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## 4级目录 Level 4 Catalog

### 初等数学 + 高等数学 Elementary Mathematics + Advanced Mathematics

1. 代数三角公式与初等函数 Algebra , Trigonometric Formulas and Elementary Functions
• 代数公式? Algebraic formula ?
• 数的扩张、分类及其基本运算规则 Expansion and Classification of Numbers and Their Basic Operation Rules
• 数的扩张与分类表? Expansion and classification of numbers ?
• 实数四则运算法则? Four algorithms for real numbers ?
• 数的三个基本运算律? The three basic arithmetic of numbers ?
• 乘方与开方? Power and square ?
• 实数进位制? Real number system ?
• 复数? Plural ?
• 复数的概念? The concept of plural ?
• 复数的表示法? Representation of plural numbers ?
• 复数的运算? The operation of complex numbers ?
• 数列与简单级数? Sequence and simple series ?
• 数列与级数的概念? The concept of series and series ?
• 等差数列与等差级数? Arithmetic series and progression series ?
• 等比数列与等比级数? Proportional series and proportional series ?
• 算术-几何级数? Arithmetic-geometric series ?
• 调和级数? Harmonic progression ?
• 高阶等差级数? Higher order difference series ?
• 某些级数的部分和? Partial sum of some series ?
• 乘法与因式分解公式? Multiplication and factorization formulas ?
• 分式 Fraction
• 分式运算? Fractional operations ?
• 部分分式? Partial fraction ?
• 比例? Proportion ?
• 不等式? Inequalities ?
• 简单不等式? Simple inequality ?
• 有关绝对值的不等式? Inequalities about absolute values ?
• 有关三角函数、指数函数、对数函数的不等式? Inequalities about trigonometric, exponential, and logarithmic functions ?
• 某些重要的不等式? Some important inequalities ?
• 二次不等式解法? Quadratic inequality solution ?
• 阶乘、排列与组合 Factorial, permutations and combinations
• 阶乘? Factorial ?
• 排列? Arrange ?
• 组合? Combination ?
• 杨辉三角形与多项式定理? Yang Hui's triangle and polynomial theorem ?
• 数学归纳法与抽屉原理? Mathematical induction and drawer principle ?
• 初等函数及其数值计算? Elementary functions and their numerical calculations ?
• 函数的概念与分类? Concept and classification of functions ?
• 幂函数与有理函数? Power function and rational function ?
• 指数函数与对数函数? Exponential and logarithmic functions ?
• 平面三角函数与反三角函数 Planar trigonometric functions and inverse trigonometric functions
• 角的度量与换算? Measurement and conversion of angle ?
• 三角函数的定义? Definition of trigonometric function ?
• 特殊角的三角函数值? Trigonometric function values ​​for special angles ?
• 三角函数基本关系和公式? Basic relations and formulas of trigonometric functions ?
• 反三角函数定义? Inverse trigonometric function definition ?
• 反三角函数的相互关系与基本公式? Interrelationships and basic formulas of inverse trigonometric functions ?
• 三角形的基本定理? The basic theorem of triangles ?
• 斜三角形解法? Oblique triangle solution ?
• 球面三角 Spherical triangle
• 球面三角有关名称及性质? What are the names and properties of spherical triangles ?
• 球面三角形基本定理与公式? Spherical triangle basic theorem and formula ?
• 球面三角形解法? Spherical triangle solution ?
• 双曲函数 Hyperbolic function
• 双曲函数的定义、图形与特征? Definition, graph and characteristics of hyperbolic function ?
• 双曲函数的相互关系和基本公式? Interrelationships and basic formulas of hyperbolic functions ?
• 反双曲函数的定义、图形与特征? Definition, graph and characteristics of inverse hyperbolic function ?
• 反双曲函数的相互关系与基本公式? Interrelation and basic formula of inverse hyperbolic function ?
• 双曲函数与三角函数的对比? Comparison of hyperbolic and trigonometric functions ?
• 反函数? Inverse function ?
• 函数图? Function graph ?
2. 初等几何图形的计算与作图? Calculation and mapping of elementary geometry ?
• 三角形各元素的计算 Calculation of triangle elements
• 三角形各元素? Triangle elements ?
• 三角形各元素计算公式? Calculation formula of each element of the triangle ?
• 三角形和四边形的面积、几何重心、转动惯量计算公式? Formulas for the area, geometric center of gravity, and moment of inertia of triangles and quads ?
• 圆与正多边形 Circles and regular polygons
• 与圆有关的各量计算公式? Calculation formulas of various quantities related to circles ?
• 与圆有关的各种图形的面积、几何重心与转动惯量计算公式? Formulas for calculating the area, geometric center of gravity, and moment of inertia of various graphics related to circles ?
• 正多边形各量换算公式与比例系数表? Regular polygon conversion formula and scale factor table ?
• 实用几何作图 Practical geometric mapping
• 正多边形作图? Regular polygon mapping ?
• 椭圆作图? Ellipse mapping ?
• 圆弧放样法? Arc loft method ?
• 几何作图问题? Geometric mapping problem ?
• 立体图形的体积、表面积、侧面积、几何重心与转动惯量计算公式 Calculation formula of volume, surface area, side area, geometric center of gravity and moment of inertia of three-dimensional graphics
• 立体图形的体积、表面积、侧面积、几何重心与转动惯量计算公式? Calculation formula of volume, surface area, side area, geometric center of gravity and moment of inertia of a solid figure ?
• 多面体? Polyhedron ?
3. 代数方程 Algebraic equation
• 二、三、四次方程的根的表达式 Expressions for roots of second, third, and fourth degree equations
• 基本概念? Basic concept ?
• 三次方程? Cubic equation ?
• 四次方程? Quartic equation ?
• 阿贝尔定理? Abel's theorem ?
• 代数方程的性质 Properties of algebraic equations
• 多项式与代数方程的一般性质? General properties of polynomials and algebraic equations ?
• 多元多项式、对称多项式、结式? Multivariate polynomial, symmetric polynomial, and result ?
• 代数方程的根的隔离? Isolation of the roots of algebraic equations ?
• 代数方程的特殊解法 Special Solutions to Algebraic Equations
• 求有理根? Seeking rational roots ?
• 解三项方程? Solve a trinomial equation ?
• 解倒数方程? Solve the inverse equation ?
• 解二项方程? Solve a binomial equation ?
• 实根的近似计算 Approximate calculation of real roots
• 秦九韶法? Qin Jiushao method ?
• 二分法? Dichotomy ?
• 迭代法? Iterative method ?
• 牛顿法 Newton method
• 一般牛顿法? General Newton's method ?
• 近似牛顿法? Approximate Newton's method ?
• 逐次压缩牛顿法? Successive compression Newton method ?
• 牛顿法解非线性方程组? Newton's method for solving nonlinear equations ?
• 弦截法(线性插值法) ? String intercept method (linear interpolation)
• 联合法(牛顿法与弦截法联合使用) ? Joint method (combined use of Newton's method and string intercept method)
• 抛物线法(穆勒法) ? Parabolic method (Muller method)
• 林士谔-赵访熊法(劈因子法) ? Lin Shizheng-Zhao Fang Bear Method (Factor Factor Method)
• 下降法? Falling method ?
4. 矩阵、行列式、线性方程组? Matrix, determinant, linear equations ?
• 矩阵与行列式 Matrix and determinant
• 矩阵及其秩? Matrix and its rank ?
• 行列式 Determinant
• 行列式及其拉普拉斯展开定理? Determinant and its Laplacian expansion theorem ?
• 行列式的性质? The determinant nature ?
• 几个特殊行列式? How many special determinants ?
• 矩阵的运算 Matrix operations
• 矩阵的相等、加、减、数乘、乘法、转置与共轭? Equality, addition, subtraction, multiplication, multiplication, transposition and conjugate of a matrix
• 矩阵的初等变换与初等矩阵? Elementary transformation of matrices and elementary matrices ?
• 矩阵的微积分? Calculus of a matrix ?
• 特殊矩阵? Special matrix ?
• 相似变换? Similarity transformation ?
• 逆矩阵? Inverse matrix ?
• 特征值与特征矢量? Eigenvalues ​​and eigenvectors ?
• 矩阵多项式与最小多项式? Matrix polynomial and minimum polynomial ?
• 线性方程组 System of linear equations
• 含有n个未知量n个方程的线性方程组解法? Solution of a system of linear equations with n unknowns and n equations ?
• 一般情形的线性方程组? General set of linear equations ?
• 整系数线性齐次方程组的整数解? Integer solution of linear homogeneous system of integer coefficients ?
• 一类线性不等式组的解(克莱姆法则) ? Solutions to a class of linear inequalities (Clem's Law)
5. 微分学 Differential
• 序列与函数的极限 Sequence and Function Limits
• 序列的极限 Limit of sequence
• 基本概念? Basic concept ?
• 序列极限存在的判别法? Discriminant method for existence of sequence limit ?
• 序列极限的基本公式? Basic formula for sequence limits ?
• 常用序列的极限? The limits of commonly used sequences ?
• 函数的极限 Function limit
• 基本概念? Basic concept ?
• 函数极限存在的判别法? A method for determining the existence of a function limit ?
• 函数极限的基本公式? Basic formula for function limits ?
• 一些重要函数的极限? The limits of some important functions ?
• 不定式的定值法-洛比达法则? The indefinite formula-Lobida's rule ?
• 函数无穷小和无穷大的阶? Order of function infinitesimal and infinity ?
• 函数的连续性 Continuity of function
• 单变量函数的连续性? Continuity of univariate functions ?
• 多变量函数的连续性? Continuity of multivariable functions ?
• 级数的收敛与运算? Convergence and operation of series ?
• 数项级数收敛的判别法 Judgment method for convergence of number series
• 基本概念与基本性质? Basic concepts and basic properties ?
• 同号级数收敛判别法? Convergence discrimination method of the same number series ?
• 变号级数收敛判别法? Variable number series convergence judgment method ?
• 函数项级数收敛的判别法 Judgment method for series convergence of function term
• 收敛与一致收敛? Convergence and Uniform Convergence ?
• 一致收敛判别法? Uniform convergence judgment method ?
• 函数级数的运算及其条件? Operation of function series and its conditions ?
• 二重级数? Double series ?
• 无穷乘积? Infinite product ?
• 微分 differential
• 单变量函数的微分 Differential of univariate functions
• 基本概念? Basic concept ?
• 求导数的基本法则? The basic rules for finding derivatives ?
• 函数的微分与高阶导数? Differential and higher-order derivatives of functions ?
• 数值导数? Numerical derivative ?
• 多变量函数的微分? Differentiation of multivariate functions ?
• 函数行列式（或雅可比式）及其性质? Function determinant (or Jacobian) and its properties ?
• 隐函数 Implicit function
• 单变量隐函数? Univariate implicit function ?
• 多变量隐函数? Multivariate implicit function ?
• 由方程组所确定的隐函数? Implicit function determined by the system of equations ?
• 微分表达式中的变量替换 Variable substitution in differential expressions
• 单变量函数? Univariate function ?
• 多变量函数? Multivariate functions ?
• 微分学的基本定理(中值定理) ? Basic theorem of differential science (median theorem)
• 泰勒公式与泰勒级数 Taylor formula and Taylor series
• 单变量函数的泰勒公式? Taylor formula for univariate functions ?
• 多变量函数的泰勒公式? Taylor formula for multivariate functions ?
• 幂级数 Power series
• 单变量的幂级数? Power series of univariate ?
• 多变量的幂级数? Multivariate power series ?
• 函数的幂级数展开式? Power series expansion of a function ?
• 实数域上函数的幂级数展开式表? Expansion table for power series of functions on real numbers
• 微分的应用(I)--函数的极值 Application of Differential (I)-Extreme Value of Function
• 单变量函数的极值? Extreme values ​​of univariate functions ?
• 多变量函数的极值? Extreme values ​​of multivariate functions ?
• 约束条件为等式的条件极值? Constraints are conditional extremes of the equation ?
• 约束条件为不等式的条件极值? Constraints are conditional extremes of inequality ?
• 微分的应用(II)--曲线的性状与作图 Application of Differentiation (II)-Characteristics and Drawing of Curves
• 曲线的性状及其条件? What are the characteristics of the curve and its conditions ?
• 奇点? Singularity ?
• 渐近线? Asymptotes ?
• 作图? Mapping ?
6. 积分学 Integral
• 单变量函数的积分 Integral of univariate function
• 积分基本概念? Basic concept of points ?
• 积分不等式? Integral inequality ?
• 原函数的求法 How to find the original function
• 不定积分法则? Indefinite integral rule ?
• 有理分式的积分? Rational fraction integral ?
• 有理函数积分的变量? Variables for integration of rational functions ?
• 不定积分表? Indefinite points table ?
• 定积分的求法? How to find a definite integral ?
• 广义积分 Generalized integral
• 广义积分的概念? The concept of generalized integral ?
• 广义积分收敛判别法? Generalized integral convergence judgment method ?
• 含参数积分 Parametric integration
• 含参数常义积分? Normal meaning integral with parameters ?
• 含参数广义积分? Generalized integral with parameters ?
• 斯蒂尔吉斯积分? Sturgis points ?
• 积分的近似计算 Approximate calculation of integrals
• 内插求积公式? Interpolation quadrature formula ?
• 多重积分、曲线积分与曲面积分 Multiple integration, curve integration and surface integration
• 多重积分 Multiple points
• 二重积分? Double points ?
• 三重积分? Triple points ?
• 多重积分? Multiple points ?
• 曲线积分? Curve integral ?
• 曲面积分? Surface integration ?
• 重积分、曲线积分与曲面积分的近似计算? Approximate calculations of double integration, curve integration and surface integration ?
• 积分的应用 Application of Points
• 求面积? Find the area ?
• 求体积? Find the volume ?
• n维空间中凸体体积公式? Formula of convex volume in n-dimensional space ?
• 求重心? Find the center of gravity ?
• 求转动惯量? Find the moment of inertia ?
• 求流体压力? Find the fluid pressure ?
• 求变力所做的功? What is the work done for variable force ?
• 区域函数 Area function
• 区域函数与密度函数? Area function and density function ?
• 密度函数的积分? Integral of the density function ?
• δ-函数概念? δ-function concept ?
7. 解析几何与微分几何 Analytic and differential geometry
• 坐标系与坐标变换 Coordinate system and coordinate transformation
• 平面坐标系及其变换表? Plane coordinate system and its transformation table ?
• 空间坐标系及其变换表? Space coordinate system and its transformation table ?
• 解析几何中的基本计算? Basic calculations in analytic geometry ?
• 平面上的直线 Straight line on the plane
• 平面上直线的方程与图形? Equations and graphics of straight lines on a plane ?
• 平面上点与直线的相互关系? The relationship between points and lines on a plane ?
• 空间中的直线与平面 Lines and planes in space
• 直线的方向? The direction of the line ?
• 平面的方程? Equation of the plane ?
• 直线的方程? Equation of a straight line ?
• 空间中点、直线、平面的相互关系? Interrelationship of points, lines and planes in space ?
• ? Round ?
• 椭圆 oval
• 椭圆的基本元素? The basic elements of an ellipse ?
• 椭圆的方程、顶点、中心与焦点? The equation, vertex, center, and focus of an ellipse ?
• 椭圆的性质? The nature of the ellipse ?
• 椭圆各量计算公式? Formula for calculating ellipse quantities ?
• 双曲线 hyperbola
• 双曲线的基本元素? The basic elements of a hyperbola ?
• 双曲线的方程、顶点、中心、焦点与渐近线? Hyperbolic equations, vertices, centers, focal points and asymptotes ?
• 双曲线的性质? The nature of the hyperbola ?
• 双曲线各量计算公式? Formula for calculating hyperbolic quantities ?
• 拋物线 parabola
• 拋物线的基本元素? The basic elements of a parabola ?
• 拋物线的方程、顶点、焦点与准线? Parabolic equations, vertices, focal points, and guidelines ?
• 拋物线的性质? What is the nature of the parabola ?
• 拋物线各量计算公式? Calculation formula for each parabola ?
• 二次曲线的一般性质? General properties of the quadratic curve ?
• 二次曲线的不变量? Invariant of a quadratic curve ?
• 二次曲线的标准方程与形状? Standard equation and shape of a quadratic curve ?
• 二次曲线的几种情况? Several cases of quadratic curve ?
• 圆锥截线? Cone intercept ?
• 球面? Spherical ?
• 椭球面? Ellipsoidal ?
• 双曲面? Hyperboloid ?
• 拋物面? Parabolic ?
• 锥面与柱面? Cone and cylinder ?
• 二次曲面的一般性质? General properties of quadrics ?
• 二次曲面的不变量? Invariants of quadrics ?
• 二次曲面的标准方程? Standard equation for quadrics ?
• 平面曲线? Flat curve ?
• Table of important plane curves ?
• 空间曲线 Space curve
• 曲线的基本概念与公式? Basic concepts and formulas of curves ?
• 螺旋线的方程与图形? Equations and graphs of spirals ?
• 螺旋面? Helix ?
• 可展曲面? Developable surface ?
• 一般曲面 General surface
• 曲面的方程与曲线坐标? Surface equations and curve coordinates ?
• 切面、法线与曲面的方向? Direction of cuts, normals, and surfaces ?
• 第一基本二次型与曲面的度量? Measurement of the first basic quadratic form and surface ?
• 第二基本二次型与曲面曲线的曲率? Curvature of the second basic quadratic form and curved surface ?
• 曲面曲线的曲率半径? The radius of curvature of a curved surface ?
• 第三基本二次型与曲面的曲率? The third basic quadratic form and the curvature of the surface ?
• 渐进曲线、共轭曲线与极小曲面? Asymptotic curves, conjugate curves and minimal surfaces ?
• 曲面的基本公式与基本方程? Basic formulas and equations for surfaces ?
• 曲面曲线的测地曲率? Geodesic curvature of surface curves ?
8. 矢量算法与场论初步、张量算法与黎曼几何初步 Vector algorithm and field theory, tensor algorithm and Riemann geometry
• 矢量算法 Vector algorithm
• 矢量代数? Vector algebra ?
• 矢量分析 Vector analysis
• 矢量微分? Vector differentiation ?
• 矢量积分? Vector integral ?
• 场论初步 Field theory preliminary
• 场论的基本概念及梯度、散度与旋度? Basic concepts of field theory and gradients, divergences and curl
• 梯度、散度、旋度在不同坐标系中的表达式 Expressions of gradient, divergence, and rotation in different coordinate systems
• 单位矢量的变换? Transformation of unit vector ?
• 矢量的坐标变换? Coordinate transformation of a vector ?
• 各种算子在不同坐标系中的表达式? Expressions of various operators in different coordinate systems ?
• 曲线积分、曲面积分与体积导数? Curve integral, surface integral, and volume derivative ?
• 矢量的积分定理? Integral theorem for vectors ?
• 仿射坐标系 Affine coordinate system
• 仿射坐标系? Affine coordinate system ?
• 逆变矢量与协变矢量? Inverse vector and covariant vector ?
• n维空间? n-dimensional space ?
• 张量算法 Tensor algorithm
• 张量概念? Tensor Concept ?
• 张量代数? Tensor algebra ?
• 张量分析? Tensor analysis ?
• 黎曼几何初步 Riemann geometry preliminary
• 黎曼空间? Riemann space ?
• 勒维—奇维塔的平行性? Levi-Civita parallelism ?
• 黎曼空间中的曲率? Curvature in Riemann space ?
9. 抽象代数、线性空间、泛函分析 Abstract algebra, linear space, functional analysis
• 抽象代数 Abstract algebra
• 基本代数系统? Basic algebra system ?
• ? Group ?
• ? Ring ?
• ? Domain ?
• 线性空间与线性子空间 Linear space and linear subspace
• 线性空间? Linear space ?
• 线性子空间? Linear subspace ?
• 线性变换 Linear transformation
• 基本概念? Basic concept ?
• 线性变换的运算? Calculation of linear transformation ?
• 对偶空间与对偶映射? Dual space versus dual mapping ?
• 酉空间酉 space
• 酉空间的定义与性质?的 The definition and nature of space ?
• 酉空间上的特殊线性变换?的 Special linear transformation on space ?
• 射影? Projective ?
• 酉空间中的度量?的 Metrics in space ?
• 埃尔米特(H)型? Hermit (H) type ?
• 方阵的若当标准形 Jordan Standard Form
• 不变子空间? Invariant subspace ?
• 方阵的标准化? Normalization of a square matrix ?
• 方阵的标准化的方法与步骤? Method and procedure of standardization of square matrix ?
• 泛函分析初步 Functional Analysis Preliminary
• 勒贝格积分 Lebesgue integral
• 测度与可测函数? Measures and measurable functions ?
• 勒贝格积分? Lebesgue points ?
• 平方可积函数? Square integrable function ?
• 希尔伯特空间? Hilbert space ?
• 巴拿赫空间? Banach space ?
10. 复变函数 Complex function
• 解析函数 Analytic Functions
• 复变函数基本概念与复变函数的导数? Basic concepts of complex functions and derivatives of complex functions ?
• 解析函数 Analytic Functions
• 解析函数的定义与柯西-黎曼方程? Definition of analytic function and Cauchy-Riemann equation ?
• 解析开拓? Analytical development ?
• 初等解析函数? Elementary analytical function ?
• 黎曼点、支点与支线? Riemann point, fulcrum and spur line ?
• 保角映射 Conformal mapping
• 保角映射及其性质? Conformal mapping and its properties ?
• 分式线性映射及其性质? Fractional linear mapping and its properties ?
• 简单分式线性映射? Simple fractional linear mapping ?
• 对称原理与多边形映射? Principle of symmetry and polygon mapping ?
• 保角映射的存在唯一性定理（黎曼定理） ? Existence and uniqueness theorem of conformal mapping (Riemann's theorem)
• 复变函数的积分 Integral of complex function
• 复变函数的积分的定义与公式? Definition and formula of integral of complex function ?
• 解析函数的积分的性质? What is the nature of the integral of the analytical function ?
• 泰勒级数、罗朗级数、留数定理 Taylor series, Laurent series, remainder theorem
• 泰勒级数与罗朗级数 Taylor series and Laurent series
• 泰勒级数? Taylor series ?
• 罗朗级数展开定理? Laurent series expansion theorem ?
• 解析函数的局部性质? The local nature of analytic functions ?
• 单值解析函数的分类? Classification of single-valued analytical functions ?
• 半纯函数的部分分式表达式? Partial fractional expressions for semi-pure functions ?
• 留数定理及其应用? Residue theorem and its application ?
11. 傅立叶级数与积分变换 Fourier series and integral transformation
• 傅立叶级数 Fourier series
• 三角级数与傅立叶级数? Triangular series and Fourier series ?
• f(x)在其他区间上的傅立叶级数? Fourier series of f (x) in other intervals ?
• 傅立叶级数的性质? What is the nature of the Fourier series ?
• 傅立叶级数的收敛性及在第一类间断点的性质? Convergence of Fourier series and properties of discontinuities in the first kind
• 傅立叶级数的逐项积分与微分? Termwise integration and differentiation of Fourier series ?
• 函数的傅立叶级数展开式表? Fourier series expansion table of functions ?
• 二重傅立叶级数? Double Fourier series ?
• 广义傅立叶级数与傅立叶-贝赛耳级数 Generalized Fourier series and Fourier-Bessel series
• 广义傅立叶级数? Generalized Fourier series ?
• 傅立叶-贝赛耳级数? Fourier-Bessel series ?
• 傅立叶变换 Fourier transform
• 傅立叶积分? Fourier integral ?
• 傅立叶变换? Fourier transform ?
• 傅立叶余弦变换? Fourier cosine transform ?
• 傅立叶正弦变换? Fourier sine transform ?
• 有限傅立叶余弦变换? Finite Fourier cosine transform ?
• 有限傅立叶正弦变换? Finite Fourier sine transform ?
• 二重傅立叶变换及其反演公式? Double Fourier transform and its inversion formula ?
• 快速傅立叶变换 Fast Fourier transform
• 有限离散傅立叶变换? Finite discrete Fourier transform ?
• 快速傅立叶变换算法? Fast Fourier transform algorithm ?
• 梅林变换? Merlin transformation ?
• 汉克尔变换? Hankel transform ?
• 勒让德变换及其反演公式? Legendre transformation and its inversion formula ?
• 由积分定义的特殊函数 Special functions defined by integrals
• 伽马函数? Gamma function ?
• 贝塔函数? Beta function ?
• 普西函数? Pusey function ?
• 菲涅耳函数? Fresnel function ?
• 概率积分（误差函数） ? Probability integral (error function)
• 正弦积分与余弦积分? Sine and cosine integrals ?
• 指数积分? Index points ?
• 对数积分? Logarithmic integral ?
• 不完全伽马函数? Incomplete gamma function ?
• 椭圆积分? Elliptic integral ?
• 正交多项式 Orthogonal polynomial
• 勒让德多项式? Legendre polynomial ?
• 第一类契贝谢夫多项式? Chebyshev polynomials of the first kind ?
• 第二类契贝谢夫多项式? Chebyshev polynomials of the second kind ?
• 拉盖尔多项式 Laguerre polynomial
• 一般拉盖尔多项式? General Laguerre polynomials ?
• 拉盖尔多项式? Laguerre polynomial ?
• 埃尔米特多项式? Hermitian polynomial ?
• 雅可比多项式? Jacobian polynomial ?
• 盖根堡多项式? Gegenberg polynomial ?
• 超几何函数 Hypergeometric function
• 高斯超几何级数? Gaussian hypergeometric series ?
• 库默尔函数（合流超几何函数) ? Kummer function (convergent hypergeometric function)
• 勒让德函数 Legendre function
• 勒让德函数的定义? Definition of Legendre function ?
• 勒让德函数的其他表达式? Other expressions of Legendre functions ?
• 勒让德函数的递推公式与有关公式? Recursive formula of Legendre function and related formulas ?
• 勒让德函数的正交性? Orthogonality of Legendre function ?
• 勒让德函数的渐近表达式与不等式? Asymptotic expressions and inequalities of Legendre functions ?
• 贝赛耳函数 Bessel function
• 第一类贝赛耳函数? Bessel functions of the first kind ?
• 第二类贝赛耳函数(诺伊曼函数) ? Bessel functions of the second kind (Neumann functions)
• 第三类贝赛耳函数(汉克尔函数) ? Bessel function of the third kind (Hankel function)
• 各类贝赛耳函数之间的关系与有关公式? The relationship between various types of Bessel functions and related formulas ?
• 变型贝赛耳函数? A modified Bessel function ?
• 椭圆函数 Elliptic function
• 椭圆函数的定义与性质? Definition and properties of elliptic functions ?
• 雅可比椭圆函数? Jacobian elliptic function ?
• 外尔斯特拉斯椭圆函数? Weilstrass elliptic function ?
• 伯努利数与伯努利多项式 Bernoulli numbers and Bernoulli polynomials
• 伯努利数? Bernoulli number ?
• 伯努利多项式? Bernoulli polynomial ?
12. 常微分方程 Ordinary differential equation
• 微分方程的一般概念? General concepts of differential equations ?
• 一阶微分方程 First order differential equation
• 一阶微分方程解的存在性和唯一性? Existence and uniqueness of solutions of first order differential equations ?
• 可积类型及其通解? Integrable types and their general solutions ?
• 奇解及其求法? Singular solution and how to find it ?
• 线性微分方程 Linear differential equation
• 一般概念? General concept ?
• 常系数线性微分方程 Constant coefficient linear differential equation
• 齐次线性微分方程通解的求法? How to find the general solution of homogeneous linear differential equation ?
• 非齐次线性微分方程特解的求法? How to find special solutions of non-homogeneous linear differential equations ?
• 欧拉方程? Euler's equation ?
• 齐次线性微分方程的幂级数解法? Power series solution for homogeneous linear differential equations ?
• 高阶微分方程与微分方程组 Higher order differential equations and differential equations
• 高阶微分方程与微分方程组的互化? Interaction of higher order differential equations and differential equations ?
• 高阶微分方程的几种可积类型及其解法? Several integrable types of higher-order differential equations and their solutions ?
• 线性微分方程组 Linear differential equations
• 齐次线性微分方程组与非齐次线性微分方程组? Homogeneous linear differential equations and non-homogeneous linear differential equations ?
• 常系数线性微分方程组? System of linear differential equations with constant coefficients ?
• 常系数非齐次线性微分方程的算子解法与方程组的算子解法(消去法) ? Operator solution of non-homogeneous linear differential equation with constant coefficients
• 稳定性理论大意 The main idea of ​​stability theory
• 稳定性的概念? The concept of stability ?
• 稳定性问题的解法? The solution to the stability problem ?
• 极限圆(或极限环) ? Limit circle (or limit cycle)
• 常微分方程的数值解法 Numerical Solution of Ordinary Differential Equations
• 一阶微分方程初值问题的数值解? Numerical solution of initial value problems for first order differential equations ?
• 一阶微分方程组初值问题的数值解? Numerical solution of initial value problems for first order differential equations ?
• 边值问题? Boundary value problem ?
• 小参数法? Small parameter method ?

• 解微分方程solve differential equation by dsolve() Solve differential equation by dsolve ()
• 验证微分方程的解test solution for differential equation by test() Test solution for differential equation by test ()

13. 偏微分方程 Partial Differential Equations
• 偏微分方程的一般概念与定解问题? General concepts of partial differential equations and problems with definite solutions ?
• 一阶偏微分方程 First order partial differential equation
• 柯西-柯娃列夫斯卡娅定理? Cauchy-Kovalevskaya theorem ?
• 一阶线性方程 First-order linear equation
• 一阶齐次线性方程? First-order homogeneous linear equation ?
• 非齐次线性方程? Non-homogeneous linear equation ?
• 一阶拟线性方程? First-order quasi-linear equations ?
• 一阶非线性方程? First-order nonlinear equation ?
• 一阶线性偏微分方程组? First-order linear partial differential equations ?
• 二阶偏微分方程 Second-order partial differential equation
• 二阶偏微分方程的分类、标准形式与特征方程? Classification, standard forms and characteristic equations of second-order partial differential equations ?
• 极值原理、能量积分、定解问题的唯一性定理? Extreme value principle, energy integration, uniqueness theorem for definite problem
• 三种典型方程 Three typical equations
• 波动方程? Wave equation ?
• 热传导方程? Heat conduction equation ?
• 拉普拉斯方程? Laplace'#39;s equation ?
• 基本解与广义解? Basic and generalized solutions ?
• 二阶偏微分方程的常用解法 Common solutions to second-order partial differential equations
• 分离变量法? Separated variable method ?
• 双曲型方程的黎曼方法? Riemann's method for hyperbolic equations ?
• 椭圆型方程的格林方法? Green's method for elliptic equations ?
• 积分变换法? Integral transformation method ?
• 偏微分方程的数值解法 Numerical Solution of Partial Differential Equations
• 差分法 Difference
• 网格与差商? Grid and difference quotient ?
• 椭圆型方程的差分方法? Difference method for elliptic equations ?
• 拋物型方程的差分方法? Difference method for parabolic equations ?
• 双曲型方程的差分方法? Difference method for hyperbolic equations ?
• 变分方法 Variational method
• 自共轭边值问题? Self-conjugate boundary value problem ?
• 变分原理与广义解? Variational Principles and Generalized Solutions ?
• 极小化序列与里兹方法? Minimizing sequences and the Ritz method ?
• 里兹方法在特征值问题上的应用? Application of Leeds method to eigenvalue problem ?
• 迦辽金方法? Galerkin method ?
14. 积分方程 Integral equation
• 积分方程一般概念与弗雷德霍姆方程 General Concepts of Integral Equations and Fred Holm Equation
• 积分方程一般概念 General concepts of integral equations
• 积分方程的定义与分类? Definition and classification of integral equations ?
• 积分方程与微分方程之间的关系? The relationship between integral equations and differential equations ?
• 格林函数及其物理意义? Green's function and its physical meaning ?
• 具有可分离核(退化核)的Fr方程? Fr equation with separable kernel (degenerate kernel) ?
• 希尔伯特-施密特的理论? Hilbert-Schmidt's theory ?
• 第二类Fr方程的逐次逼近法与诺伊曼级数解? Successive approximation of the second type of Fr equation and Neumann series solution ?
• 弗雷德霍姆的理论? Fred Holm's theory ?
• 奇异积分方程 Singular integral equation
• 奇异积分方程的定义与例子? Definition and example of singular integral equation ?
• 具有柯西核和希尔伯特核的积分方程? Integral equation with Cauchy and Hilbert kernels ?
• 沃尔泰拉积分方程? Volterra integral equation ?
• 积分方程的近似解法? Approximate solution of integral equation ?
• 非线性积分方程? Non-linear integral equation ?
15. 概率统计与随机过程 Probability Statistics and Stochastic Processes
• 概率论? Probability theory ?
• 事件与概率 Events and probability
• 随机事件及其运算关系? Random events and their operational relationships ?
• 概率的几种定义? Several definitions of probability ?
• 概率的基本性质? The basic nature of probability ?
• 概率的计算公式? Formula for calculating probability ?
• 随机变量与分布函数? Random variables and distribution functions ?
• 随机变量的数字特征? Numerical characteristics of random variables ?
• 概率母函数、矩母函数、特征函数? Probability generating function, moment generating function, characteristic function ?
• 常用分布函数 Commonly used distribution functions
• 常用离散型分布? Commonly used discrete distributions ?
• 常用连续型分布? Common continuous distribution ?
• 大数法则与中心极限定理? The law of large numbers and the central limit theorem ?
• 正态分布表的用途? What is the normal distribution table ?
• 数理统计方法? Mathematical statistics method ?
• 总体参数的估计 Estimation of population parameters
• 总体(母体)与样本(子样) ? Population (parent) and sample (subsample)
• 样本特征数与总体数字特征对照表? Comparison table between the number of sample features and the overall number features ?
• 总体参数的点估计? Point estimates for population parameters ?
• 样本的频率分布? The frequency distribution of the sample ?
• 总体参数的区间估计? Interval estimation of population parameters ?
• 统计假设检验 Statistical hypothesis test
• 统计假设检验的步骤? Steps for statistical hypothesis testing ?
• 正态总体参数的统计假设检验表? Statistical hypothesis test table for normal population parameters ?
• 总体分布函数的统计假设检验? Test of statistical hypothesis for population distribution function ?
• 两个样本是否来自同分布总体的统计假设检验? Do the two samples come from a statistical hypothesis test of the same population ?
• 方差分析? Analysis of variance ?
• 回归分析 regression analysis
• 最小二乘法原理? The principle of least squares ?
• 一元线性回归? Unary linear regression ?
• 抛物线回归? Parabolic regression ?
• 可化成线性回归的曲线回归? Curve regression that can be transformed into linear regression ?
• 二元线性回归? Binary linear regression ?
• 多元线性回归? Multiple linear regression ?
• 正交实验设计? Orthogonal experimental design ?
• 抽样检验? Sampling inspection ?
• 质量评估(工序控制)方法? Quality assessment (process control) method ?
• 随机过程 stochastic process
• 一般随机过程? General stochastic process ?
• 马尔科夫过程 Markov process
• 转移概率? Transition probability ?
• 马尔科夫链? Markov chain ?
• 时间连续、状态离散的马尔科夫过程? Time-continuous, discrete-state Markov processes ?
• 扩散过程? The diffusion process ?
• 平稳随机过程? Stationary random process ?
16. 误差理论与实验数据处理 Error theory and experimental data processing
• 误差理论 Error theory
• 观测误差? Observation error ?
• 平均值及其精密度指标? The average value and its precision index ?
• 误差的表示法? Representation of error ?
• 高斯误差定律? Gauss error law ?
• 误差与有效数字? Errors and significant figures ?
• 插值公式 Interpolation formula
• 不等距节点插值公式(差商插值多项式) ? Unequally spaced node interpolation formula (difference quotient interpolation polynomial)
• 等距节点插值公式(差分公式) ? Isometric node interpolation formula (difference formula)
• 拉格朗日插值多项式? Lagrange interpolation polynomial ?
• 三次样条(Spline)内插公式? Cubic spline interpolation formula ?
• 其它插值公式? Other interpolation formulas ?
• 曲线拟合的圆弧法与平均法 Arc method and average method for curve fitting
• 曲线拟合的圆弧法? Arc method for curve fitting ?
• 曲线拟合的平均法? Curve fitting average method ?
• 实验曲线的光滑法 Smoothing method
• 实验曲线的光滑与观测值的修匀? The smoothness of the experimental curve and the smoothing of the observed values ?
• 直线的滑动平均法? Linear moving average method ?
• 二次抛物线的滑动平均法? Moving average method of quadratic parabola ?
• 三次抛物线的滑动平均法? Moving average method of cubic parabola ?
• 滤波 Filtering
• 最小二乘滤波? Least square filtering ?
• 维纳滤波? Wiener filtering ?
• 卡尔曼滤波? Kalman filtering ?
17. 最优化方法 optimal method
• 单变量函数极值问题解法(直接法) ? Solution to the extreme value problem of a univariate function (direct method)
• 多变量函数极值问题解法(直接法) ? Multivariable function extreme value solution (direct method)
• 无条件极值问题解法? Unconditional extreme value solution ?
• 条件极值问题解法? The solution to the conditional extreme value problem ?
• 变分法 Variational method
• 泛函的变分与泛函的极值? Functional Variations and Extreme Values ​​of Functionals ?
• 不动边界的泛函的极值、欧拉方程? Extreme values ​​of functionals with fixed boundaries, Euler's equation ?
• 可动边界的泛函的极值? Extreme value of a functional with a moving boundary ?
• 条件极值问题? Conditional extremes ?
• 变分问题的直接方法? A direct method for variational problems ?
• 最小(大)值原理? Principle of minimum (large) value ?
• 动态规划? Dynamic programming ?
18. 有限元法 Finite element method
• 一般原理与解算步骤 General principles and solution steps
• 变分原理与有限元法? Variational principle and finite element method ?
• 在弹性力学问题上的应用(位移法) ? Application on elastic mechanics (displacement method)
• 有限元解法的主要步骤? The main steps of the finite element method ?
• 基本单元与线性插值 Basic unit and linear interpolation
• 基本单元与型函数? Basic unit and type function ?
• 直线段单元? Linear segment element ?
• 三边形单元? Triangular element ?
• 四面体单元? Tetrahedral element ?
• 六面体单元? Hexahedral element ?
• 三棱柱单元? Triangular prism unit ?
• 基本单元的特点? What are the characteristics of the basic unit ?
• 等参数单元与高次插值 Isoparametric units and higher-order interpolation
• 等参数单元? Isoparametric units ?
• 多节点线元上的插值? Interpolation on multi-node line elements ?
• 平面等参数单元的型函数? Type function of plane and other parametric elements ?
• 空间等参数单元的型函数? Type function of space and other parameter units ?
• 等参数单元的特点? Characteristics of isoparametric units ?
• 拟协调单元 Quasi-coordination unit
• 协调问题与拟协调单元? Coordination issues and proposed coordination units ?
• 一维单元的高次插值? Higher-order interpolation of one-dimensional units ?
• 三边形单元的高次插值? Higher-order interpolation of triangular cells ?
• 弹性理论与有限元解法 Elasticity theory and finite element method
• 三维的弹性问题? Three-dimensional elasticity ?
• 二维的弹性问题? Two-dimensional elasticity problem ?
• 一维的弹性问题? One-dimensional elasticity ?
• 与有限元解法有关的问题? Questions related to finite element solutions ?
• 有限元法中的数值积分? Numerical integration in the finite element method ?
19. 初等数论 Elementary number theory
• 整数? Integer ?
• 连分数? Continue score ?
• 同余式? Congruence ?
• 数论函数? Number theory function ?
• 多项式? Polynomial ?
• 代数数? Algebraic number ?
20. 集论与一般拓扑学? Set theory and general topology
• 集(集合) ? Set (set)
• 集的定义 Definition of set
• 集的古典定义? The classical definition of a set ?
• 罗素怪异? Russell weird ?
• ZFC公理系统与BNG公理系统? ZFC axiom system and BNG axiom system ?
• 变换、集的一般表示法、标号集? Transformations, general representations of sets, label sets ?
• 公理系统规定的集? The set of axiom rules ?
• 序数与基数 Ordinal and cardinality
• 排队(良序)集? Queued (well-ordered) set ?
• 序数? Ordinal number ?
• 正整数、超限序数、超限归纳法? Positive integer, transfinite ordinal, transfinite induction ?
• 选择公理与排队定理? Choice axioms and queueing theorems ?
• 序数算术? Ordinal arithmetic ?
• 基数? Base ?
• 基数算术? Cardinal arithmetic ?
• 拓扑空间 Topological space
• 基本概念? Basic concept ?
• 点集的基本拓扑概念? Basic topology concepts for point sets ?
• 拓扑空间的分离程度、可数公理 Degree of separation in topological space, countable axioms
• 不同分离程度的拓扑空间? Topological space with different degrees of separation ?
• 可数性? Countability ?
• 极限与连续? Limit and continuous ?
• 点网? Dot network ?
• 尺度空间与一致空间 Scale space and uniform space
• 尺度空间? Scale space ?
• 一致空间? Consistent space ?
• 紧致点集与联结点集 Compact point set and connection point set
• 紧致点集? Compact point set ?
• 联结点集? Set of connection points ?
• 流形? Manifold ?
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