Year 13 CIE Further Maths: Key Vocabulary & Terminology Quick Memorisation Guide | CIE 进阶数学：词汇术语速记指南

📚 Year 13 CIE Further Maths: Key Vocabulary & Terminology Quick Memorisation Guide | CIE 进阶数学：词汇术语速记指南

Mastering the precise language of CIE A-Level Further Mathematics is half the battle. This guide breaks down essential terms from complex numbers to reduction formulae, pairing clear English definitions with their Chinese equivalents and mnemonics to help you memorise them quickly and apply them accurately in your Year 13 exams.

掌握 CIE A-Level 进阶数学的精准语言是成功的一半。本指南将复数、递推公式等核心术语一一拆解，给出了清晰的英文定义、中文对应以及助记技巧，帮助你在 Year 13 考试中快速记忆并准确运用这些词汇。

1. Complex Numbers | 复数

Complex number (z = a + bi): A number of the form a + bi where a, b ∈ ℝ and i² = –1. The real part is Re(z) = a, the imaginary part Im(z) = b.

复数 (z = a + bi)：形如 a + bi 的数，其中 a, b 为实数且 i² = –1。实部记作 Re(z) = a，虚部记作 Im(z) = b。

Modulus |z|: The distance from the origin in the Argand diagram, calculated as √(a² + b²). It represents the ‘magnitude’ of the complex number.

模 |z|：复数在阿根图上到原点的距离，计算为 √(a² + b²)，代表复数的“大小”。

Argument arg(z): The angle θ measured from the positive real axis, usually in (–π, π]. For a point in the first quadrant, arg(z) = tan⁻¹(b/a).

辐角 arg(z)：从正实轴开始测量的角度θ，通常落在 (–π, π] 内。第一象限中，arg(z) = tan⁻¹(b/a)。

Complex conjugate z*: The reflection of z across the real axis, written as z* = a – bi. Key property: z·z* = |z|².

共轭复数 z*：复数关于实轴的反射，写作 z* = a – bi。重要性质：z·z* = |z|²。

Polar form: z = r(cosθ + i sinθ) = r e^iθ, with r = |z| and θ = arg(z). This links algebra to geometry.

极坐标形式：z = r(cosθ + i sinθ) = r e^iθ，其中 r = |z|，θ = arg(z)。这一形式将代数与几何联系起来。

De Moivre’s theorem: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ). It is used to find powers and roots of complex numbers.

棣莫弗定理：(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)，用于求复数的幂与根。

Roots of unity: The n distinct solutions of zⁿ = 1, given by e^2kπi/n for k = 0,1,…,n–1. They lie symmetrically on the unit circle.

单位根：方程 zⁿ = 1 的 n 个不同解，表示为 e^2kπi/n（k = 0,1,…,n–1），它们对称地分布在单位圆上。

Mnemonic: “Modulus = Magnitude; Argument = Angle” — both begin with ‘A’. Think of the conjugate as a ‘copy’ reflected along the real axis.

助记：“模 = 大小；辐角 = 角度”——Argument 与 Angle 都以 A 开头。把共轭想象成沿实轴的“镜像复制”。

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2. Matrices & Eigenvalues | 矩阵与特征值

Determinant (det A or |A|): A scalar value that indicates whether a square matrix is invertible. For a 2×2 matrix [[a,b],[c,d]], det = ad – bc.

行列式（det A 或 |A|）：一个标量值，用于判断方阵是否可逆。对 2×2 矩阵 [[a,b],[c,d]]，行列式为 ad – bc。

Singular vs non-singular: A matrix is singular if det = 0 (no inverse); non-singular if det ≠ 0 (inverse exists).

奇异与非奇异：若行列式为零，矩阵奇异（不可逆）；若行列式非零，矩阵非奇异（可逆）。

Inverse matrix A⁻¹: The matrix such that A·A⁻¹ = I. For 2×2, A⁻¹ = (1/det)[[d,–b],[–c,a]].

逆矩阵 A⁻¹：满足 A·A⁻¹ = I 的矩阵。对 2×2 矩阵，A⁻¹ = (1/det)[[d,–b],[–c,a]]。

Eigenvalue λ and eigenvector v: A scalar λ and non-zero vector v satisfying Av = λv. The direction of v is unchanged by the transformation.

特征值 λ 与特征向量 v：标量 λ 与非零向量 v 满足 Av = λv。特征向量的方向在变换下保持不变。

Characteristic equation: det(A – λI) = 0. Its roots are the eigenvalues.

特征方程：det(A – λI) = 0，其根即为特征值。

Diagonalisation: Writing A = PDP⁻¹, where D is a diagonal matrix of eigenvalues and P has the corresponding eigenvectors. Possible only if eigenvectors are linearly independent.

对角化：将 A 写成 A = PDP⁻¹，其中 D 是由特征值组成的对角阵，P 包含对应的特征向量。仅当特征向量线性无关时可行。

Mnemonic: ‘Eigen’ comes from German for ‘own’ — an eigenvector keeps its own direction. For 2×2 inverse, remember ‘swap d and a, flip signs of b and c, divide by det’.

助记：“Eigen”源于德语“自身的”——特征向量保持自身方向。逆矩阵记忆技巧：“d 和 a 交换，b 和 c 变号，再除以行列式”。

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3. Vectors in 3D | 三维向量

Position vector: A vector from the origin to a point P, written as OP or r = xi + yj + zk.

位置向量：从原点到点 P 的向量，写作 OP 或 r = xi + yj + zk。

Magnitude: The length |r| = √(x² + y² + z²).

模（长度）：|r| = √(x² + y² + z²)。

Scalar (dot) product: a · b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃. Used to find angles and test orthogonality.

数量积（点乘）：a · b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃，用于求角度和判断垂直。

Vector (cross) product: a × b produces a vector perpendicular to both a and b. Magnitude |a×b| = |a||b|sinθ.

向量积（叉乘）：a × b 产生一个垂直于 a 和 b 的向量，其模为 |a×b| = |a||b|sinθ。

Triple scalar product: a · (b × c) gives the volume of the parallelepiped defined by a, b, c. Equals the determinant of the matrix formed by the vectors.

三重数积：a · (b × c) 给出由 a, b, c 三向量确定的平行六面体的体积，也等于由这些向量构成的矩阵的行列式。

Equation of a line: r = a + λd, where a is a point on the line and d is the direction vector.

直线方程：r = a + λd，其中 a 是直线上一点，d 是方向向量。

Equation of a plane: Scalar form r·n = p or cartesian form ax + by + cz = d, where n is the normal vector.

平面方程：数量形式 r·n = p，或笛卡儿形式 ax + by + cz = d，其中 n 为法向量。

Mnemonic: Dot product gives a scalar (dot → single output). Cross product gives a vector (cross → new direction). Triple scalar product = ‘box product’ giving volume.

助记：点乘得到标量（点 → 唯一输出），叉乘得到向量（叉 → 新方向）；三重数积 = “盒子积”求体积。

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4. Hyperbolic Functions | 双曲函数

sinh x and cosh x: sinh x = (eˣ – e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2. They are the odd and even parts of eˣ.

双曲正弦与双曲余弦：sinh x = (eˣ – e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2。它们分别是 eˣ 的奇分量和偶分量。

tanh x: tanh x = sinh x / cosh x = (e²ˣ – 1)/(e²ˣ + 1). Range is (–1, 1).

双曲正切：tanh x = sinh x / cosh x = (e²ˣ – 1)/(e²ˣ + 1)，值域为 (–1, 1)。

Key identity: cosh²x – sinh²x = 1. Compare with trigonometric identity cos²θ + sin²θ = 1 — note the sign change.

核心恒等式：cosh²x – sinh²x = 1。对比三角恒等式 cos²θ + sin²θ = 1，注意符号变化。

Inverse hyperbolic functions: arsinh x = ln(x + √(x²+1)); arcosh x = ln(x + √(x²–1)), x ≥ 1; artanh x = ½ ln((1+x)/(1–x)), |x| < 1.

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。

Derivatives: d/dx(sinh x) = cosh x; d/dx(cosh x) = sinh x; d/dx(tanh x) = sech²x. No negative sign, unlike trig derivatives.

导数：d/dx(sinh x) = cosh x；d/dx(cosh x) = sinh x；d/dx(tanh x) = sech²x。与三角导数不同，这里没有负号。

Mnemonic: Think ‘h’ for ‘hyperbolic’ — the ‘h’ functions are like trig without minus signs. Sinh is the odd part of eˣ (like sin is odd), cosh is even (like cos).

助记：“h”代表 hyperbolic（双曲）——带 h 的函数就像没有负号的三角函数。sinh 是 eˣ 的奇部（类似 sin 是奇函数），cosh 是偶部（类似 cos）。

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5. Differential Equations | 微分方程

Ordinary differential equation (ODE): An equation involving derivatives of a function y(x) with respect to one independent variable x.

常微分方程 (ODE)：包括未知函数 y(x) 对其唯一自变量 x 的导数的方程。

Order: The highest derivative present, e.g. d²y/dx² + 3 dy/dx = 0 is second order.

阶数：方程中最高阶导数的阶数，例如 d²y/dx² + 3 dy/dx = 0 是二阶的。

Linear vs non-linear: A linear ODE has y and its derivatives to the first power only, with coefficients depending on x at most.

线性与非线性：线性 ODE 中的 y 及其导数仅有一次幂，系数最多依赖于 x。

Homogeneous vs non-homogeneous: A linear ODE is homogeneous if the RHS is 0; otherwise it is non-homogeneous.

齐次与非齐次：线性 ODE 若右端为 0 则是齐次的，否则为非齐次。

Complementary function (CF): The general solution of the associated homogeneous equation, containing arbitrary constants.

补函数 (CF)：对应齐次方程的通解，含有任意常数。

Particular integral (PI): Any solution of the full non-homogeneous equation, found by trial functions or other methods.

特积分 (PI)：完整非齐次方程的任一特解，通过试函数或其他方法求得。

Auxiliary equation: For a constant-coefficient ODE a y” + b y’ + c y = 0, the auxiliary equation is am² + bm + c = 0. Its roots determine the form of the CF.

辅助方程：对常系数 ODE a y” + b y’ + c y = 0，辅助方程为 am² + bm + c = 0。其根决定了 CF 的形式。

Integrating factor: For a first-order linear ODE dy/dx + P(x)y = Q(x), multiply by e^{∫P dx} to make the LHS an exact derivative.

积分因子：对一阶线性 ODE dy/dx + P(x)y = Q(x)，乘以 e^{∫P dx} 可使左端成为恰当导数。

General solution: y = CF + PI. Apply boundary/initial conditions to find specific constants.

通解：y = CF + PI。代入边界/初始条件可求出特定常数。

Mnemonic: ‘Complementary’ sounds like ‘complete’ of the homogeneous part; ‘Particular Integral’ picks out ‘a particular’ solution. Think ‘Auxiliary equation’ helps to find the ‘auxiliary’ (helping) roots.

助记：“Complementary” 听起来像齐次部分的“补充”；“Particular Integral” 挑选出一个“特定的”解。“Auxiliary equation” 就是帮助找出根的辅助工具。

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6. Series & Expansions | 级数与展开

Maclaurin series: f(x) = f(0) + f ‘(0)x + f ”(0)x²/2! + f ”'(0)x³/3! + … . Expansion about x = 0.

麦克劳林级数：f(x) = f(0) + f ‘(0)x + f ”(0)x²/2! + f ”'(0)x³/3! + …，在 x = 0 附近展开。

Taylor series: f(x) = f(a) + f ‘(a)(x–a) + f ”(a)(x–a)²/2! + … . Maclaurin is a special case with a = 0.

泰勒级数：f(x) = f(a) + f ‘(a)(x–a) + f ”(a)(x–a)²/2! + …。麦克劳林级数是 a = 0 的特例。

Summation notation Σ: Σ (from r=1 to n) u_r represents u₁ + u₂ + … + u_n. Commonly used with standard results for Σr, Σr², Σr³.

求和符号 Σ：Σ (r=1 to n) u_r 表示 u₁ + u₂ + … + u_n。常与 Σr, Σr², Σr³ 的标准结果一起使用。

Method of differences: If u_r = f(r) – f(r–1), then Σ u_r = f(n) – f(0). Useful for telescoping sums.

差分法：若 u_r = f(r) – f(r–1)，则 Σ u_r = f(n) – f(0)，适用于裂项相消求和。

Radius of convergence: For a power series Σ a_n xⁿ, the series converges for |x| < R and diverges for |x| > R. Found via ratio test.

收敛半径：对幂级数 Σ a_n xⁿ，级数在 |x| < R 时收敛，|x| > R 时发散，通常用比值法求得。

Mnemonic: Maclaurin = x = 0 (zero ‘o’ in Maclaurin? Or think ‘Mac’ starts near origin). Taylor = any ‘a’. Method of differences: ‘differences’ cancel like a collapsing telescope.

助记：麦克劳林在零处展开（Maclaurin 中含有 ‘a’ 并非 a，但可以联想“麦当劳零度可乐”）。泰勒在 a 处展开。差分法就像伸缩望远镜一样前后相消。

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7. Polar Coordinates | 极坐标

Pole & initial line: The fixed origin O is the pole; the positive x-axis is the initial line. A point P is defined by (r, θ), where r is distance from O and θ is the angle from the initial line.

极点与极轴：固定原点 O 为极点，正 x 轴为极轴。点 P 表示为 (r, θ)，r 为到 O

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