Year 13 CIE Mathematics: Vocabulary & Terminology Quick Reference Guide | Year 13 CIE 数学：词汇术语速记指南

📚 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。

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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 lim_x→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 lim_x→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 lim_x→a nction is continuous over an interval if it is continuous at every point in that interval.

点连续性：若 lim_x→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: 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。

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2. Differentiation Rules | 微分法则

Derivative: f'(x) = lim_h→0 [f(x+h) − f(x)] / h, the instantaneous rate of change of f with respect to x.

导数：f'(x) = lim_h→0 [f(x+h) − f(x)] / h，表示 f 关于 x 的瞬时变化率。

e: 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）。

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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: ∫_a^b Represents the signed area between the curve y = f(x) and the x-axis from x = a to x = b.

定积分：∫_a^b 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。

(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。

. 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。

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4. Differential Equations | 微分方程

rst-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。

nstant 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。

tor: 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。

s 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）。

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5. Vectors in 3D | 三维向量

nt 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）。

d, 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。

tor (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。

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6. Complex Numbers | 复数

Imaginary unit: i = √−1 = −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。

ugate 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 e^iθ, 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 e^iθ，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: e^2πik/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：e^2πik/n，k = 0, 1, …, n−1。

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7. Algebra and Functions | 代数与函数

Modulus function: |x| ulus 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 ∘ 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 . 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。

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8. Numerical Methods | 数值方法

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 x_n+1 = x_n − f(x_n) / f'(x_n). 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 x_n+1 = x_n − f(x_n) / f'(x_n)。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 x_n+1 = g(x_n). 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 x_n+1 = g(x_n) 进行迭代。收敛条件为在根附近 |g'(x)| < 1。

Trapezium rule: approximate ∫_a^b es of trapezoids. With n strips of width h = (b−a)/n, area ≈ h/2 [y₀ + 2(y₁ + y₂ + ... + y_n−1) + y_n].

梯形法则：用一系列梯形逼近定积分 ∫_a^b f(x) dx。将区间 n 等分，步长 h = (b−a)/n，面积 ≈ h/2 [y₀ + 2(y₁ + y₂ + ... + y_n−1) + y_n]。

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9. Parametric Equations | 参数方程

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。

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10. Sequences and Series | 序列与级数

Arithmetic progression (AP): a, a+d, a+2d, ... with common difference d. n^th term: u_n = a + (n−1)d. Sum of first n terms: S_n = n/2 [2a + (n−1)d].

等差数列：a, a+d, a+2d, ...，公差为 d。第 n 项：u_n = a + (n−1)d。前 n 项和：S_n = n/2 [2a + (n−1)d]。

Geometric progression (GP): a, ar, ar², ... with common ratio r. n^th term: u_n = arⁿ⁻¹. Sum of first n terms: S_n = 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：u_n = arⁿ⁻¹。前 n 项和：S_n = 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! x^r + ... , 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! x^r + ...，当 |x| < 1 时成立。

Sigma notation: Σ_k=1ⁿ a_k denotes the sum a₁ + a₂ + ... + a_n.

西格玛符号：Σ_k=1ⁿ a_k 表示 a₁ + a₂ + ... + a_n 的和。

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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)。

n 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。

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12. Probability and Statistics Terminology | 概率统计术语

Discrete random variable: variable X that takes a countable number of values x_i with probabilities P(X = x_i) = p_i. Σ p_i = 1.

离散随机变量：取值 x_i 具有可数个的变量 X，其概率为 P(X = x_i) = p_i，且 Σ p_i = 1。

Expectation (expected value): E(X) = Σ x_i p_i. 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 x_i p_i。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 p^r (1−p)^n−r. Mean = np, variance = np(1−p).

二项分布：X ~ B(n, p)。P(X = r) = ⁿCᵣ p^r (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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