📚 Vocabulary and Terminology Quick-Reference Guide for Year 13 Edexcel Further Maths | Year 13 Edexcel 进阶数学：词汇术语速记指南
Mastering the vocabulary of Year 13 Edexcel Further Maths is essential for interpreting exam questions accurately and articulating solutions with precision. This guide breaks down key terms from core pure, further statistics, further mechanics and decision mathematics into bite-sized English–Chinese explanations, helping you build instant recall and deeper understanding.
掌握Year 13 Edexcel进阶数学的词汇对于准确理解考题和精确表述解答至关重要。本指南将核心纯数、进阶统计、进阶力学和决策数学中的关键术语拆解为简明扼要的英中对照解释，帮助你建立即时回忆与更深刻的理解。
1. Complex Numbers (Core Pure) | 复数（核心纯数）
Real part and imaginary part – For a complex number z = x + i y, the real part Re(z) is x and the imaginary part Im(z) is y, where i² = –1. The Cartesian form naturally separates these two components.
实部与虚部 – 对于复数 z = x + i y，实部 Re(z) 为 x，虚部 Im(z) 为 y，其中 i² = –1。笛卡儿形式自然地将这两个部分分开。
Modulus and argument – The modulus |z| = √(x² + y²) represents the distance from the origin on the Argand diagram. The argument arg(z) = θ is the angle measured from the positive real axis, usually in the range (–π, π]. The principal argument must be used with care.
模与辐角 – 模 |z| = √(x² + y²) 表示复数在阿根图上到原点的距离。辐角 arg(z) = θ 是从正实轴量起的角度，通常取主值范围 (–π, π]。使用主辐角时需格外小心。
Complex conjugate – The conjugate of z = x + i y is z* = x – i y. It reflects the point in the real axis, satisfies z z* = |z|², and is vital for division and finding real-valued polynomials.
共轭复数 – z = x + i y 的共轭为 z* = x – i y。它在实轴上反射该点，满足 z z* = |z|²，在作除法和求实系数多项式中至关重要。
de Moivre’s theorem – (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ for integer n. It is central for powers, roots, and trigonometric identities.
棣莫弗定理 – 对整数 n，有 (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。它是处理幂、方根和三角恒等式的核心。
Roots of unity – The nᵗʰ roots of unity are solutions of zⁿ = 1, given by z = e^(2πik/n) for k = 0, 1, …, n–1. They lie equally spaced on the unit circle.
单位根 – n次单位根是方程 zⁿ = 1 的解，由 z = e^(2πik/n)，k = 0, 1, …, n–1 给出。它们在单位圆上等距分布。
2. Matrices and Linear Transformations (Core Pure) | 矩阵与线性变换（核心纯数）
Determinant – For a 2×2 matrix M = [[a, b], [c, d]], det(M) = ad – bc. A non-zero determinant indicates an invertible matrix and scales area by |det(M)| under the associated linear transformation.
行列式 – 对于 2×2 矩阵 M = [[a, b], [c, d]]，det(M) = ad – bc。非零行列式表明矩阵可逆，并且在相应的线性变换下面积按因子 |det(M)| 缩放。
Inverse matrix – M⁻¹ = (1/det(M)) [[d, –b], [–c, a]] when det(M) ≠ 0. It reverses the transformation and satisfies M M⁻¹ = I.
逆矩阵 – 当 det(M) ≠ 0 时，M⁻¹ = (1/det(M)) [[d, –b], [–c, a]]。它逆转变换并满足 M M⁻¹ = I。
Eigenvalue and eigenvector – For a square matrix A, if A v = λ v for a non-zero vector v, then λ is an eigenvalue and v is the corresponding eigenvector. They define invariant lines and are found from det(A – λ I) = 0.
特征值与特征向量 – 对于方阵 A，若存在非零向量 v 满足 A v = λ v，则 λ 为特征值，v 为对应的特征向量。它们定义不变直线，通过 det(A – λ I) = 0 求得。
Orthogonal matrices – A matrix Q is orthogonal if Qᵗ Q = I. Its columns form an orthonormal set, and it preserves lengths and angles (rotations or reflections).
正交矩阵 – 若 Qᵗ Q = I，则矩阵 Q 为正交矩阵。其列构成标准正交基，并且它保持长度和角度（对应旋转或反射）。
3. Hyperbolic Functions (Core Pure) | 双曲函数（核心纯数）
Definitions – sinh x = (eˣ – e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x. These mirror trig functions but lack periodic behaviour.
定义 – sinh x = (eˣ – e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x。它们与三角函数类似但无周期性。
Osborn’s rule – To convert a trig identity into a hyperbolic identity, replace any product of two sine terms by –sinh² etc., essentially changing the sign when a product of two sines appears.
奥斯本法则 – 将三角恒等式转换为双曲恒等式时，将两个正弦的乘积替换为 –sinh² 等形式，本质是在出现两个正弦乘积时改变符号。
Inverse hyperbolic functions – arsinh x = ln(x + √(x²+1)), arcosh x = ln(x + √(x²–1)) for x ≥ 1, artanh x = ½ ln((1+x)/(1–x)) for |x| < 1. Differentiating these logarithmic forms is common in exams.
反双曲函数 – arsinh x = ln(x + √(x²+1))，arcosh x = ln(x + √(x²–1))（x ≥ 1），artanh x = ½ ln((1+x)/(1–x))（|x| < 1）。在考试中常对这些对数形式求导。
4. Polar Coordinates (Core Pure) | 极坐标（核心纯数）
Polar equation – A curve is given as r = f(θ). The angle θ is measured from the initial line, and r is the radial distance. Conversion: x = r cos θ, y = r sin θ.
极坐标方程 – 曲线表示为 r = f(θ)。角度 θ 从极轴计量，r 是径向距离。转换公式：x = r cos θ, y = r sin θ。
Area enclosed by a polar curve – Area = ½ ∫ r² dθ between suitable limits. The sector formula is fundamental and requires careful limit selection when curves have loops.
极曲线围成面积 – 面积 = ½ ∫ r² dθ，在适当积分限之间。该扇形公式是基础，当曲线有环路时需要仔细选择积分限。
Common shapes – Cardioid: r = a(1 + cos θ); Limaçon with inner loop: r = a + b cos θ (b>a); Rose curves: r = a cos(nθ) for n odd gives n petals, for n even gives 2n petals.
常见形状 – 心脏线：r = a(1 + cos θ)；带内环的蜗线：r = a + b cos θ (b>a)；玫瑰线：r = a cos(nθ)，n为奇数时有 n 瓣，n为偶数时有 2n 瓣。
5. Differential Equations (Core Pure) | 微分方程（核心纯数）
Order and degree – The order is the highest derivative; the degree is the power of the highest derivative after clearing fractional powers. Linear ODEs have the form aₙ(x) y⁽ⁿ⁾ + … + a₀(x) y = f(x).
阶与次 – 阶是最高导数的阶数；次是清除分数幂后最高导数的次数。线性常微分方程的形式为 aₙ(x) y⁽ⁿ⁾ + … + a₀(x) y = f(x)。
Integrating factor for first-order linear ODEs – For dy/dx + P(x) y = Q(x), multiply by IF = e^∫ P dx. Then d/dx (y × IF) = Q(x) × IF, leading to the general solution.
一阶线性常微分方程的积分因子 – 对于 dy/dx + P(x) y = Q(x)，两边乘以积分因子 IF = e^∫ P dx，则 d/dx (y × IF) = Q(x) × IF，从而求出通解。
Second-order constant coefficient homogeneous ODE – a y” + b y’ + c y = 0 has auxiliary equation a m² + b m + c = 0. Real distinct roots → y = A e^(m₁x) + B e^(m₂x); repeated root → y = (A + Bx) e^(mx); complex roots α ± iβ → y = e^(αx) (A cos βx + B sin βx).
二阶常系数齐次常微分方程 – a y” + b y’ + c y = 0 的辅助方程为 a m² + b m + c = 0。相异实根 → y = A e^(m₁x) + B e^(m₂x)；重根 → y = (A + Bx) e^(mx)；复根 α ± iβ → y = e^(αx) (A cos βx + B sin βx)。
Particular integral (PI) and complementary function (CF) – For a non-homogeneous ODE, y_GS = CF (solution to homogeneous part) + PI (a function that satisfies the full equation). Trial functions depend on f(x): polynomial, exponential, or trigonometric forms.
特解（PI）与余函数（CF） – 对于非齐次常微分方程，通解 y_GS = CF（齐次部分的解）+ PI（满足整个方程的一个函数）。试函数取决于 f(x) 形式：多项式、指数或三角形式。
6. Further Vectors (Core Pure) | 向量进阶（核心纯数）
Scalar triple product – For vectors a, b, c, the scalar triple product a · (b × c) gives the volume of the parallelepiped. If a · (b × c) = 0, the vectors are coplanar.
标量三重积 – 对于向量 a, b, c，标量三重积 a · (b × c) 给出平行六面体的体积。若 a · (b × c) = 0，则向量共面。
Equation of a plane – Vector form: r = a + λ b + μ c, or r · n = a · n where n is normal. Cartesian form: ax + by + cz = d.
平面方程 – 向量形式：r = a + λ b + μ c，或 r · n = a · n，其中 n 为法向量。笛卡儿形式：ax + by + cz = d。
Angle between planes and lines – The acute angle θ between two planes with normals n₁, n₂ satisfies cos θ = |n₁ · n₂| / (|n₁||n₂|). For a line and plane, use sin θ = |direction · n| / (|direction||n|).
平面与直线的夹角 – 两平面（法向量为 n₁, n₂）的锐角 θ 满足 cos θ = |n₁ · n₂| / (|n₁||n₂|)。对于直线与平面，用 sin θ = |方向向量 · n| / (|方向向量||n|)。
Vector product and geometry – a × b is perpendicular to both a and b; its magnitude |a × b| = |a||b| sin θ gives the area of the parallelogram they span.
向量积与几何 – a × b 垂直于 a 和 b 两者；其大小 |a × b| = |a||b| sin θ 给出它们所张平行四边形的面积。
7. Further Probability Distributions (Further Stats) | 进阶概率分布（进阶统计）
Poisson distribution – X ~ Po(λ) models the number of random events in a fixed interval. E(X) = Var(X) = λ. The sum of independent Poissons is Poisson: Po(λ₁) + Po(λ₂) = Po(λ₁+λ₂).
泊松分布 – X ~ Po(λ) 用于建模固定区间内随机事件的数量。期望 E(X) = Var(X) = λ。独立泊松变量之和仍为泊松：Po(λ₁) + Po(λ₂) = Po(λ₁+λ₂)。
Geometric distribution – X ~ Geo(p) counts trials until first success. P(X = x) = p(1-p)^(x-1) for x = 1, 2, … . E(X) = 1/p, Var(X) = (1–p)/p².
几何分布 – X ~ Geo(p) 计数直到首次成功的试验次数。P(X = x) = p(1-p)^(x-1)，x = 1, 2, … 。期望 E(X) = 1/p，方差 Var(X) = (1–p)/p²。
Negative binomial distribution – NB(r, p) counts trials until r successes. This generalises the geometric and is used when a fixed number of successes is required.
负二项分布 – NB(r, p) 计数达到 r 次成功所需的试验次数。它推广了几何分布，用于需要固定成功次数的情况。
Continuous uniform distribution – X ~ U(a, b) has pdf f(x) = 1/(b-a) for a ≤ x ≤ b, mean (a+b)/2, variance (b–a)²/12. The cumulative distribution function rises linearly.
连续均匀分布 – X ~ U(a, b) 的概率密度函数为 f(x) = 1/(b-a)，a ≤ x ≤ b，均值 (a+b)/2，方差 (b–a)²/12。累积分布函数线性增加。
8. Hypothesis Testing (Further Stats) | 假设检验（进阶统计）
Null and alternative hypotheses (H₀ and H₁) – H₀ is the assumption of no effect or status quo; H₁ states what the test aims to support. Testing is based on assuming H₀ is true.
原假设与备择假设 (H₀ 与 H₁) – H₀ 是无效应或维持现状的假设；H₁ 陈述检验试图支持的结论。检验基于假设 H₀ 成立而进行。
Significance level and critical region – The significance level α (often 5 %) is the probability of rejecting H₀ when true (Type I error). The critical region contains sample results leading to rejection.
显著性水平与拒绝域 – 显著性水平 α（常用5%）是当 H₀ 为真时拒绝它的概率（第I类错误）。拒绝域包含会导致拒绝 H₀ 的样本结果。
p-value – The probability of obtaining a test statistic at least as extreme as observed, given H₀. Reject H₀ if p-value < α.
p值 – 在 H₀ 成立下，获得至少与实际观测一样极端的检验统计量的概率。若 p 值 < α，则拒绝 H₀。
Contingency tables and chi-squared test – Used for testing independence of categorical variables. The test statistic Σ (O-E)²/E follows χ² distribution with (rows–1)(cols–1) degrees of freedom. Expected frequencies E must be ≥ 5.
列联表与卡方检验 – 用于检验分类变量的独立性。检验统计量 Σ (O-E)²/E 服从 χ² 分布，自由度为 (行数–1)(列数–1)。期望频数 E 必须 ≥ 5。
9. Kinematics in Further Mechanics (Further Mechanics) | 进阶力学中的运动学（进阶力学）
Variable acceleration – When acceleration a is a function of time t, velocity v = ∫ a dt and displacement s = ∫ v dt. When a = f(v) or a = g(s), use the appropriate chain rule: a = v dv/ds.
变加速度 – 当加速度 a 是时间 t 的函数时，速度 v = ∫ a dt，位移 s = ∫ v dt。若 a = f(v) 或 a = g(s)，则使用链式法则 a = v dv/ds。
Projectiles with resistive forces – Air resistance modelled by a force proportional to speed (or squared speed) leads to differential equations that must be solved to find velocity–time relationships.
受阻力作用的抛体运动 – 将空气阻力建模为与速度（或速度平方）成正比的力，会引出必须求解的微分方程，以找到速度–时间关系。
Further vector kinematics – For motion in two or three dimensions, displacement r(t), velocity v = dr/dt, acceleration a = dv/dt, all expressed as vectors. Integration of vector equations requires initial conditions.
进阶向量运动学 – 对于二维或三维运动，位移 r(t)，速度 v = dr/dt，加速度 a = dv/dt，全部以向量表示。积分向量方程需要初始条件。
10. Algorithms and Graph Theory (Decision Maths) | 算法与图论（决策数学）
Graph terminology – A simple graph has no loops or multiple edges. A path has no repeated vertices; a cycle is a closed path. A tree is a connected graph with no cycles; a spanning tree connects all vertices with the minimum number of edges.
图论术语 – 简单图无环或多重边。路径中顶点不重复；回路是一条闭合路径。树是无回路的连通图；生成树用最少的边连接所有顶点。
Kruskal’s and Prim’s algorithms – Both find a minimum spanning tree. Kruskal’s algorithm adds the cheapest edge that does not create a cycle; Prim’s builds from a start vertex, always adding the cheapest edge connecting the tree to a new vertex.
克鲁斯卡尔算法与普里姆算法 – 两者均用于寻找最小生成树。克鲁斯卡尔算法添加不产生回路的最便宜边；普里姆算法从起始顶点开始，始终添加连接树到新顶点的最便宜边。
Dijkstra’s algorithm – Finds the shortest path from a start node to all others in a weighted graph with non-negative weights. Temporary labels are updated until all nodes are permanently labelled.
迪杰斯特拉算法 – 在具有非负权重的加权图中，寻找从起始节点到所有其他节点的最短路径。临时标号不断更新，直到所有节点都被永久标号。
Critical path analysis (CPA) – Determines the minimum project completion time. An activity on node diagram shows earliest start time (EST), latest finish time (LFT), and float. The critical path consists of activities with zero total float.
关键路径分析 (CPA) – 确定项目的最短完成时间。节点活动图显示最早开始时间 (EST)、最晚完成时间 (LFT) 和时差。关键路径由总时差为零的活动组成。
Published by TutorHao | Further Maths Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导