia n. 惯性, 惯量attribute 特性 momentum n. 动量 proportional adj. 比例的, 成比例的, 相称的, 均衡的 designate 指明 negligible adj. 可以忽略的, 不予重视的 projectile n. 射弹 adj. 发射的 ballistics n. 弹道学, 发射学 intractable adj. 难处理的 {Mechanics of a Particle in consequence of adv. 由于的...缘故 exert vt. 尽(力), 施加(压力等), 努力 v. 发挥, 竭尽全力, 尽 galaxy n. 星系, 银河, 一群显赫的人, 一系列光彩夺目的东furnish vt. 供应, 提供, 装备, 布置 v. 供给 torque n. 扭矩, 转矩 moment 力矩的friction 摩擦 dissipation n. 消散, 分散, 挥霍, 浪费, 消遣, 放荡, 狂饮 infer v.推断Hooke s Law and Its Consequences elasticity n. 弹力, 弹性 constitutive adj. 构成的, 制定的 atomistic adj. 原子论的 crack n. 裂缝, 噼啪声 v. (使)破裂, 裂纹, (使)爆裂 continuum mechanics n. 连续介质力学 superposition n. 重叠, 重合, 叠合strain n. 过度的疲劳, 紧张, 张力, 应变 vt. 扭伤, 损伤 v. 拉紧, 扯紧, (使)紧张, 尽力 thermodynamics n. [物] 热力学 reckon vt. 计算, 总计, 估计, 猜想 vi. 数, 计算, 估计, 依赖, 料想 lesson 20 strength 强度 load 载荷 empirical 以经验为依据的member 构件 isolated 孤立的 segment 部分、段、节 stress 应力 strain 应变 tension 拉伸 shear 剪切 bend 弯曲 torsion 扭转、扭力 insofar 在……范围 cohesive 内聚性的 tensile 拉力、张力 stiffness 硬度 furnish 供给 Lesson 23 Fluid Mechanics eruption 喷发、爆发 turbulent 湍流 laminar 层流 isothermal 等温 isotropic 各向同性 prevalent 普遍的、流行的 tornado 旋风、飓风 eddy 旋涡 viscosity 粘性、粘度nonviscous 无粘性的 rotation 旋转 adiabatic 绝热的 reversible 可逆的 isentropic 等熵的 instant 瞬时的 streamline 流线 stream tube 流管 tangential 切线的incompressible 不可压缩的 resultant 合成的,组合的 downstream 下游的,顺流的elbow 弯管,肘 similitude 相似性 hydraulic 水力的,水力学的 predominante 占主导地位 spillway (河或水坝的)放水道,泄洪道 prototype 原型,样板 Lesson 24 Mechanical Vibration repetitive 重复的,反复的 periodic 周期的,定期的 tidal 潮的,像潮的stationary 固定的,不动的 vibratory 振动的,摆动的 propagation 传播 couple v .连接,连合 acoustic 听觉的,声学的 annoyance 烦恼,困惑 adjacent 接近的,邻近的 damp 阻尼,衰减 restore 复职,归还 neutral 平衡 exciting force 激励力 resonant adj. 共振的,谐振的 stiffness 刚度,刚性 proportionality 成比例地 inclusion 包含,包括magnitude 数值,大小 substantially adv. 实质上的 perturb 干扰,扰乱 resonance n. 共振 vibratory adj. 振动的, 可知的 perceptible 可见的,可知的 adudible 听得见的,可闻的 foregoing 前述的 impulsive 冲击的 shock 冲击 Fourier series 傅里叶级数excitation 激发,激励 discrete 分离,离散的 contend with 向…作斗争 compressor 压气机 fatigue 疲劳 perceptible 可见的,可知觉的 shredder 切菜器 disposal 处理urban 都市的 metropolitan 大都市的 at-grade 在同一水平面上 elevated 高架的guideway 导轨 Lesson 25 A prefect to the Continuum Mechanics preface 序言 continuum 连续 pl. continuua rigid body 刚体 contemporary 当代的,同时期的 widespread 分布广的, 普及的 accommodate 容纳,使适应 medium 介质 plasticity 塑性 residual 剩余的,残留的 creep 蠕变,爬行,塑性变形 aging 老化 polymeric聚合(物)的 sandy 沙的,沙质的 aubterranean 地下的,隐藏的 essence 精髓,本质 thermodynamics 热力学self-similar 自相似 expedient 方便的 sonsolidate 把…联合为一体,统一 justify 证明…有理 radically 根本地,本质上 deliberate 从容不迫的,深思熟虑 Lesson 33 what is a computer Attribute v. 赋予 medieval 中世纪的 astronomer 天文学家 Mars 火星resemble vt. 像,相似 tedious adj. 冗长乏味的 pulp 浆状物,果肉 filter vt.过滤underlying adj. 潜在的, 基本的 ore n. 矿沙,矿石 perceive v. 察觉,看见intervention n. 干涉,插入 intelligent adj. 有智力的,聪明的 Lesson 34 A computer system manipulate vt. 操纵,使用 chip n. 芯片 etch vt. 蚀刻,蚀镂 fingernail 指甲mount vt. 安装,安置 assemble vt. 集合,聚集 cabinet 橱柜 execute vt. 执行,实现paycheck n.支付薪金的支票 bar chart 直方图 joystick 游戏杆 encounter vt. 遇到,遇上Mathematical Modeling industry n. 工业, 产业, 行业, 勤奋 commerce n. 商业complexity n. 复杂(性), 复杂的事物, 复杂性 career n. (原意:道路, 轨道)事业, 生涯, 速度 outset n. 开端, 开始 essence n. 基本, [哲]本质, 香精 advocation n.(=advocacy)拥护支持 provision n. 供应, (一批)供应品, 预备, 防备, 规定 publicize v. 宣扬 roundabout adj. 迂回的, 转弯抹角的 n. 道路交叉处的环形路, 迂回路线, 兜圈子的话 trial-error vt. n. 试制, 试生产 maneuverability n. 可操作性, 机动性vehicle n. 交通工具, 车辆, 媒介物, 传达手段 junction n. 连接, 接合, 交叉点, 汇合处 ponder v. 沉思, 考虑 contrive v. 发明, 设计, 图谋 snooker n. (=snooker pool)彩色台球, 桌球 context n. 上下文, 文章的前后关系 deviation n. 背离。

《线性代数》英文专业词汇序号英文中文1LinearAlgebra线性代数2determinant行列式3row行4column列5element元素6diagonal对角线7principaldiagona主对角线8auxiliarydiagonal次对角线9transposeddeterminant转置行列式10triangulardeterminants三角行列式11thenumberofinversions逆序数12evenpermutation奇排列13oddpermutation偶排列14parity奇偶性15interchange互换16absolutevalue绝对值17identity恒等式18n-orderdeterminantsn阶行列式19evaluationofdeterminant行列式的求值20Laplace’sexpansiontheorem拉普拉斯展开定理21cofactor余子式22Algebracofactor代数余子式23theVandermondedeterminant范德蒙行列式24bordereddeterminant加边行列式25reductionoftheorderofadeterminant降阶法26methodofRecursionrelation递推法27induction归纳法28Cramer′s rule克莱姆法则29matrix矩阵30rectangular矩形的31thezeromatrix零矩阵32theidentitymatrix单位矩阵33symmetric对称的序号英文中文34skew-symmetric反对称的35commutativelaw交换律36squareMatrix方阵37amatrixoforder m×n矩阵m×n38thedeterminantofmatrixA方阵A的行列式39operationsonMatrices矩阵的运算40atransposedmatrix转置矩阵41aninversematrix逆矩阵42anconjugatematrix共轭矩阵43andiagonalmatrix对角矩阵44anadjointmatrix伴随矩阵45singularmatrix奇异矩阵46nonsingularmatrix非奇异矩阵47elementarytransformations初等变换48vectors向量49components分量50linearlycombination线性组合51spaceofarithmeticalvectors向量空间52subspace子空间53dimension维54basis基55canonicalbasis规范基56coordinates坐标57decomposition分解58transformationmatrix过渡矩阵59linearlyindependent线性无关60linearlydependent线性相关61theminorofthe k thorderk阶子式62rankofaMatrix矩阵的秩63rowvectors行向量64columnvectors列向量65themaximallinearlyindependentsubsystem最大线性无关组66Euclideanspace欧几里德空间67Unitaryspace酉空间序号英文中文68systemsoflinearequations线性方程组69eliminationmethod消元法70homogenous齐次的71nonhomogenous非齐次的72equivalent等价的73component-wise分式74necessaryandsufficientcondition充要条件75incompatiable无解的76uniquesolution唯一解77thematrixofthecoefficients系数矩阵78augmentedmatrix增广矩阵79generalsolution通解80particularsolution特解81trivialsolution零解82nontrivialsolution非零解83thefundamentalsystemofsolutions基础解系84eigenvalue特征值85eigenvector特征向量86characteristicpolynomial特征多项式87characteristicequation特征方程88scalarproduct内积89normedvector单位向量90orthogonal正交的91orthogonalization正交化92theGram-Schmidtprocess正交化过程93reducingamatrixtothediagonalform对角化矩阵94orthonormalbasis标准正交基95orthogonaltransformation正交变换96lineartransformation线性变换97quadraticforms二次型98canonicalform标准型99thecanonicalformofaquadraticform二次型的标准型100themethodofseparatingperfectsquares配完全平方法101thesecond-ordercurve二次曲线102coordinatetransformation坐标变换。

数学符号公式缩写的英文发音数学缩写列表this article is a listing of abbreviated names of mathematical functions, function-like operators and other mathematical terminology.这篇文章是一个数学函数，类似于函数的操作符和其他的数学术语的缩写名列表。

this list is limited to abbreviations of two or more letters. the capitalization of some of these abbreviations is not standardized – different authors use different capitalizations.this list is inplete; you can help by expanding it.这个列表受限于两个或更多字母的缩略语。

其中，一些缩略语字母大写并不是标准的 - 不同的使用不同的大写形式。

ac – axiom of choice.[1] 选择公理a.c. – absolutely continuous. 绝对连续的acrd – inverse chord function. 逆弦函数adj – adjugate of a matrix. 矩阵的伴随矩阵a.e. – almost everywhere. 殆遍，几乎处处ai – airy function. 艾里函数al – action limit. 处置界限alt – alternating group (alt(n) is also written as an.) 交错群a.m. – arithmetic mean. 算数平均数arccos – inverse cosine function. 反余弦函数arccosec – inverse cosecant function. (also written as arccsc.) 反余割函数arccot – inverse cotangent function. 反余切函数arccsc – inverse cosecant function. (also written as arccosec.) 反余割函数arcexc – inverse excosecant function. (also written as arcexcsc, arcexcosec.) 反外余割函数arcexcosec – inverse excosecant function. (also written as arcexcsc, arcexc.) 反外余割函数arcexcsc – inverse excosecant function. (also written as arcexcosec, arcexc.) 反外余割函数arcexs – inverse exsecant function. (also written as arcexsec.) 反外正割函数arcexsec – inverse exsecant function. (also written as arcexs.) 反外正割函数arcosech – inverse hyperbolic cosecant function. (also written as arcsch.) 反双曲余割函数arcosh – inverse hyperbolic cosine function. 反双曲余弦函数arcoth – inverse hyperbolic cotangent function. 反双曲余切函数arcsch – inverse hyperbolic cosecant function. (also written as arcosech.) 反双曲余割函数arcsec – inverse secant function. 反正割函数arcsin – inverse sine function. 反正弦函数arctan – inverse tangent function. 反正切函数arctan2 – inverse tangent function with two arguments. (also written as atan2.) 带有2个参数的反正切函数arg – argument of a plex number.[2] 复数的参数arg max – argument of the maximum. 最大值时的参数arg min – argument of the minimum. 最小值时的参数arsech – inverse hyperbolic secant function. 反双曲正割函数arsinh – inverse hyperbolic sine function. 反双曲正弦函数artanh – inverse hyperbolic tangent function. 反双曲正切函数a.s. – almost surely. 殆必，几乎必然atan2 – inverse tangent function with two arguments. (also written as arctan2.) 同 arctan2，带有两个参数的反正切函数a.p. – arithmetic progression. 等差数列aut – automorphism group. 自同构群bd – boundary. 边界（拓扑学）bi – airy function of the second kind. 第二类艾里函数bias – bias of an estimator 估计器偏置card – cardinality of a set.[3] (card(x) is also written #x, ♯x or |x|.) 集合的势cdf – cumulative distribution function. 累积分布函数c.f. – cumulative frequency. 累积频率char – characteristic of a ring. 环的特征chi – hyperbolic cosine integral function. 双曲余弦积分函数ci – cosine integral function. 余弦积分函数cis – cos + i sin function. 欧拉公式函数cl – conjugacy class. 共轭类cl – topological closure. 拓扑学闭包cod, codom – codomain. 到达域cok, coker – cokernel. 上核，余核cor – corollary. 推论，余定理corr – correlation. 相关cos – cosine function. 余弦函数割函数cosech – hyperbolic cosecant function. (also written as csch.) 双曲余割函数cosh – hyperbolic cosine function. 双曲余弦函数cosiv – coversine function. (also written as cover, covers, cvs.) 余矢函数cot – cotangent function. (also written as ctg.) 余切函数coth – hyperbolic cotangent function. 双曲余切函数cov – covariance of a pair of random variables. 协方差cover – coversine function. (also written as covers, cvs, cosiv.) 余矢函数covercos – covercosine function. (also written as cvc.) 正矢函数covers – coversine function. (also written as cover, cvs, cosiv.) 余矢函数crd – chord function. 弦（几何）函数csc – cosecant function. (also written as cosec.) 余割函数csch – hyperbolic cosecant function. (also written as cosech.) 双曲余割函数函数curl – curl of a vector field. (also written as rot.) 向量场的旋度cvc – covercosine function. (also written as covercos.) 余余矢函数cvs – coversine function. (also written as cover, covers, cosiv.) 正余矢函数def – define or definition. 定义deg – degree of a polynomial. (also written as ∂.)多项式的次数del – del, a differential operator. (also writtenas .) 微分运算符det – determinant of a matrix or linear transformation. 矩阵或线性变换的行列式dim – dimension of a vector space. 向量空间的维度div – divergence of a vector field. 向量场的散度dkl – decalitre 公斗。