Pre-U WJEC Further Mathematics: Key Terminology Quick Memorisation Guide | Pre-U WJEC 进阶数学：词汇术语速记指南

📚 Pre-U WJEC Further Mathematics: Key Terminology Quick Memorisation Guide | Pre-U WJEC 进阶数学：词汇术语速记指南

Mastering the specialised vocabulary of Further Mathematics is as crucial as solving the problems themselves. The WJEC Pre-U specification demands precise use of language, and confusing terms like 'argument' and 'modulus' or 'hyperbolic sine' and 'inverse hyperbolic sine' can cost marks in explanation questions. This guide groups essential terms by topic and pairs English definitions with Chinese explanations, weaving in simple memory hooks to help you retain them efficiently.

掌握进阶数学的专业词汇与解题本身同等重要。WJEC Pre-U 大纲要求准确使用术语，如果混淆 "argument" 和 "modulus" 或 "hyperbolic sine" 与 "inverse hyperbolic sine"，在解释题中就会失分。本指南按主题梳理核心术语，将英文定义与中文解释配对，并融入简易记忆钩子，助你高效记住它们。

1. Complex Numbers Terminology | 复数术语

Imaginary unit i: Defined by i² = –1; it allows square roots of negative numbers to be expressed, e.g. √–9 = 3i.

虚数单位 i：由 i² = –1 定义；它使得负数开方能表示，如 √–9 = 3i。联想：i 像一根小小的旗杆，插在实数世界的负侧。

Complex number z = x + iy: A number with a real part x = Re(z) and an imaginary part y = Im(z).

复数 z = x + iy：具有实部 x = Re(z) 和虚部 y = Im(z) 的数。助记：Real 和 Imaginary 的首字母 R 与 I 对应 Re 和 Im。

Argand diagram: A plane where the x-axis represents the real part and the y-axis represents the imaginary part of a complex number.

阿甘图（复平面）：以 x 轴表示实部、y 轴表示虚部的平面。想到 "Argand" 如同 "argument and diagram"，用于观察角度的图。

Modulus |z|: The distance from the origin to the point representing z, calculated from |z| = √x² + y².

模 |z|：原点到表示 z 的点的距离，由 |z| = √x² + y² 计算。同音 "魔" 如魔戒大小，衡量复数的长度。

Argument arg(z): The angle θ between the positive real axis and the line joining the origin to z, usually given in (–π, π] or [0, 2π).

辐角 arg(z)：正实轴与连接原点和 z 的直线之间的夹角 θ，常取 (–π, π] 或 [0, 2π)。记忆：argument 本义争论，争论总要有个角度（angle）。

Complex conjugate z̅: If z = x + iy, then z̅ = x – iy; reflects the point in the real axis.

共轭复数 z̅：若 z = x + iy，则 z̅ = x – iy；点关于实轴反射。助记：conjugate 音似 "共轭"，像孪生兄弟，只是虚部反号。

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

Order of a matrix: Described as m × n where m is the number of rows and n is the number of columns.

矩阵的阶：表示为 m × n，其中 m 是行数，n 是列数。记作 "行 × 列"，字母顺序 r × c。

Identity matrix I leading diagonal and 0 elsewhere; it serves as the multiplicative identity: A I = I A = A.

单位矩阵 I：主对角线元素为 1、其余为 0 的方阵；满足乘法单位作用：A I = I A = A。联想：I 像数字 1 在矩阵世界的化身。

Determinant det(A) or |A| hat determines whether a matrix is invertible; for a 2×2 matrix [[a, b], [c, d]], det = ad – bc.

行列式 det(A) 或 |A|：决定矩阵是否可逆的标量；对 2×2 矩阵 [[a, b], [c, d]]， det = ad – bc。记住 "交叉相减"：左上乘右下减去右上乘左下。

Singular matrix: A square matrix with determinant equal to zero, meaning it has no inverse.

奇异矩阵：行列式为零的方阵，因此没有逆矩阵。Singular 表示 "单一"，零行列式使它变得 "孤立无逆"。

Linear transformation: A mapping T: ℝⁿ → ℝᵐ that preserves vector addition and scalar multiplication; representable by a matrix.

线性变换：满足向量加法和数乘保持的映射 T: ℝⁿ → ℝᵐ；可由矩阵表示。线性的本质就是 "加和倍" 不变。

Eigenvalue λ and eigenvector v a square matrix A, a non-zero vector v such that Av = λv; v is stretched by the factor λ.

特征值 λ 与特征向量 v：对于方阵 A，存在非零向量 v 满足 Av = λv；v 被拉伸因子 λ 缩放。Eigen- 源自德语 "自身的"，它们就是矩阵自身固有的方向和伸缩。

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3. Vectors in Three Dimensions | 三维向量

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

位置向量：从原点到点 P 的向量，记作 r = xi + yj + zk。它确定了空间中的 "位置"。

Unit vectors i, j, k: Mutually perpendicular vectors of length 1 along the x-, y- and z-axes respectively.

单位向量 i, j, k：沿 x、y、z 轴方向长度 1 且两两垂直的向量。用字母顺序 i, j, k 对应 x, y, z。

Scalar (dot) product a·b: Defined as |a||b| cos θ, or for components a₁b₁ + a₂b₂ + a₃b₃; used to find angles and to test orthogonality.

标量积（点积）a·b：定义为 |a||b| cos θ，或分量形式 a₁b₁ + a₂b₂ + a₃b₃；用于求角与判断垂直。点积结果是 scalar，记住 "点 ⇒ 标量"。

Vector (cross) product a × b: Produces a vector perpendicular to both a and b, with magnitude |a||b| sin θ; components found using 3×3 determinant.

向量积（叉积）a × b：得到垂直于 a 与 b 的向量，大小为 |a||b| sin θ；可用三阶行列式计算。叉号 × 暗示结果仍是向量。

Direction ratios and direction cosines: A set of three numbers proportional to the components of a vector; direction cosines are the cosines of the angles with the axes.

方向数与方向余弦：一组与向量分量成比例的数；方向余弦是向量与各轴夹角的余弦。前者 'ratio' 比例，后者 'cosine' 余弦。

Vector equation of a line t d, where a is a point on the line, d is a direction vector and t is a scalar parameter.

直线的向量方程：r = a + t d，其中 a 为线上一点，d 为方向向量，t 为参数。参数 t 滑出整条直线。

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4. Hyperbolic Functions | 双曲函数

Hyperbolic sine sinh x: Defined as (eˣ – e⁻ˣ)/2; it is odd and its graph resembles a slanted S-curve through the origin.

双曲正弦 sinh x：定义为 (eˣ – e⁻ˣ)/2；奇函数，图像过原点呈倾斜 S 形。sinh 念作 "shine"，记住 'sh' 有 h 代表双曲 (hyperbolic)。

Hyperbolic cosine cosh x: Defined as (eˣ + e⁻ˣ)/2; it is even and its graph is a suspended chain (catenary), minimum value 1 at x = 0.

双曲余弦 cosh x：定义为 (eˣ + e⁻ˣ)/2；偶函数，图像为悬链线，在 x = 0 取得最小值 1。cosh 像 cush，它永远大于等于 1。

Hyperbolic tan tanh x: Defined as sinh x / cosh x; its graph lies between y = –1 and y = 1, analogous to the trigonometric tan but bounded.

双曲正切 tanh x：定义为 sinh x / cosh x；图像介于 y = –1 与 y = 1 之间，类似三角正切但有界。

Osborn’s rule: To convert a trigonometric identity into a hyperbolic identity, replace any product (or implied product) of two sines by the negative of the corresponding hyperbolic sinh product.

奥斯本法则：将三角恒等式转换为双曲恒等式时，将两个正弦的乘积（或隐含乘积）替换为对应双曲正弦乘积的相反数。简记：sin² 变 –sinh²。

Inverse hyperbolic functions: 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：arsinh x = ln(x + √x²+1)，arcosh x = ln(x + √x²–1)（x ≥ 1），artanh x = ½ ln((1+x)/(1–x))（|x|<1）。注意前缀 ar- 而非 arc-，记忆：area 级数相关。

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5. Polar Coordinates | 极坐标

Pole and initial line: The fixed point O (origin) and the fixed ray (positive x-axis) used to define polar coordinates (r, θ).

极点与极轴：用于定义极坐标 (r, θ) 的定点 O（原点）和固定射线（正 x 轴）。Pol- 表示 "极"，像地极。

Polar coordinates (r, θ): r is the distance from the pole, θ is the angle measured anticlockwise from the initial line.

极坐标 (r, θ)：r 为到极点的距离，θ 为从极轴逆时针测量的角度。r 代表 radius，θ 代表 angle。

Cardioid r = a(1 + cos θ): A heart-shaped curve; the name comes from 'kardia', the Greek for heart.

心形线 r = a(1 + cos θ)：心脏形状的曲线；cardioid 源自希腊语 "kardia"（心脏）。记住当 cos θ = 1 时 r 最大，形成一个尖。

Area enclosed by a polar curve: The area of a sector is ½ ∫ r² dθ between limits α and β.

极曲线所围面积：扇形面积公式为 ½ ∫ₐᵝ r² dθ。联想此式与圆形面积 ½r²θ 的推广。

Arc length in polar coordinates: The length of a curve given by r = f(θ) is ∫ √(r² + (dr/dθ)²) dθ.

极坐标下的弧长：曲线 r = f(θ) 的长度为 ∫ √(r² + (dr/dθ)²) dθ。形式上类似参数方程弧长，用 r 和 dr/dθ。

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6. Differential Equations | 微分方程

Order of a differential equation ghest derivative that appears in the equation; e.g. d²y/dx² + 3 dy/dx = 0 is second order.

微分方程的阶：方程中出现的最高阶导数；例如 d²y/dx² + 3 dy/dx = 0 是二阶的。

General solution: A solution containing arbitrary constants equal in number to the order of the equation.

通解：包含任意常数且常数个数等于方程阶数的解。通解是 "家族"，特解是 "个体"。

Particular integral (PI): Any particular solution of a non-homogeneous linear differential equation, found by trying a function similar to the RHS.

特解积分 (PI)：非齐次线性微分方程的任何一个特解，通过尝试与右边类似形式的函数得到。

Complementary function (CF): The general solution of the corresponding homogeneous equation (RHS = 0).

补充函数 (CF)：对应齐次方程（右边为 0）的通解。CF + PI = 通解，像搭配套餐。

Integrating factor: For a first order linear ODE dy/dx + P(x)y = Q(x), the factor is e^{∫ P dx}.

积分因子：对于一阶线性常微分方程 dy/dx + P(x)y = Q(x)，积分因子为 e^{∫ P dx}。用它乘方程，左边变为精确导数。

Separation of variables: Rewriting dy/dx = g(x)h(y) as (1/h(y)) dy = g(x) dx and integrating both sides.

变量分离法：将 dy/dx = g(x)h(y) 改写为 (1/h(y)) dy = g(x) dx 并两边积分。把 y 跟 dy 绑一起，x 跟 dx 绑一起。

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7. Series and Sequences | 级数与数列

Maclaurin series: A power series expansion of a function f(x) about x = 0: f(0) + f'(0)x + f''(0)x²/2! + …

麦克劳林级数：函数 f(x) 在 x = 0 处的幂级数展开：f(0) + f'(0)x + f''(0)x²/2! + …。Maclaurin 专指零点展开。

Taylor series: A series expansion about a point x = a: f(a) + f'(a)(x–a) + f''(a)(x–a)²/2! + …

泰勒级数：关于点 x = a 的级数展开：f(a) + f'(a)(x–a) + f''(a)(x–a)²/2! + …。Maclaurin 是 a = 0 的 Taylor。

Radius of convergence: The distance from the centre of a power series to the nearest singularity; within this radius the series converges absolutely.

收敛半径：从幂级数中心到最近奇点的距离；在此半径内级数绝对收敛。用 ratio test 求极限决定。

Method of differences: A technique to sum series by writing the r-th term as f(r) – f(r+1) so that terms cancel telescopically.

差分法：将级数的第 r 项写成 f(r) – f(r+1)，使项前后抵消以求和的技巧。像望远镜收缩，故名 telescoping。

Summation of finite series using standard results: ∑ r = n(n+1)/2, ∑ r² = n(n+1)(2n+1)/6, ∑ r³ = n²(n+1)²/4.

用标准结果求有限级数和：∑ r = n(n+1)/2，∑ r² = n(n+1)(2n+1)/6，∑ r³ = n²(n+1)²/4。这些是必背公式。

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8. Proof by Induction | 数学归纳法

Inductive hypothesis: Assume the statement P(k) is true for some integer k ≥ base value.

归纳假设：假设对于某个整数 k ≥ 起始值，命题 P(k) 为真。这是证明的 "假设台阶"。

Base case: Verify that P(1) or the smallest applicable integer is true.

基础情形：验证 P(1) 或最小适用整数为真。通常 n = 1，有时 n = 0。

Inductive step: Prove that if P(k) is true then P(k+1) is true, using the inductive hypothesis.

归纳步骤：利用归纳假设证明若 P(k) 真则 P(k+1) 真。多米诺递推的核心环节。

Conclusion by induction: Since P(base) is true and P(k) ⇒ P(k+1), P(n) holds for all integers n ≥ base.

归纳结论：因基础成立且 P(k) ⇒ P(k+1)，对所有 n ≥ base 的整数 P(n) 成立。最后必须给出标准结语。

Strong induction P(k+1); used where the result for k+1 depends on more than just the immediate predecessor.

强归纳法：假设对所有 j ≤ k P(j) 真以证明 P(k+1)；当 k+1 的结果依赖多个前驱时使用。

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9. Further Calculus Techniques | 进阶微积分技巧

Reduction formula of Iₙ₋₁, Iₙ₋₂, etc., usually derived via integration by parts. Example: Iₙ = ∫ sinⁱ x dx.

递推公式：将积分 Iₙ 用 Iₙ₋₁、Iₙ₋₂ 等表示，通常通过分部积分导出。例如 Iₙ = ∫ sinⁱ x dx。Reduction 指降低幂次。

Arc length of a curve y = f(x): From a to b, the length is ∫ √(1 + (dy/dx)²) dx.

曲线 y = f(x) 的弧长：从 a 到 b，长度为 ∫ √(1 + (dy/dx)²) dx。由勾股定理对小段 ds 推导。

Surface area of revolution out the x-axis, area = 2π ∫ y √(1 + (dy/dx)²) dx; about y-axis: 2π ∫ x √(1 + (dx/dy)²) dy.

旋转体表面积：绕 x 轴转：面积 = 2π ∫ y √(1 + (dy/dx)²) dx；绕 y 轴：2π ∫ x √(1 + (dx/dy)²) dy。记忆：2π × 半径 × 弧长微元。

Improper integral: An integral with an infinite limit or an integrand with a vertical asymptote; evaluated by taking a limit.

瑕积分：有无穷上限或被积函数有垂直渐近线的积分；通过取极限来求值。Improper 表示 "不标准"。

Mean value of a function: The average value of f(x) on [a, b] is (1/(b–a)) ∫ f(x) dx over the interval.

函数的平均值：f(x) 在 [a, b] 上的平均值为 (1/(b–a)) ∫ₐᵇ f(x) dx。类比离散平均推广到连续。

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10. Numerical Methods | 数值方法

Change of sign method y finding an interval [a, b] where f(a) and f(b) have opposite signs; requires continuity.

变号法：通过寻找 f(a) 与 f(b) 异号的区间 [a, b] 来定位 f(x)=0 的根；要求函数连续。简单但必须验证连续性以免漏根。

Newton-Raphson method: Iteration formula: xₙ₊₁ = xₙ – f(xₙ)/f'(xₙ); provides rapid convergence near a root.

牛顿-拉弗森法：迭代公式：xₙ₊₁ = xₙ – f(xₙ)/f'(xₙ)；在根附近收敛极快。用切线逼近。

Fixed-point iteration: Rearrange f(x)=0 into the form x = g(x) and iterate xₙ₊₁ = g(xₙ); converges if |g'(x)| < 1 near the root.

不动点迭代：将 f(x)=0 改写为 x = g(x) 并迭代 xₙ₊₁ = g(xₙ)；若在根附近 |g'(x)| < 1 则收敛。不动点即 x = g(x) 的解。

Mid-ordinate rule ∫ f(x) dx by dividing the interval into n strips of width h, and summing f(midpoint) × h.

中纵线法则：把区间分为宽为 h 的 n 条，用各条中点函数值乘以 h 求和来近似积分。矩形高度取自中点。

Simpson’s rule: Uses parabolic arcs to estimate the integral, with weighting pattern h/3 (y₀ + 4y₁ + 2y₂ + 4y₃ + … + yₙ) for even n.

辛普森法则：利用抛物线弧估计积分，当 n 为偶数时加权模式为 h/3 (y₀ + 4y₁ + 2y₂ + 4y₃ + … + yₙ)。系数 4, 2, 4 轮换，比梯形法更精确。

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