📚 Year 13 CIE Mathematics: Vocabulary & Terminology Quick Reference Guide | Year 13 CIE 数学：词汇术语速记指南
Mastering the precise language of CIE A-Level Mathematics is half the battle. This guide walks you through the essential terminology for Year 13 topics — from limits and complex numbers to vectors and differential equations — with clear explanations and immediate Chinese translations to speed up your revision.
掌握 CIE A-Level 数学的精确语言是成功的一半。本指南带你遍历 Year 13 各个主题的核心术语——从极限、复数到向量与微分方程——并配有清晰解释和即时中文翻译，帮助你快速复习。
1. Limits and Continuity | 极限与连续性
Limit of f(x) as x approaches a: the value that f(x) gets arbitrarily close to when x is near a, written as limx→a f(x) = L.
函数 f(x) 在 x 趋近于 a 时的极限：当 x 无限靠近 a 时 f(x) 趋近的值 L，记作 limx→a f(x) = L。
One-sided limit: the left-hand limit (x → a⁻) and right-hand limit (x → a⁺); the full limit exists only when both one-sided limits exist and are equal.
单侧极限：左极限 (x → a⁻) 与右极限 (x → a⁺)；只有当两个单侧极限都存在且相等时，极限才存在。
Continuity at a point: f is continuous at a if limx→a f(x) = f(a). A function is continuous over an interval if it is continuous at every point in that interval.
点连续性：若 limx→a f(x) = f(a)，则 f 在 a 点连续。如果一个函数在区间内每一点都连续，则它在该区间上连续。
Asymptote: a line that a curve approaches but never touches. Vertical asymptote: x = a if f(x) → ±∞ as x → a. Horizontal asymptote: y = L if f(x) → L as x → ±∞.
渐近线：曲线无限接近但永不相交的直线。垂直渐近线：若 x → a 时 f(x) → ±∞，则 x = a 是一条垂直渐近线。水平渐近线：若 x → ±∞ 时 f(x) → L，则 y = L 是一条水平渐近线。
2. Differentiation Rules | 微分法则
Derivative: f'(x) = limh→0 [f(x+h) − f(x)] / h, the instantaneous rate of change of f with respect to x.
导数：f'(x) = limh→0 [f(x+h) − f(x)] / h，表示 f 关于 x 的瞬时变化率。
Chain rule: if y = f(u) and u = g(x), then dy/dx = (dy/du) × (du/dx). Used for composite functions.
链式法则：若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du) × (du/dx)，用于复合函数的求导。
Product rule: (uv)’ = u’v + uv’. Quotient rule: (u/v)’ = (u’v − uv’) / v².
乘积法则：(uv)’ = u’v + uv’。商法则：(u/v)’ = (u’v − uv’) / v²。
Implicit differentiation: differentiate both sides of an equation with respect to x, treating y as a function of x, and then solve for dy/dx.
隐函数微分：方程两边同时对 x 求导，将 y 视为 x 的函数，然后解出 dy/dx。
Second derivative: d²y/dx², the derivative of dy/dx. It indicates concavity and is used to classify stationary points (maximum, minimum, point of inflection).
二阶导数：d²y/dx²，即 dy/dx 的导数。用以判断曲线的凹凸性并用来对驻点进行分类（极大值、极小值、拐点）。
3. Integration Techniques | 积分技巧
Indefinite integral: ∫ f(x) dx = F(x) + C, where F'(x) = f(x). C is the constant of integration.
不定积分：∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)，C 为积分常数。
Definite integral: ∫ab f(x) dx = F(b) − F(a). Represents the signed area between the curve y = f(x) and the x-axis from x = a to x = b.
定积分：∫ab f(x) dx = F(b) − F(a)。表示曲线 y = f(x) 与 x 轴在 x = a 到 x = b 之间的带符号面积。
Integration by substitution: substitute u = g(x), convert dx and the limits (for definite integrals), then integrate with respect to u.
换元积分法：令 u = g(x)，将 dx 转换为 du，并改变积分上下限（定积分），然后对 u 积分。
Integration by parts: ∫ u dv = uv − ∫ v du. Often used when the integrand is a product of a polynomial and an exponential, trigonometric, or logarithmic function.
分部积分法：∫ u dv = uv − ∫ v du。常用于被积函数为多项式与指数、三角或对数函数的乘积。
Improper integral: an integral where the integrand has a vertical asymptote within the interval, or the interval of integration is infinite. Evaluate using limits.
反常积分：在积分区间内被积函数存在垂直渐近线，或积分区间为无限的积分。需通过取极限来计算。
4. Differential Equations | 微分方程
Separable first-order ODE: dy/dx = f(x)g(y). Solve by separating variables: ∫ (1/g(y)) dy = ∫ f(x) dx.
可分离一阶常微分方程：dy/dx = f(x)g(y)。通过分离变量求解：∫ (1/g(y)) dy = ∫ f(x) dx。
General solution: contains an arbitrary constant C. Particular solution: obtained by applying an initial condition, e.g. y(x₀) = y₀.
通解：含任意常数 C 的解。特解：由初始条件（如 y(x₀) = y₀）确定的解。
Integrating factor: for a linear first-order ODE dy/dx + P(x)y = Q(x), the integrating factor is μ(x) = e∫ P(x) dx. Multiply both sides by μ(x) to solve.
积分因子：对于一阶线性微分方程 dy/dx + P(x)y = Q(x)，积分因子为 μ(x) = e∫ P(x) dx。方程两边同乘 μ(x) 后求解。
Second-order homogeneous ODE: a d²y/dx² + b dy/dx + c y = 0. Solve using the auxiliary equation ar² + br + c = 0. Roots r₁, r₂ give the form of the complementary function (real distinct, repeated, complex conjugate).
二阶齐次常微分方程：a d²y/dx² + b dy/dx + c y = 0。借助辅助方程 ar² + br + c = 0 求解。根 r₁, r₂ 决定余函数的类型（相异实根、重根、共轭复根）。
5. Vectors in 3D | 三维向量
Position vector: the vector from the origin O to a point P, written as OP = ai + bj + ck, where i, j, k are unit vectors along the x, y, z axes.
位置向量：从原点 O 指向点 P 的向量，记作 OP = ai + bj + ck，其中 i, j, k 分别是沿 x, y, z 轴的单位向量。
Magnitude: |v| = √(a² + b² + c²) for v = ai + bj + ck.
模：对于 v = ai + bj + ck，|v| = √(a² + b² + c²)。
Dot (scalar) product: u · v = u₁v₁ + u₂v₂ + u₃v₃ = |u||v| cos θ. Used to find angles, test perpendicularity (u · v = 0).
点积（标量积）：u · v = u₁v₁ + u₂v₂ + u₃v₃ = |u||v| cos θ。用于求夹角及判断垂直（u · v = 0）。
Vector equation of a line: r = a + λd, where a is a point on the line, d is the direction vector, and λ is a scalar parameter.
直线的向量方程：r = a + λd，其中 a 是直线上一点的位置向量，d 是方向向量，λ 是标量参数。
Vector (cross) product: a × b produces a vector perpendicular to both a and b, with magnitude |a||b| sin θ. Used only in 3D.
向量积（叉积）：a × b 得到同时垂直于 a 与 b 的向量，模长为 |a||b| sin θ。仅用于三维空间。
6. Complex Numbers | 复数
Imaginary unit: i = √(−1), i² = −1. A complex number z = a + bi, with real part Re(z) = a and imaginary part Im(z) = b.
虚数单位：i = √(−1)，i² = −1。复数 z = a + bi，实部 Re(z) = a，虚部 Im(z) = b。
Complex conjugate: z* = a − bi. Conjugate pairs are used to simplify division and to find real polynomials with complex roots.
共轭复数：z* = a − bi。共轭对用于化简除法，以及构造具有复根的实系数多项式。
Modulus: |z| = √(a² + b²), the distance from the origin in the Argand diagram. Argument (arg z): the angle θ measured from the positive real axis, usually in (−π, π].
模：|z| = √(a² + b²)，即复平面上点到原点的距离。辐角 (arg z)：从正实轴开始度量的角度 θ，通常取值区间为 (−π, π]。
Polar form: z = r(cos θ + i sin θ) = r eiθ, where r = |z| and θ = arg z. de Moivre’s theorem: (cos θ + i sin θ)n = cos(nθ) + i sin(nθ).
极坐标形式：z = r(cos θ + i sin θ) = r eiθ，其中 r = |z|，θ = arg z。棣莫弗定理：(cos θ + i sin θ)n = cos(nθ) + i sin(nθ)。
Roots of unity: solutions to zn = 1. The n distinct roots are equally spaced on the unit circle: e2πik/n for k = 0, 1, …, n−1.
单位根：zn = 1 的解。n 个不同的根在单位圆上等距分布：e2πik/n，k = 0, 1, …, n−1。
7. Algebra and Functions | 代数与函数
Modulus function: |x| = x if x ≥ 0, −x if x < 0. Modulus equations and inequalities often require splitting into cases or squaring both sides.
绝对值函数：|x| = x (若 x ≥ 0)，−x (若 x < 0)。绝对值方程和不等式常常需要分情况讨论或两边平方。
Partial fractions: expressing a rational function as a sum of simpler fractions. Types: distinct linear factors (A/(ax+b) + B/(cx+d)), repeated linear factors, and irreducible quadratic factors.
部分分式：将有理函数表示为较简单分式的和。类型包括：不同一次因式 (A/(ax+b) + B/(cx+d))、重复一次因式和不可约二次因式。
Composition of functions: (f ∘ g)(x) = f(g(x)). Domain and range restrictions must be checked carefully for composite functions.
复合函数：(f ∘ g)(x) = f(g(x))。复合函数的定义域与值域需仔细检验。
Inverse function: f−1(x) such that f(f−1(x)) = x. Only one-to-one functions have inverses; their graphs are reflections in the line y = x.
反函数：满足 f(f−1(x)) = x 的函数 f−1(x)。只有一一映射的函数才有反函数，其图像关于直线 y = x 对称。
8. Numerical Methods | 数值方法
Sign-change method (bisection): to locate a root of f(x) = 0, find an interval [a, b] where f(a) and f(b) have opposite signs. Repeatedly halve the interval.
符号改变法（二分法）：为寻找 f(x) = 0 的根，找到一个区间 [a, b] 使得 f(a) 与 f(b) 符号相反，然后反复对分区间。
Newton-Raphson method: iterative formula xn+1 = xn − f(xn) / f'(xn). Converges quadratically if initial guess is sufficiently close to the root and f'(x) ≠ 0.
牛顿-拉夫森法：迭代公式 xn+1 = xn − f(xn) / f'(xn)。若初始猜测足够接近根且 f'(x) ≠ 0，该方法以平方收敛的速度趋于根。
Fixed-point iteration: rearrange f(x) = 0 into x = g(x), then iterate xn+1 = g(xn). Convergence requires |g'(x)| < 1 near the root.
不动点迭代：将 f(x) = 0 改写为 x = g(x)，然后利用 xn+1 = g(xn) 进行迭代。收敛条件为在根附近 |g'(x)| < 1。
Trapezium rule: approximate ∫ab f(x) dx using a series of trapezoids. With n strips of width h = (b−a)/n, area ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yn−1) + yn].
梯形法则：用一系列梯形逼近定积分 ∫ab f(x) dx。将区间 n 等分，步长 h = (b−a)/n，面积 ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yn−1) + yn]。
9. Parametric Equations | 参数方程
Parametric equations: x = f(t), y = g(t), where t is the parameter. They describe a curve by giving each coordinate as a function of t.
参数方程：x = f(t), y = g(t)，其中 t 为参数。它们通过将各坐标表示为 t 的函数来描述一条曲线。
Gradient: dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0.
斜率：dy/dx = (dy/dt) / (dx/dt)，前提是 dx/dt ≠ 0。
Second derivative: d²y/dx² = d/dx (dy/dx) = [d/dt (dy/dx)] / (dx/dt).
二阶导数：d²y/dx² = d/dx (dy/dx) = [d/dt (dy/dx)] / (dx/dt)。
Area under parametric curve: ∫ y dx = ∫ y(t) (dx/dt) dt, with limits in terms of t.
参数曲线下的面积：∫ y dx = ∫ y(t) (dx/dt) dt，积分上下限需转换为 t 的取值范围。
10. Sequences and Series | 序列与级数
Arithmetic progression (AP): a, a+d, a+2d, … with common difference d. nth term: un = a + (n−1)d. Sum of first n terms: Sn = n/2 [2a + (n−1)d].
等差数列：a, a+d, a+2d, …，公差为 d。第 n 项：un = a + (n−1)d。前 n 项和：Sn = n/2 [2a + (n−1)d]。
Geometric progression (GP): a, ar, ar², … with common ratio r. nth term: un = arn−1. Sum of first n terms: Sn = a(1−rn)/(1−r) for r ≠ 1. Sum to infinity: S∞ = a/(1−r), valid when |r| < 1.
等比数列：a, ar, ar², …，公比为 r。第 n 项：un = arn−1。前 n 项和：Sn = a(1−rn)/(1−r)（r ≠ 1）。无穷项和：S∞ = a/(1−r)，仅当 |r| < 1 时有效。
Binomial expansion for rational n: (1 + x)n = 1 + nx + n(n−1)/2! x² + … + n(n−1)…(n−r+1)/r! xr + … , valid for |x| < 1.
有理数指数二项展开式：(1 + x)n = 1 + nx + n(n−1)/2! x² + … + n(n−1)…(n−r+1)/r! xr + …，当 |x| < 1 时成立。
Sigma notation: Σk=1n ak denotes the sum a₁ + a₂ + … + an.
西格玛符号：Σk=1n ak 表示 a₁ + a₂ + … + an 的和。
11. Trigonometric Identities and Equations | 三角恒等式与方程
Reciprocal identities: sec θ = 1/cos θ, csc (cosec) θ = 1/sin θ, cot θ = 1/tan θ.
倒数恒等式：sec θ = 1/cos θ，csc (cosec) θ = 1/sin θ，cot θ = 1/tan θ。
Compound angle formulae: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B).
和差角公式：sin(A ± B) = sin A cos B ± cos A sin B；cos(A ± B) = cos A cos B ∓ sin A sin B；tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)。
Double-angle identities: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ; tan 2θ = 2 tan θ/(1 − tan²θ).
倍角公式：sin 2θ = 2 sin θ cos θ；cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ；tan 2θ = 2 tan θ/(1 − tan²θ)。
Harmonic form: a sin θ + b cos θ = R sin(θ ± α) or R cos(θ ± α), where R = √(a² + b²) and α = arctan(b/a). Essential for solving equations and finding maximum/minimum values.
谐波形式：a sin θ + b cos θ 可写成 R sin(θ ± α) 或 R cos(θ ± α)，其中 R = √(a² + b²)，α = arctan(b/a)。这是解方程和求极值的重要工具。
12. Probability and Statistics Terminology | 概率统计术语
Discrete random variable: variable X that takes a countable number of values xi with probabilities P(X = xi) = pi. Σ pi = 1.
离散随机变量：取值 xi 具有可数个的变量 X，其概率为 P(X = xi) = pi，且 Σ pi = 1。
Expectation (expected value): E(X) = Σ xi pi. Variance: Var(X) = E[(X−μ)²] = E(X²) − [E(X)]². Standard deviation: σ = √Var(X).
期望（数学期望）：E(X) = Σ xi pi。方差：Var(X) = E[(X−μ)²] = E(X²) − [E(X)]²。标准差：σ = √Var(X)。
Binomial distribution: X ~ B(n, p). P(X = r) = ⁿCᵣ pr (1−p)n−r. Mean = np, variance = np(1−p).
二项分布：X ~ B(n, p)。P(X = r) = ⁿCᵣ pr (1−p)n−r。均值 = np，方差 = np(1−p)。
Normal distribution: X ~ N(μ, σ²). Z = (X − μ)/σ gives the standard normal Z ~ N(0, 1). Use tables or technology to find probabilities.
正态分布：X ~ N(μ, σ²)。通过 Z = (X − μ)/σ 转化为标准正态分布 Z ~ N(0, 1)，查表或使用计算工具求概率。
Conditional probability: P(A|B) = P(A ∩ B)/P(B). Independent events: P(A ∩ B) = P(A) × P(B).
条件概率：P(A|B) = P(A ∩ B)/P(B)。独立事件：P(A ∩ B) = P(A) × P(B)。
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