📚 Pre-U AQA Further Mathematics: Vocabulary & Terminology Quick Memorization Guide | Pre-U AQA 进阶数学：词汇术语速记指南
In AQA Further Mathematics, mastering precise vocabulary and terminology is essential for interpreting questions correctly and articulating solutions clearly. This quick memorization guide collects key terms from across the syllabus, pairing each with a concise definition, memory hook, and its Chinese equivalent. Whether you are revising complex numbers, matrices, hyperbolic functions, or polar coordinates, having instant recall of the language will boost your confidence and accuracy in exams.
在AQA进阶数学中，掌握精确的词汇术语对于正确理解题意和清晰表达解题过程至关重要。本速记指南汇集了考纲中的核心术语，每个术语都配有简明定义、记忆提示和中文对照。无论你是在复习复数、矩阵、双曲函数还是极坐标，快速回忆起这些术语将极大提升你在考试中的信心和准确度。
1. Fundamental Symbols & Core Terms | 基础符号与核心术语
Sigma notation (Σ) Σ (from r=1 to n) of r means 1+2+...+n. Always identify the index, lower and upper limits.
求和符号 (Σ)：紧凑表示累加的形式。例如 Σ(r=1 to n) r 表示 1+2+...+n。需认准指标、下限与上限。速记：Σ 像一个 S(tandard) 的总和机。
Discriminant (Δ) Δ>0 (two distinct real roots), Δ=0 (one repeated real root), Δ<0 (complex conjugate pair).
判别式 (Δ)：对 ax²+bx+c，Δ = b²-4ac。决定根的性质：Δ>0 两不等实根，Δ=0 重实根，Δ<0 共轭复根。联想：Δ就像一位侦探(detective)，从系数中判别结果。
Asymptote: a line that a curve approaches arbitrarily closely but never touches. Horizontal, vertical, and oblique asymptotes are common; they reveal end behaviour and undefined points.
渐近线：曲线无限接近但永不相交的直线。常见有水平、垂直和斜渐近线，揭示了曲线的远端行为和未定义点。记忆：asymptote 像一条“渐行渐近”的线。
Binomial coefficient (nCr or n choose r): number of ways to choose r items from n without order, given by n!/(r!(n-r)!). Appears in binomial expansion (1+x)ⁿ and Pascal's triangle.
二项式系数 (nCr)：从 n 个中无序选取 r 个的方法数，公式 n!/(r!(n-r)!)。见于 (1+x)ⁿ 展开式和杨辉三角。速记：组合数，顺序不重要。
Rational function x), where P and Q are polynomials. Partial fractions decompose such functions into simpler sums for easier integration.
有理函数：形如 P(x)/Q(x) 的函数，P、Q 是多项式。部分分式可将它分解成简单分式之和，利于积分。联想：有理数→有理函数，“有理”即有道理的分式结构。
2. Complex Numbers | 复数
Real part Re(z) & imaginary part Im(z) rt, b (a real number) is the imaginary part, not including i. Im(z) is a real coefficient.
实部 Re(z) 与虚部 Im(z)：对 z = a + bi，a 为实部，b 为虚部（不带 i）。Im(z) 是一个实数系数。注意：Im(z) 不会输出带 i 的形式。
Modulus |z|: the distance from the origin to z on the Argand diagram, calculated as √a²+b². It is always non‑negative and represents the 'size' of the complex number.
模 |z|：Argand 图上原点到复数 z 的距离，计算 √a²+b²。始终非负，代表复数的“长度”。记忆：模如绝对值，距离自然非负。
Argument arg(z) senting z. Usually taken in (-π, π] as the principal argument. Write z = r(cosθ + i sinθ).
辐角 arg(z)：从正实轴逆时针旋转到表示 z 的向量所成角 θ，通常主值取 (-π, π]。可写为 z = r(cosθ + i sinθ)。联想：argument 在这里争论角度，arg(z) 即角度代号。
Complex conjugate z* or z̄ given z = a+bi, the conjugate is a-bi. It reflects the vector across the real axis; z × z̄ = a²+b² = |z|². Useful for division and finding real denominators.
共轭复数 z̄：z = a+bi 的共轭为 a-bi。它将复数向量关于实轴反射；z·z̄ = a²+b² = |z|²。在复数除法与有理化分母中极为实用。记忆：共轭如同镜像，虚部变号。
Euler's formula eiθ = cosθ + i sinθ: this connects exponential and trigonometric functions. Any complex number can be written in exponential form reiθ, which simplifies multiplication and powers.
欧拉公式 eiθ = cosθ + i sinθ：架起指数与三角的桥梁。复数可写成指数式 reiθ，使乘法和乘方极方便。联想：Euler 拉近了指数与三角的距离，e 和 i 的魔法。
De Moivre's theorem (r(cosθ + i sinθ))ⁿ = rⁿ(cos nθ + i sin nθ) to raise complex numbers to integer powers and to find nth roots. For roots, formula w = r1/n[cos(θ+2kπ)/n + i sin(θ+2kπ)/n].
棣莫弗定理 (cosθ + i sinθ)ⁿ = cos nθ + i sin nθ：用于复数整数次幂和求 n 次方根。开方公式：w = r1/n[cos(θ+2kπ)/n + i sin(θ+2kπ)/n]。速记：幂指数搬进角度，模幂按实部处理。
3. Matrices & Linear Transformations | 矩阵与线性变换
Order of a matrix er of rows, n is the number of columns. A square matrix has m = n; a column vector is n×1.
矩阵的阶：m×n 表示 m 行 n 列。方阵满足 m=n；列向量为 n×1 矩阵。记忆：行在前，列在后，如同 RC 遥控车先说行。
Determinant det(A) or |A| nes invertibility: if det=0 the matrix is singular and has no inverse. For 3×3, use Sarrus' rule or expansion.
行列式 det(A)：2×2 阵 [[a,b],[c,d]] 的行列式为 ad-bc。它决定可逆性：det=0 则矩阵奇异，无逆阵。3×3 可用沙路法或展开计算。联想：ad-bc 就像交叉相乘求差，可逆与否全系此值。
Inverse matrix A⁻¹ A⁻¹ = 1/(ad-bc) [[d,-b],[-c,a]]. Exists only when det≠0. Solving Ax = b becomes x = A⁻¹b.
逆矩阵 A⁻¹：满足 A⁻¹A = AA⁻¹ = I。2×2 逆阵公式需除以行列式。仅当 det≠0 存在。解 Ax=b 可化为 x = A⁻¹b。速记：互换 a,d 并变号 b,c，再除以行列式。
Linear transformation rix columns are the images of basis vectors. Common 2D transformations: rotation by θ is [[cosθ,-sinθ],[sinθ,cosθ]], reflection in x-axis is [[1,0],[0,-1]].
线性变换：用矩阵乘法 T(x)=Mx 表达的映射。矩阵的各列是基向量的像。常见二维变换：旋转 θ 角的矩阵为 [[cosθ,-sinθ],[sinθ,cosθ]]，以 x 轴反射为 [[1,0],[0,-1]]。记忆：变换矩阵直接作用于位置向量，形象如推土机。
Eigenvalue λ and eigenvector v Solved from det(A-λI)=0. Eigenvalues indicate scaling factors along invariant directions.
特征值 λ 与特征向量 v：若对非零向量 v 有 Av = λv，则 λ 为 A 的特征值，v 为对应特征向量。由 det(A-λI)=0 求解。特征值表明沿不变方向伸缩的比例。联想：特征向量保持方向，只被“特”别拉伸。
4. Vectors & 3D Geometry | 向量与三维几何
Unit vector: a vector with magnitude 1. Often indicated with a hat, e.g., â = a/|a|. The basis i, j, k are mutually orthogonal unit vectors along the x, y, z axes.
单位向量：模长为1的向量，常戴帽子如 â = a/|a|。基向量 i, j, k 分别为沿 x, y, z 轴的正交单位向量。记忆：单位即长度为 1，帽子表示标准化。
Scalar (dot) product a·b: a·b = |a||b| rs, test orthogonality (a·b=0), and compute projection. In component form: a₁b₁+a₂b₂+a₃b₃.
数量积（点乘）a·b：a·b = |a||b|cosθ，结果为标量。用于求夹角、判断垂直(a·b=0)及投影计算。分量式：a₁b₁+a₂b₂+a₃b₃。联想：点乘得数，cosθ 藏于其中。
Vector (cross) product a×b: a×b = |a||b|sinθ n̂, where n̂ and rule). Result is a vector. For 3D components, use determinant. Area of parallelogram = |a×b|.
向量积（叉乘）a×b：a×b = |a||b|sinθ n̂，n̂ 为右手定则下垂直于 a 和 b 的单位向量，结果为向量。分量式用行列式。平行四边形面积 = |a×b|。速记：叉乘得向量，方向用右手；面积由 sin 获得。
Vector equation of a line: r = a + λd, where a is a point on the line, d is a direction vector, and λ ∈ ℝ ametric equations x=a₁+λd₁, etc., and to Cartesian form (x-a₁)/d₁ = (y-a₂)/d₂ = (z-a₃)/d₃.
直线的向量方程：r = a + λd，a 为线上定点，d 为方向向量，λ 为实数参数。可化为参数方程及对称式。记忆：直线上每一点都由定点加 λ 倍方向得到。
Plane equation sian form n₁x+n₂y+n₃z = d. To find angle between planes, use dot product of their normals.
平面方程：点法式 r·n = a·n，或笛卡式 n₁x+n₂y+n₃z = d，其中 n 为法向量。求两平面夹角使用法向量的点乘。联想：平面由法向量唯一确定，法向量像一把垂直的伞柄。
5. Sequences & Series | 数列与级数
Arithmetic series on difference d. nth term uₙ = a+(n-1)d; sum to n terms Sₙ = n/2 (2a+(n-1)d) or n/2 (a+l).
等差级数：首项 a，公差 d 的 AP 求和。第 n 项 uₙ = a+(n-1)d；前 n 项和 Sₙ = n/2 (2a+(n-1)d) 或 n/2 (a+l)。速记：平均项=(首+末)/2。
Geometric series e r is the common ratio. Sum to n terms Sₙ = a(1-rⁿ)/(1-r) for r≠1. Infinite sum exists if |r|<1: S∞ = a/(1-r).
等比级数：通项 uₙ = arⁿ⁻¹。前 n 项和 Sₙ = a(1-rⁿ)/(1-r)，|r|<1 时有无限和 S∞ = a/(1-r)。记忆：无限和公式分母为 1-r，源于极限 rⁿ→0。
Maclaurin series: a Taylor expansion about x=0. f(x)=f(0)+f'(0)x+f''(0)/2! x²+... . Common expansions: eˣ, sin x, cos x, ln(1+x), (1+x)ⁿ. Used to approximate functions and evaluate limits.
麦克劳林级数：在 x=0 处的泰勒展开。f(x)=f(0)+f'(0)x+f''(0)/2! x²+... 。常见展开式如 eˣ, sin x, cos x, ln(1+x), (1+x)ⁿ。用于逼近函数和求极限。联想：Maclaurin 是 0 处的 Taylor，系数由导数决定。
Method of differences s where terms cancel in pairs. Write uᵣ = f(r)-f(r+1), then Σ uᵣ collapses to f(1)-f(n+1).
差分法：用于求和可裂项相消的级数，写成 uᵣ = f(r)-f(r+1)，求和后只留首尾差。速记：将通项拆为前后项之差，像多米诺骨牌倒下一片。
6. Advanced Calculus Techniques | 高等微积分技巧
Chain rule: d/dx f(g(x))=f'(g(x))g'(x). Essential for differentiating composites like esin x, ln(cos x). Write in Leibniz form dy/dx = dy/du * du/dx.
链式法则：复合函数求导规则，d/dx f(g(x))=f'(g(x))g'(x)。对 esin x、ln(cos x) 等必不可少。记法：dy/dx = dy/du × du/dx，如链条环环相扣。
Product & quotient rules: d/dx(uv)=u'v+uv'; d/dx(u/v)=(u'v-uv')/v². Used when two differentiable functions multiply or divide. Recognise by pattern: 'first-differentiate-second plus...'
积商法则：d/dx(uv)=u'v+uv'；d/dx(u/v)=(u'v-uv')/v²。两可导函数相乘除时应用。口诀：前导后不导 + 前不导后导，商的分子对应变号。
Implicit differentiation ides w.r.t. x, treating dy/dx as a variable. E.g., x²+y²=1 → 2x+2y dy/dx=0 → dy/dx = -x/y.
隐函数求导：当 y 未显式表达为 x 的函数时，对方程两边关于 x 求导，将 dy/dx 视为变量。如 x²+y²=1 求导得 2x+2y dy/dx=0。记忆：逐项求导，y 自带 dy/dx 尾巴。
Integration by substitution: ∫f(g(x))g' y integrand, then replace dx with du/(du/dx). Always adjust limits for definite integrals.
换元积分法：∫f(g(x))g'(x)dx = ∫f(u) du，令 u = g(x)。关键在于用 du = g'(x)dx 替换，定积分同时更换上下限。速记：将被积表达式与微分一起转换，像打包托运。
Integration by parts e. Use LIATE rule to choose u (Log, Inverse trig, Algebraic, Trig, Exponential). Repeated by parts for x²eˣ etc.
分部积分法：∫u dv = uv - ∫v du，源于乘积求导法则。按 LIATE 顺序选 u：对、反、幂、三、指。常用于 ∫x²eˣ dx 等。联想：乘法的积分还是乘法，多出一项交换舞伴。
7. Hyperbolic Functions | 双曲函数
Definitions in terms of eˣ: sinh x = (eˣ - e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x. They model hanging cables and appear in integration.
用指数定义双曲函数：sinh x = (eˣ - e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x。它们在悬链线和积分中出现。记忆：双曲函数带 h，指数平均分。
Hyperbolic identity cosh²x - sinh²x = 1 her identities: 1 - tanh²x = sech²x, sinh 2x = 2 sinh x cosh x, cosh 2x = cosh²x + sinh²x.
双曲恒等式 cosh²x - sinh²x = 1：与 cos²θ+sin²θ=1 类似但变号。另有 1 - tanh²x = sech²x，倍角公式 sinh 2x = 2 sinh x cosh x。速记：基本恒等式记住符号差，Osborn定则可转换三角恒等式。
Osborn's rule gn of any product (or implied product) of two sines. Example: cos2θ=1-2sin²θ → cosh 2x = 1+2sinh²x.
Osborn 法则：将三角恒等式转为双曲恒等式时，替换函数并在每出现两个 sin 的乘积时变号。如 cos2θ=1-2sin²θ 变成 cosh 2x = 1+2sinh²x。联想：两个 sinh 相乘要反号，就像两负数相乘得正号偏移。
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. Their derivatives are algebraic, e.g., d/dx arsinh x = 1/√x²+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。其导数简洁：d/dx arsinh x = 1/√x²+1。速记：反双曲用 ln 表示，求导
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