📚 Pre-U OCR Further Mathematics: Quick Memorisation Guide to Key Terminology | Pre-U OCR 进阶数学：词汇术语速记指南
Mastering further mathematics begins with a confident command of the language of the subject. This guide provides concise, bilingual explanations of the essential terms and notation found across the Pre-U OCR Further Mathematics syllabus, helping you recall definitions quickly, avoid common misconceptions, and approach examinations with clarity.
掌握进阶数学，从自如运用学科的语言开始。本指南提供 Pre-U OCR 进阶数学大纲中关键术语与符号的简明中英双语解释，帮助您快速回忆定义、避免常见误解，并以清晰的头脑迎接考试。
1. Sets and Logic | 集合与逻辑
A set is a well-defined collection of objects, called elements. Standard notation includes ∅ for the empty set, ℕ for natural numbers, ℤ for integers, ℚ for rational numbers, ℝ for real numbers, and ℂ for complex numbers. The complement A' refers to all elements not in A relative to a universal set ℰ.
集合是明确定义的对象的全体，称为元素。标准符号包括 ∅ 表示空集，ℕ 自然数，ℤ 整数，ℚ 有理数，ℝ 实数，ℂ 复数。补集 A' 指相对于全集 ℰ 不在 A 中的所有元素。
Logical symbols include ∀ (for all), ∃ (there exists), ∧ (and), ∨ (or), ¬ (not), ⇒ (implies), and ⇔ (if and only if). A contrapositive of P ⇒ Q is ¬Q ⇒ ¬P, which is logically equivalent to the original statement.
逻辑符号包括 ∀（对所有），∃（存在），∧（且），∨（或），¬（非），⇒（蕴含）和 ⇔（当且仅当）。P ⇒ Q 的逆否命题是 ¬Q ⇒ ¬P，与原命题逻辑等价。
- ⊆ subset – every element of A is in B; ⊂ proper subset – A ⊆ B but A ≠ B.
- ∩ intersection – elements in both A and B; ∪ union – elements in either A or B.
- ⊆ 子集 – A 的每个元素都在 B 中；⊂ 真子集 – A ⊆ B 但 A ≠ B。
- ∩ 交集 – 同时在 A 和 B 中的元素；∪ 并集 – 在 A 或 B 中的元素。
2. Proof Techniques | 证明方法
Direct proof starts from known facts and uses logical steps to arrive at the statement. Proof by contradiction assumes the negation of the statement and shows this leads to an impossibility.
直接证明从已知事实出发，通过逻辑步骤推出结论。反证法假设命题的否定成立，并推出矛盾。
Proof by induction case (n = 1) and an inductive step: assuming true for n = k, prove true for n = k + 1. The induction hypothesis is the assumption P(k) is true.
数学归纳法包含归纳奠基（n = 1）和归纳递推：假设 n = k 时成立，证明 n = k + 1 时成立。归纳假设指假设 P(k) 为真。
Proof by exhaustion checks every possible case. Disproof by counterexample requires just one case where the statement fails.
穷举证明检验所有可能情形。举反例证伪只需一个使命题不成立的例子。
3. Functions and Graphs | 函数与图像
A function f : A → B maps each element of domain A to exactly one element of codomain B. The range is the set of actual output values. A function is injective (one-to-one) if f(a) = f(b) ⇒ a = b. It is surjective (onto) if its range equals the codomain. A bijection is both injective and surjective, allowing an inverse function f⁻¹.
函数 f : A → B 将定义域 A 的每个元素映射到陪域 B 的唯一元素。值域是实际输出值的集合。若 f(a) = f(b) ⇒ a = b，则函数是单射（一一映射）。若值域等于陪域，则为满射（到上映射）。双射既是单射又是满射，因而存在反函数 f⁻¹。
Odd functions satisfy f(−x) = −f(x) (rotational symmetry about the origin). Even functions satisfy f(−x) = f(x) (reflective symmetry in the y‑axis). A periodic function repeats with period T: f(x + T) = f(x).
奇函数满足 f(−x) = −f(x)（关于原点中心对称）。偶函数满足 f(−x) = f(x)（关于 y 轴对称）。周期函数以 T 为周期重复：f(x + T) = f(x)。
Composite function gf means apply f first, then g. The modulus function |x| gives the absolute value. An asymptote is a line the graph approaches but never touches.
复合函数 gf 表示先作用 f，再作用 g。模函数 |x| 给出绝对值。渐近线是图像无限接近但永不触及的直线。
4. Complex Numbers | 复数
number z = x + iy has real part Re(z) = x and imaginary part Im(z) = y, where i² = −1. The complex conjugate z* = x − iy.
复数 z = x + iy 的实部 Re(z) = x，虚部 Im(z) = y，其中 i² = −1。共轭复数 z* = x − iy。
The modulus |z| = √x² + y² gives the distance from the origin in the Argand diagram. The argument arg(z) = θ is the angle with the positive real axis, usually given in (−π, π].
模 |z| = √x² + y² 表示在阿尔冈图中到原点的距离。辐角 arg(z) = θ 是与正实轴的夹角，通常取值 (−π, π]。
Polar form: z = r(cos θ + i sin θ) where r = |z|. Exponential form: z = reⁱᶿ using Euler's formula eⁱᶿ = cos θ + i sin θ. 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)，其中 r = |z|。指数形式：z = reⁱᶿ，应用欧拉公式 eⁱᶿ = cos θ + i sin θ。棣莫弗定理：(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。
Roots of unity: the nᵗʰ roots of 1 are given by e²ᵨi ᵏ/ⁿ for k = 0, 1, …, n−1. The principal root is 1.
单位根：1 的 n 次方根为 e²ᵨi ᵏ/ⁿ，k = 0, 1, …, n−1。主根为 1。
5. Matrices and Linear Transformations | 矩阵与线性变换
A matrix is a rectangular array of numbers. An identity matrix I has 1s on the leading diagonal and 0s elsewhere; AI = IA = A. A square matrix A has inverse A⁻¹ satisfying AA⁻¹ = I, provided det A ≠ 0.
矩阵是数字的矩形阵列。单位矩阵 I 主对角线为 1，其余为 0；AI = IA = A。方阵 A 若 行列式 det A ≠ 0，则存在逆矩阵 A⁻¹ 满足 AA⁻¹ = I。
The determinant − bc. For 3×3 matrices, use Sarrus’ rule or cofactor expansion. A matrix with det A = 0 is singular (non‑invertible).
2×2 矩阵 [a b; c d] 的行列式为 ad − bc。对 3×3 矩阵使用萨吕法则或余子式展开。行列式为零的矩阵是奇异矩阵（不可逆）。
A linear transformation cT(v). The transformation matrix maps the basis vectors (1, 0) and (0, 1) to its columns. Rotation by θ counter‑clockwise uses [cos θ −sin θ; sin θ cos θ]. Reflection in the x‑axis uses [1 0; 0 −1].
线性变换 T 满足 T(u + v) = T(u) + T(v) 和 T(cv) = cT(v)。变换矩阵将基向量 (1, 0) 和 (0, 1) 映射到矩阵的列向量。旋转 θ 角逆时针的矩阵为 [cos θ −sin θ; sin θ cos θ]；关于 x 轴的反射使用 [1 0; 0 −1]。
Eigenvalues λ and eigenvectors v satisfy Av = λv. Solve det(A − λI) = 0 for λ, then substitute to find v.
特征值 λ 和特征向量 v 满足 Av = λv。解 det(A − λI) = 0 求 λ，再代入求 v。
6. Series and Sequences | 级数与数列
An arithmetic sequence has common difference d: term aₙ = a + (n−1)d. Sum Sₙ = n/2 (a + l) = n/2 (2a + (n−1)d).
等差数列公差为 d：通项 aₙ = a + (n−1)d。求和 Sₙ = n/2 (a + l) = n/2 (2a + (n−1)d)。
A geometric sequence tio r: term aₙ = arⁿ⁻¹. Sum to n terms Sₙ = a(1 − rⁿ)/(1 − r), and the sum to infinity for |r| < 1 is S∞ = a/(1 − r).
等比数列公比为 r：通项 aₙ = arⁿ⁻¹。前 n 项和 Sₙ = a(1 − rⁿ)/(1 − r)，当 |r| < 1 时无穷项和为 S∞ = a/(1 − r)。
The Maclaurin series expands f(x) as f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + … . The series for eˣ, sin x, cos x, and ln(1 + x) must be known.
麦克劳林级数将 f(x) 展开为 f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + … 。必须熟记 eˣ, sin x, cos x, ln(1 + x) 的级数。
Method of differences lets series collapse through cancellation. Binomial expansion for rational n: (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + … , valid for |x| < 1.
差分法通过抵消使级数坍塌求和。二项展开式对有理数 n：(1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + … ，有效范围 |x| < 1。
7. Calculus Techniques | 微积分技巧
Differentiation from first principles: f'(x) = limₕ→₀ [f(x + h) − f(x)] / h. The chain rule: if y = f(u) and u = g(x), then dy/dx = (dy/du) × (du/dx).
第一原理求导：f'(x) = limₕ→₀ [f(x + h) − f(x)] / h。链式法则：若 y = f(u)，u = g(x)，则 dy/dx = (dy/du) × (du/dx)。
Product rule: d(uv)/dx = u'v + uv'. Quotient rule: d(u/v)/dx = (u'v − uv') / v². Implicit differentiation treats y as a function of x, differentiating both sides and using chain rule for y terms.
乘法法则：d(uv)/dx = u'v + uv'。除法法则：d(u/v)/dx = (u'v − uv') / v²。隐函数求导将 y 视为 x 的函数，等式两边求导并对 y 项使用链式法则。
Integration by substitution reverses the chain rule; integration by parts uses ∫ u dv = uv − ∫ v du. Definite integrals compute area; improper integrals have infinite limits or unbounded integrands, requiring limits.
换元积分逆用链式法则；分部积分法用 ∫ u dv = uv − ∫ v du。定积分计算面积；反常积分有无限限或被积函数无界，需要取极限。
Separable differential equations: rewrite dy/dx = g(x)h(y) as ∫ dy/h(y) = ∫ g(x) dx. An integrating factor for dy/dx + P(x)y = Q(x) is e∫ ᴾ ᵈˣ.
可分离微分方程：将 dy/dx = g(x)h(y) 改写为 ∫ dy/h(y) = ∫ g(x) dx。对 dy/dx + P(x)y = Q(x)，积分因子为 e∫ ᴾ ᵈˣ。
8. Vectors | 向量
A vector has magnitude and direction. In 3D, position vector r = xi + yj + zk. Scalar product (dot product) a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃. If a · b = 0, vectors are perpendicular.
向量有大小和方向。在三维中，位置向量 r = xi + yj + zk。数量积（点积）a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。若 a · b = 0，则向量垂直。
Vector product (cross product) a × b gives a vector perpendicular to both, magnitude |a||b| sin θ. The scalar triple product a · (b × c) gives the volume of the parallelepiped.
向量积（叉积）a × b 得到垂直于 a 和 b 的向量，模为 |a||b| sin θ。标量三重积 a · (b × c) 等于平行六面体的体积。
The plane equation is r · n = a · n or r = a + λb + μc. Distance from point to plane uses |(r₁ − a) · n| / |n|.
直线方程为 r = a + tb。平面方程为 r · n = a · n 或 r = a + λb + μc。点到平面的距离公式为 |(r₁ − a) · n| / |n|。
9. Hyperbolic Functions | 双曲函数
Hyperbolic functions are defined: sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ r many trigonometric identities with occasional sign changes, e.g., cosh² x − sinh² x = 1.
双曲函数定义为：sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x。它们类似三角恒等式，但偶有符号差异，例如 cosh² x − sinh² x = 1。
Inverse hyperbolic functions can be expressed in logarithms: 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.
反双曲函数可用对数表示：arsinh x = ln(x + √x² + 1)，arcosh x = ln(x + √x² − 1)（x ≥ 1），artanh x = ½ ln((1 + x)/(1 − x))（|x| < 1）。
inh x)/dx = cosh x; d(cosh x)/dx = sinh x; d(tanh x)/dx = sech² x. For use in integration, √x² + a² substitutions suggest hyperbolic identities.
导数：d(sinh x)/dx = cosh x；d(cosh x)/dx = sinh x；d(tanh x)/dx = sech² x。积分时遇到 √x² + a² 常借助双曲恒等式。
10. Mechanics – Kinematics and Dynamics | 力学 – 运动学与动力学
Displacement s is the vector change in position; velocity v = ds/dt; acceleration quations apply: v = u + at, s = ut + ½ at², s = vt − ½ at², v² = u² + 2as, s = ½ (u + v)t.
位移 s 是位置的变化矢量；速度 v = ds/dt；加速度 a = dv/dt = d²s/dt²。匀加速运动应用 SUVAT 公式：v = u + at, s = ut + ½ at², s = vt − ½ at², v² = u² + 2as, s = ½ (u + v)t。
Newton’s laws without net force. Second law: F = dp/dt, and for constant mass F = ma. Third law: action‑reaction pairs are equal and opposite. Momentum p = mv; impulse = change in momentum = ∫ F dt.
牛顿定律：第一定律 – 无合外力时速度恒定。第二定律 – F = dp/dt，质量恒定时 F = ma。第三定律 – 作用与反作用大小相等、方向相反。动量 p = mv；冲量 = 动量的变化 = ∫ F dt。
Work done by constant force F over displacement s is F·s. Kinetic energy = ½ m v²; potential energy = mgh near Earth’s surface. Power = rate of work = F·v.
功：恒力 F 经过位移 s 做功为 F·s。动能 = ½ m v²；势能 近地表为 mgh。功率 = 做功速率 = F·v。
Projectile motion: horizontal motion u cos θ constant, vertical motion y = u sin θ t − ½ g t². Range = (u² sin 2θ)/g; time of flight = (2u sin θ)/g.
抛体运动：水平方向速度 u cos θ 恒定，垂直方向 y = u sin θ t − ½ g t²。射程 = (u² sin 2θ)/g；飞行时间 = (2u sin θ)/g。
11. Mechanics – Centre of Mass and Moments | 力学 – 质心与力矩
The centre of mass / Σ mᵢ. For a uniform lamina, it lies on axes of symmetry. Composite bodies use the weighted position formula. Lamina is a 2D rigid body with negligible thickness.
质点系的质心坐标 r = (Σ mᵢ rᵢ) / Σ mᵢ。均匀薄片的质心位于对称轴上。组合体使用加权位置公式。薄片是厚度可忽略的二维刚体。
Moment of a force about a point = force × perpendicular distance. Couple is a pair of equal, opposite, parallel forces producing rotation without translation. Torque vector τ = r × F.
力矩：力对点的力矩 = 力 × 垂直距离。力偶是一对大小相等、方向相反、作用线平行的力，产生纯转动而无平动。力矩矢量 τ = r × F。
Equilibrium conditions for a rigid body: resultant force = 0 and resultant moment about any point = 0. Tipping/sliding analysis involves comparing limiting friction μR with the applied force.
刚体平衡条件：合外力为零 → ∑F = 0，对任一点的合力矩为零 → ∑M = 0。倾覆/滑动分析需比较极限摩擦力 μR 与施加力。
12. Statistics – Probability and Distributions | 统计学 – 概率与分布
Random variable X assigns a numerical value to outcomes. Probability density function (pdf) f(x) for continuous variable satisfies f(x) ≥ 0 and ∫ f(x) dx = 1 over its support. Cumulative distribution function F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt.
随机变量 X 将数值赋予结果。概率密度函数 f(x) 对连续变量满足 f(x) ≥ 0 且在全定义域积分为 1。累积分布函数 F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt。
Expectation (mean) E(X) = ∫ x f(x) dx; variance Var(X) = E(X²) − [E(X)]². The standard deviation is √Var(X).
期望（均值）E(X) = ∫ x f(x) dx；方差 Var(X) = E(X²) − [E(X)]²。标准差为 √Var(X)。
Binomial distribution B(n, p): P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ, mean np, variance np(1 − p). Poisson distribution Po(λ): P(X = r) = e⁻ʸ λʳ / r!, mean = variance = λ.
二项分布 B(n, p)：P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ，均值 np，方差 np(1 − p)。泊松分布 Po(λ)：P(X = r) = e⁻ʸ λʳ / r!，均值 = 方差 = λ。
Normal distribution N(μ, σ²) has bell‑shaped curve. Standardise using Z = (X − μ)/σ. The Central Limit Theorem: for large n, sample mean approximately follows N(μ, σ²/n).
正态分布 N(μ, σ²) 呈钟形曲线。标准化使用 Z = (X − μ)/σ。中心极限定理：当 n 大时，样本均值近似服从 N(μ, σ²/n)。
Hypothesis testing: null hypothesis H₀, alternative H₁. P‑value is the probability of observing the sample result, or more extreme, given H₀ is true. Reject H₀ if p < significance level α.
假设检验：零假设 H₀，备择假设 H₁。P 值是假定 H₀ 为真时观察到当前样本结果或更极端结果的概率。若 p < 显著性水平 α，则拒绝 H₀。
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