A Guide to Memorising Key Terminology for Cambridge Year 13 Further Mathematics | 剑桥 Year 13 进阶数学核心术语速记指南

📚 A Guide to Memorising Key Terminology for Cambridge Year 13 Further Mathematics | 剑桥 Year 13 进阶数学核心术语速记指南

Mastering Year 13 Further Mathematics for Cambridge hinges not only on calculation skills but also on precise, fluent use of technical vocabulary. This guide picks out high-frequency terms from the pure, mechanics and statistics components and pairs each with a simple mnemonic or memory hook. Use it for active recall, self-quizzing and last-minute revision – every entry connects a term to its meaning in seconds.

掌握剑桥 Year 13 进阶数学不仅依靠计算能力，更需要对专业词汇的准确、流畅使用。本指南从纯数、力学和统计各模块挑选高频术语，并为每一个配上一个简明的助记法或记忆挂钩。可用于主动回忆、自测和考前速览——每个词条都能让你在几秒内将术语与意义绑定。

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1. Complex Numbers Terminology | 复数术语

The imaginary unit i satisfies i² = −1. Mnemonic: ‘i’ stands for ‘imaginary’ – it is not real.

虚数单位 i 满足 i² = −1。速记：i 代表‘imaginary’（想象的）——它不是实数。

A complex number z = x + iy has real part Re(z) = x and imaginary part Im(z) = y. Think: Real on the x‑axis, Imaginary on the y‑axis, just like coordinates.

复数 z = x + iy，实部 Re(z) = x，虚部 Im(z) = y。联想：实部对横轴，虚部对纵轴，和坐标一样。

Modulus |z| = √(x² + y²) is the distance from the origin. Mnemonic: ‘modulus’ sounds like ‘module’ – measure the length.

模 |z| = √(x² + y²) 是到原点的距离。助记：‘modulus’ 发音像‘module’（模块）——量一量长度。

The argument arg(z) = θ where tanθ = y/x, measured from the positive real axis. Mnemonic: you can ‘argue’ about the angle.

辐角 arg(z) = θ，满足 tanθ = y/x，从正实轴开始度量。速记：‘argue’（争论）那个角度。

The complex conjugate of z is z̅ = x − iy. Memory hook: just flip the sign of i – conjugate helps to rationalise denominators.

共轭复数为 z̅ = x − iy。记忆钩子：把 i 前的符号反过来——共轭常用于分母有理化。

Polar form: z = r(cosθ + i sinθ) and Euler’s form z = reⁱᶿ. Link: think of a polar bear circling the origin.

极形式：z = r(cosθ + i sinθ) 和欧拉形式 z = reⁱᶿ。联想：想象一只北极熊 (polar bear) 绕原点转圈。

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2. Matrices and Linear Transformations | 矩阵与线性变换

The order of a matrix is rows × columns. Think: ‘RC’ – rows come first, just like reading a book.

矩阵的阶为行数×列数。助记：‘RC’——先行后列，就像看书。

The identity matrix I behaves like the number 1: AI = IA = A. Mnemonic: ‘I’ is the identical twin that leaves everything unchanged.

单位阵 I 像数字 1：AI = IA = A。速记：‘I’ 是那个不变的同卵双胞胎。

A matrix is singular if det(A) = 0, meaning no inverse. Link: determinant ‘determines’ singularity – zero is a dead end.

若 det(A) = 0，矩阵奇异，不可逆。联想：行列式 (determinant) 决定 (determine) 奇异性——零就是死胡同。

An orthogonal matrix satisfies AᵀA = I, so its columns are orthonormal. Mnemonic: orthogonal = ortho (right) + gonal (angle) – columns meet at right angles.

正交矩阵满足 AᵀA = I，列向量为标准正交基。词根：ortho（正）+ gonal（角）——列之间直角相交。

An eigenvalue λ and eigenvector v satisfy Av = λv. Memory aid: ‘eigen’ means ‘own’ in German – the vector keeps its own direction after transformation.

特征值 λ 与特征向量 v 满足 Av = λv。记忆法：‘eigen’ 在德语里是‘自己的’——变换后向量方向保持不变。

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3. Hyperbolic Functions | 双曲函数

sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2. Mnemonic: sinh has a minus, so it is ‘shy’; cosh has a plus, so it is ‘cosy’.

sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2。助记：sinh 中有减号，像‘shy’（害羞）；cosh 中有加号，像‘cosy’（舒适）。

Osborn’s rule: to convert a trig identity, replace every trig function with its hyperbolic counterpart and flip the sign of any term containing a product of two sines. Shortcut: ‘Osborn stares at the sin² sign’.

Osborn 法则：将三角恒等式中的三角函数换成双曲函数，并把含有两个正弦之积的项变号。口诀：‘Osborn 盯着 sin² 的符号’。

Inverse hyperbolic functions can be written as logs: arsinh x = ln(x + √(x²+1)). Memory: ‘arsinh looks like arcsin but gives a log.’

反双曲函数可用对数表示：arsinh x = ln(x + √(x²+1))。速记：‘arsinh 看起来像 arcsin，但结果是对数。’

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4. Polar Coordinates | 极坐标

A point is given by (r, θ) with pole O and initial line. Mnemonic: polar picture – radius and angle sweep like a radar.

点用 (r, θ) 表示，原点为极点 O，极轴为起始线。联想：极坐标就像雷达扫描——半径和角度。

Area enclosed by a polar curve: ½ ∫ r² dθ. Remember: ‘half the square of r, integrated over angle’ – similar to circular sector area.

极曲线所围面积：½ ∫ r² dθ。速记：‘r 的平方的一半对角积分’——与扇形面积公式相似。

The tangent at a point on a polar curve: use dy/dx = (r sinθ derivative) / (r cosθ derivative). Mnemonic: Treat it as a parametric curve with parameter θ.

极坐标曲线上一点的切线：利用 dy/dx = (r sinθ 导数) / (r cosθ 导数)。助记：把曲线看作以 θ 为参数的参数曲线。

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5. Differential Equations | 微分方程

An ordinary differential equation (ODE) involves derivatives with respect to one variable. Mnemonic: ‘ordinary’ = one independent variable.

常微分方程 (ODE) 仅含关于一个自变量的导数。速记：‘ordinary’（平常）→单个自变量。

The order is the highest derivative; the degree is the power of that derivative after clearing fractions. Say: Order is how many dashes; degree is the power of the highest dash.

阶数是最高导数的阶，次数是去分母后最高导数的幂。口诀：阶看几阶导数，次看它头上的指数。

A first‑order linear ODE: dy/dx + P(x)y = Q(x). The integrating factor is e∫ᴾ ᵈˣ. Memory: Multiply by e to the integral of P – it ‘integrates’ the equation into solvable form.

一阶线性 ODE：dy/dx + P(x)y = Q(x)。积分因子为 e∫ᴾ ᵈˣ。记忆：乘上 e 的 P 积分，便可将方程‘积分’成可解形式。

Second‑order linear ODE with constant coefficients: a d²y/dx² + b dy/dx + cy = f(x). The auxiliary equation is am² + bm + c = 0. Mnemonic: Auxiliary (helper) equation finds the complementary function.

常系数二阶线性 ODE：a d²y/dx² + b dy/dx + cy = f(x)。辅助方程为 am² + bm + c = 0。助记：辅助 (auxiliary) 方程帮忙找齐次通解。

The complementary function (CF) solves the homogeneous equation; the particular integral (PI) fits the RHS. Think: Full solution = CF (from roots) + PI (from trial form).

余函数 (CF) 是齐次方程的解，特积分 (PI) 匹配右边项。牢记：全解 = CF（根决定）+ PI（试凑形式）。

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6. Vectors and 3D Geometry | 向量与三维几何

The scalar (dot) product a·b = |a||b| cosθ gives a number. Mnemonic: ‘dot’ produces a scalar – a dot is tiny (scalar quantity).

标量积（点积）a·b = |a||b| cosθ 结果为一个数。助记：‘点’积出标量——点很小，就是标量。

The vector (cross) product a×b yields a vector perpendicular to both. Shortcut: Cross product = ‘cross’ out the common plane – it points out of the page.

向量积（叉积）a×b 得出一个垂直于两者的向量。口诀：叉积‘叉’开公共平面，指向纸外。

The distance from a point to a line: |(AP)×d| / |d|. Memory: It’s the cross product magnitude over direction magnitude – think ‘area of parallelogram / base’.

点到直线的距离：|(AP)×d| / |d|。速记：叉积的模除以方向向量的模——‘平行四边形面积除以底’。

A unit vector has magnitude 1: â = a / |a|. Mnemonic: ‘unit’ means one – any vector divided by its length becomes a unit.

单位向量模长为 1：â = a / |a|。助记：‘unit’就是 1——任何向量除以其长度即得单位向量。

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7. Further Calculus | 进阶微积分

The Maclaurin series expands f(x) about 0: f(x) = Σ f⁽ⁿ⁾(0) xⁿ / n!. Mnemonic: Maclaurin makes a polynomial portrait of a function – all terms evaluated at zero.

麦克劳林级数在 x=0 处展开：f(x) = Σ f⁽ⁿ⁾(0) xⁿ / n!。助记：Maclaurin 为函数画多项式肖像——所有项都在 0 处取值。

An improper integral has an infinite limit or an unbounded integrand. Remember: ‘Improper’ means it goes beyond a proper interval – handle with limits.

反常积分有无穷限或被积函数无界。速记：‘Improper’（不正常的）意味着超出正常区间——用极限处理。

A reduction formula expresses an integral Iₙ in terms of Iₙ₋₁ or Iₙ₋₂. Mnemonic: It ‘reduces’ the power index step by step, like walking downstairs.

递推公式将积分 Iₙ 用 Iₙ₋₁ 或 Iₙ₋₂ 表示。助记：它一步一步‘缩减’指数，好像下楼梯。

The arc length for a curve y=f(x): L = ∫ √(1 + (dy/dx)²) dx. Memory: It’s Pythagoras on an infinitely small triangle: ds² = dx² + dy².

曲线 y=f(x) 的弧长：L = ∫ √(1 + (dy/dx)²) dx。记忆：无限小三角形的勾股定理：ds² = dx² + dy²。

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8. Further Mechanics | 进阶力学

Momentum p = mv. Mnemonic: Momentum = mass in motion – think of ‘moving mass’.

动量 p = mv。助记：动量 = 运动的质量——想想‘moving mass’。

The impulse of a constant force F over time t is Ft, equal to change in momentum. Memory: Impulse punches momentum: force × time = Δp.

常力 F 作用 t 时间的冲量为 Ft，等于动量变化量。速记：冲量打动量：力 × 时间 = Δp。

Conservation of linear momentum: total momentum before collision = total after, no external forces. Mnemonic: ‘Conservation’ – kept unchanged, like a conserved recipe.

线动量守恒：无外力时碰撞前后总动量不变。助记：‘Conservation’（守恒）——像保存的食谱，不增不减。

The coefficient of restitution e = relative speed after / relative speed before. 0 ≤ e ≤ 1. Memory: ‘e’ for elasticity – e=1 super bouncy, e=0 gooey.

恢复系数 e = 碰后相对速率 / 碰前相对速率，0 ≤ e ≤ 1。速记：‘e’代表弹性——e=1 超弹，e=0 粘滞。

In circular motion, centripetal acceleration a = rω² = v²/r. Mnemonic: ‘Centripetal’ points center – think of ‘centripetal force pulling you in’.

在圆周运动中，向心加速度 a = rω² = v²/r。助记：‘向心’指向中心——向心力把你往里拉。

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9. Further Probability and Statistics | 进阶概率与统计

The Poisson distribution: X ~ Po(λ), where λ = mean = variance. Memory: ‘Poisson’ sounds like ‘fish’ – average number of fish caught per hour!

泊松分布：X ~ Po(λ)，λ = 均值 = 方差。速记：‘Poisson’ 听上去像 ‘fish’——平均每小时钓到的鱼数！

The probability generating function (PGF) is Gₓ(t) = E(tˣ). Mnemonic: PGF ‘generates’ probabilities – coefficient of tᵏ gives P(X=k).

概率生成函数 (PGF) 为 Gₓ(t) = E(tˣ)。助记：PGF‘生成’概率——tᵏ 的系数就是 P(X=k)。

A continuous uniform distribution on [a,b] has pdf f(x)=1/(b−a). Link: flat as a board – all intervals of equal length have the same probability.

连续型均匀分布在 [a,b] 上的概率密度函数为 f(x)=1/(b−a)。联想：像平板一样平坦——等长区间等概率。

The moment generating function (MGF) Mₓ(t) = E(eᵗˣ) yields raw moments. Memory: MGF ‘generates’ moments by differentiation at 0.

矩生成函数 (MGF) 为 Mₓ(t) = E(eᵗˣ)，可求原矩。速记：MGF 通过在 t=0 处求导‘生成’矩。

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10. Proof by Induction and Contradiction | 归纳法与反证法

Mathematical induction: prove a statement P(n) for all n by base case, inductive hypothesis and inductive step. Mnemonic: domino effect – knock the first, and the rest fall.

数学归纳法：证明对所有 n 成立 P(n)，需基础步、归纳假设和归纳步。助记：多米诺骨牌——推倒第一个，其余的随之倒下。

Base case: verify P(1) or P(0) is true. Say: ‘base’ is the first stone.

基础步：验证 P(1) 或 P(0) 为真。口诀：‘基础’是第一块骨牌。

Inductive step: assume P(k) true and prove P(k+1) true. Memory: Assume for k, then climb to k+1 – one step at a time.

归纳步：假设 P(k) 真，推出 P(k+1) 真

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