📚 AS AQA Further Maths: Quick Terminology Memorization Guide | AS AQA 进阶数学：词汇术语速记指南
Mastering AS AQA Further Maths requires not only problem-solving skills but also a firm grasp of the subject’s specialist vocabulary. This guide breaks down the essential terms from pure and applied modules into bite-sized English-Chinese explanations, complete with memory hooks to speed up revision. Use it to build confidence with technical language and to avoid losing marks through misinterpretation in exams.
掌握 AS AQA 进阶数学不仅需要解题能力，还要求牢固掌握学科的专业词汇。本指南将纯数及应用模块中的核心术语拆解为英中对照的精简解释，并配有记忆窍门，帮助快速复习。通过熟练使用专业语言，避免因考试中误解术语而丢分。
1. Complex Numbers Terminology | 复数术语
An imaginary unit i satisfies i² = –1. A complex number z = a + bi has a real part Re(z) = a and an imaginary part Im(z) = b. The complex conjugate of z is z* = a – bi. The modulus |z| = √(a² + b²) gives the distance from the origin, and the argument arg(z) = θ is the angle made with the positive real axis. Multiplying by i corresponds to a 90° rotation anticlockwise.
虚数单位 i 满足 i² = –1。复数 z = a + bi 有实部 Re(z) = a 和虚部 Im(z) = b。z 的共轭复数为 z* = a – bi。模 |z| = √(a² + b²) 表示到原点的距离，辐角 arg(z) = θ 是与正实轴的夹角。乘以 i 相当于逆时针旋转 90°。
Memory aid: Think of ‘conjugate’ as ‘twin with a flipped sign of i’, and ‘modulus’ as the magnitude of the vector from 0 to z.
记忆口诀：共轭就是‘虚部变号的双胞胎’，模是‘复数向量的长度’。
2. Matrices and Transformations | 矩阵与变换
A matrix is a rectangular array of elements. The order is rows × columns. The identity matrix I has 1s on the leading diagonal and 0s elsewhere; it leaves a vector unchanged under multiplication. A matrix is singular if its determinant is zero, meaning it has no inverse. The transpose of a matrix A, written Aᵀ, swaps rows and columns. In transformation geometry, a 2×2 matrix can represent rotations, reflections, enlargements and shears. Invariant points satisfy Mx = x, while invariant lines are lines which map onto themselves under the transformation.
矩阵是一个矩形元素阵列。阶数为行数×列数。单位矩阵 I 的主对角线上全为 1，其余为 0；与向量相乘时保持向量不变。若矩阵的行列式为零，则该矩阵是奇异的（不可逆）。矩阵 A 的转置 Aᵀ 行列互换。在变换几何中，2×2 矩阵可表示旋转、反射、放大和剪切。不变点满足 Mx = x，不变线是变换下映射为自身的直线。
To remember ‘singular’, think: ‘Single, no partner → no inverse’. For ‘transpose’, picture rows dropping down to become columns.
助记：‘奇异’联想到‘单身，无逆’；‘转置’想象行翻转为列。
3. Roots of Polynomials | 多项式根
For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum of roots is α + β = –b/a, and the product is αβ = c/a. The discriminant Δ = b² – 4ac determines the nature of the roots: Δ > 0 gives two distinct real roots; Δ = 0 gives a repeated real root; Δ < 0 gives a pair of complex conjugate roots.
对于二次方程 ax² + bx + c = 0，若根为 α 和 β，则根之和 α + β = –b/a，根之积 αβ = c/a。判别式 Δ = b² – 4ac 决定根的性质：Δ > 0 有两个不等实根；Δ = 0 有一个重根；Δ < 0 有一对共轭复根。
For higher-degree polynomials, symmetrical sums of roots can be expressed in terms of coefficients. The relationship is: sum of roots = –aₙ₋₁/aₙ, sum of product of roots taken two at a time = aₙ₋₂/aₙ, etc., alternating signs.
高次多项式中，根的对称和可用系数表示：所有根之和 = –aₙ₋₁/aₙ，所有可能的两个根乘积之和 = aₙ₋₂/aₙ，依次变号。
Mnemonics: ‘Sum –B over A, Product C over A’ (for quadratics). For signs, recall the pattern + – + – … starting from the highest degree.
记忆口诀：二次式‘和负B除以A，积C除以A’；高次式符号从最高次开始正负交替。
4. Vectors in 3D | 三维向量
A 3D vector a = xi + yj + zk has magnitude |a| = √(x² + y² + z²). The direction cosines are cos α = x/|a|, cos β = y/|a|, cos γ = z/|a|. The scalar (dot) product a·b = |a||b|cos θ = x₁x₂ + y₁y₂ + z₁z₂. Two vectors are perpendicular if a·b = 0. The vector (cross) product a×b produces a vector perpendicular to both a and b, with magnitude |a||b|sin θ. The triple scalar product a·(b×c) gives the volume of the parallelepiped; a value of zero indicates coplanarity.
三维向量 a = xi + yj + zk，模长 |a| = √(x² + y² + z²)。方向余弦 cos α = x/|a|，cos β = y/|a|，cos γ = z/|a|。数量（点）积 a·b = |a||b|cos θ = x₁x₂ + y₁y₂ + z₁z₂。两向量垂直 ⇔ a·b = 0。向量（叉）积 a×b 得到同时垂直于 a 和 b 的向量，大小为 |a||b|sin θ。三重纯量积 a·(b×c) 等于平行六面体体积；为零则三向量共面。
Memory: Dot product → ‘dot the i’s’ → component-wise sum. Cross product → ‘cross your arms’ → perpendicular result. Triple scalar → ‘box volume’.
记忆：点积→同向分量相乘求和；叉积→双臂交叉，结果为垂直向量；三重积→‘盒子体积’。
5. Calculus Techniques | 微积分技巧
The chain rule: if y = f(u) and u = g(x), then dy/dx = (dy/du)(du/dx). The product rule: d(uv)/dx = u dv/dx + v du/dx. The quotient rule: d(u/v)/dx = (v du/dx – u dv/dx)/v². When integrating, integration by substitution reverses the chain rule, and integration by parts uses ∫ u dv = uv – ∫ v du. The Fundamental Theorem of Calculus links differentiation and integration: ∫ₐᵇ f ‘(x) dx = f(b) – f(a).
链式法则：若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du)(du/dx)。乘积法则：d(uv)/dx = u dv/dx + v du/dx。商法则：d(u/v)/dx = (v du/dx – u dv/dx)/v²。积分时，换元积分法是链式法则的逆运算，分部积分法使用 ∫ u dv = uv – ∫ v du。微积分基本定理联系微分与积分：∫ₐᵇ f ‘(x) dx = f(b) – f(a)。
To recall integration by parts, use the ‘LIATE’ rule for picking u (Log, Inverse trig, Algebraic, Trig, Exponential). The quotient rule can be remembered as ‘low d-high minus high d-low, over low-squared’.
分部积分用‘LIATE’顺序选 u（对数、反三角、代数、三角、指数）。商法则口诀‘下导上减上导下，除以下平方’。
6. Polar Coordinates | 极坐标
A point P is given by (r, θ) where r is the distance from the pole O and θ is the angle measured anticlockwise from the initial line. Conversion formulas: x = r cos θ, y = r sin θ, and r² = x² + y², tan θ = y/x. The area enclosed by a polar curve r = f(θ) between θ = α and θ = β is A = ½ ∫ₐᵝ r² dθ. The arc length of a polar curve is s = ∫ √(r² + (dr/dθ)²) dθ.
点 P 用 (r, θ) 表示，r 为到极点 O 的距离，θ 为从极轴逆时针测量的角度。转换公式：x = r cos θ, y = r sin θ, 且 r² = x² + y², tan θ = y/x。极坐标曲线 r = f(θ) 在 θ = α 到 β 之间围成的面积为 A = ½ ∫ₐᵝ r² dθ。曲线的弧长为 s = ∫ √(r² + (dr/dθ)²) dθ。
Think of the area formula as ‘half the integral of radius squared’ – like summing tiny circular sectors. For conversion, recall ‘x is r cos, like a cosine projection’.
面积公式像‘半径平方积分的一半’——求和无数小扇形；转换中，x 是 r cos，类似余弦投影。
7. Differential Equations | 微分方程
A first-order differential equation involves dy/dx. It is called separable if it can be written as g(y) dy = f(x) dx. An integrating factor is used for linear equations of the form dy/dx + P(x)y = Q(x). The integrating factor is e^{∫ P dx}. A general solution contains an arbitrary constant; a particular solution satisfies an initial condition. Second-order homogeneous linear equations with constant coefficients ay” + by’ + cy = 0 are solved using the auxiliary equation am² + bm + c = 0.
一阶微分方程含有 dy/dx。若能写成 g(y) dy = f(x) dx，则称可分离变量。对于形如 dy/dx + P(x)y = Q(x) 的线性方程，可用积分因子 e^{∫ P dx} 求解。通解含任意常数；特解满足初始条件。常系数二阶齐次线性方程 ay” + by’ + cy = 0 用辅助方程 am² + bm + c = 0 求解。
To set up the integrating factor, remember ‘integrate P, exponentiate it’. For auxiliary equations, the roots m give the solution form: real distinct m₁, m₂ → Ae^{m₁x} + Be^{m₂x}; repeated m → (A + Bx)e^{mx}; complex α ± iβ → e^{αx} (A cos βx + B sin βx).
积分因子：‘先积 P，再取指数’。辅助方程的根决定解的形式：不等实根→指数组合；重根→乘 x；复根→指数乘三角。
8. Hyperbolic Functions | 双曲函数
The hyperbolic sine and cosine are defined as sinh x = (eˣ – e⁻ˣ)/2 and cosh x = (eˣ + e⁻ˣ)/2. The hyperbolic tangent is tanh x = sinh x / cosh x. Key identities mirror trigonometric ones but with sign differences: cosh²x – sinh²x = 1, sinh(2x) = 2 sinh x cosh x, cosh(2x) = cosh²x + sinh²x. The inverse hyperbolic functions arsinh, arcosh, artanh can be expressed in logs, e.g., arsinh x = ln(x + √(x² + 1)).
双曲正弦和双曲余弦定义为 sinh x = (eˣ – e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2。双曲正切 tanh x = sinh x / cosh x。关键恒等式类似三角但符号有别：cosh²x – sinh²x = 1，sinh(2x) = 2 sinh x cosh x，cosh(2x) = cosh²x + sinh²x。反双曲函数 arsinh、arcosh、artanh 可用对数表示，如 arsinh x = ln(x + √(x² + 1))。
Memory: ‘cosh² – sinh² = 1’ is like ‘cos² + sin² = 1’ but with a minus. Think of ‘h’ for ‘half’ in definitions: sinh is half the difference of exponentials.
助记：‘cosh² – sinh² = 1’ 类比三角但用减号。定义中 sinh 是 eˣ 与 e⁻ˣ 差的一半。
9. Series and Sequences | 级数与序列
The Maclaurin series expands a function about x = 0: f(x) = f(0) + f ‘(0)x + f ”(0)x²/2! + f ”'(0)x³/3! + … . The method of differences simplifies sums by expressing terms as differences, often using partial fractions. The sigma notation Σ indicates summation. For arithmetic sequences: uₙ = a + (n–1)d, Sₙ = n/2 [2a + (n–1)d]. For geometric sequences: uₙ = arⁿ⁻¹, sum to infinity S∞ = a/(1 – r) for |r| < 1.
麦克劳林级数在 x = 0 处展开函数： f(x) = f(0) + f ‘(0)x + f ”(0)x²/2! + f ”'(0)x³/3! + …。差分法通过将项表示为差分（常用部分分式）来化简求和。Σ 符号表示求和。等差数列：uₙ = a + (n–1)d，Sₙ = n/2 [2a + (n–1)d]。等比数列：uₙ = arⁿ⁻¹，当 |r| < 1 时无穷和 S∞ = a/(1 – r)。
Formula for sum of squares Σn² = n(n+1)(2n+1)/6; sum of cubes Σn³ = [n(n+1)/2]². The differences method is useful when terms cancel like a telescopic series.
平方和公式 Σn² = n(n+1)(2n+1)/6；立方和 Σn³ = [n(n+1)/2]²。当项像望远镜般相消时，差分法特别有用。
10. Proof by Induction | 数学归纳法
Mathematical induction is a proof technique for statements about natural numbers n. It has four steps: Basis: verify the statement for n = 1. Assumption: assume true for n = k. Inductive step: prove true for n = k+1 using the assumption. Conclusion: state that by induction the statement holds for all n ∈ ℕ. Common applications include proving summation formulae, divisibility, matrix powers, and inequalities.
数学归纳法是一种关于自然数 n 的命题证明技术。分四步：奠基：验证 n = 1 时命题成立。假设：假设 n = k 时成立。归纳递推：利用假设证明 n = k+1 时成立。结论：由归纳法断言命题对所有 n ∈ ℕ 成立。常见应用：求和公式、整除性、矩阵幂、不等式证明。
A typical structure: ‘Let P(n) be …’. Show P(1) true. Assume P(k) true. Then show P(k+1) true. Finally, ‘Hence by mathematical induction, P(n) is true for all positive integers n.’
标准结构：‘设 P(n) 为……’，证明 P(1) 真，假设 P(k) 真，推证 P(k+1) 真，最后‘由数学归纳法，对所有正整数 n，P(n) 均成立’。
11. Mechanics: Kinematics | 运动学
Kinematics describes motion with quantities: displacement (s), velocity (v), acceleration (a). For motion in a straight line with constant acceleration, the SUVAT equations apply: v = u + at; s = ut + ½at²; s = ½(u+v)t; v² = u² + 2as; s = vt – ½at². Here u is initial velocity, v final velocity, s displacement, a acceleration, t time. Variable acceleration uses calculus: v = ds/dt, a = dv/dt = d²s/dt², and s = ∫ v dt, v = ∫ a dt.
运动学用位移 (s)、速度 (v)、加速度 (a) 描述运动。对于匀加速直线运动，有 SUVAT 公式：v = u + at；s = ut + ½at²；s = ½(u+v)t；v² = u² + 2as；s = vt – ½at²。这里 u 初速度，v 末速度，s 位移，a 加速度，t 时间。变加速度用微积分：v = ds/dt，a = dv/dt = d²s/dt²，s = ∫ v dt，v = ∫ a dt。
The SUVAT letters: S – displacement, U – initial velocity, V – final velocity, A – acceleration, T – time. ‘v² = u² + 2as’ is like an energy equation without mass.
SUVAT 字母：S 位移，U 初速，V 末速，A 加速度，T 时间。公式 v² = u² + 2as 类似无质量版本的能量方程。
12. Mechanics: Forces and Equilibrium | 力与平衡
Newton’s laws: First law: an object remains at rest or uniform motion unless acted upon by a resultant force. Second law: F = ma. Third law: action and reaction are equal and opposite. Forces can be resolved into components. An object is in equilibrium if the vector sum of all forces is zero and the sum of moments about any point is zero. Friction F ≤ μR, where μ is the coefficient of friction and R is the normal reaction. The limiting equilibrium occurs when an object is just about to move. For connected particles, equations of motion are written for each mass, often involving tension T in light inextensible strings.
牛顿定律：第一定律：物体在无合力时保持静止或匀速运动。第二定律：F = ma。第三定律：作用力与反作用力大小相等、方向相反。力可分解为分量。若所有力的矢量和为零，且对任一点力矩和为零，则物体处于平衡。摩擦力 F ≤ μR，μ 为摩擦系数，R 为法向反力。极限平衡是物体即将运动的状态。对于连接体，为各质量写出运动方程，常涉及轻质不可伸长的绳子中的张力 T。
For resolving on an inclined plane: component of weight down the slope = mg sin θ, perpendicular to slope = mg cos θ. Friction opposes motion or tendency to move.
斜面分解：重力沿斜面向下分量为 mg sin θ，垂直斜面为 mg cos θ。摩擦力阻碍运动或运动趋势。
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