📚 Year 12 AQA Further Maths: Vocabulary Terminology Quick Memorisation Guide | AQA 进阶数学：词汇术语速记指南
Mastering Year 12 AQA Further Maths depends just as much on fluently using precise terminology as on algebraic skill. This guide organises essential vocabulary by topic and attaches short memory hooks, so you can review definitions, properties and applications in one place. Read it aloud, test yourself daily, and train your brain to recognise every term instantly.
掌握 AQA Year 12 进阶数学不仅依赖代数技巧，同样需要熟练使用精确的术语。本指南按主题整理核心词汇并附上简短的记忆钩子，让你能在一处复习定义、性质和应用。每天读出声来、自我测试，训练大脑即刻识别每个术语。
1. Complex Numbers Terminology | 复数术语
The imaginary unit i is defined by i2 = – ex number z = a + bi has real part Re(z) = a and imaginary part Im(z) = b (not bi). Think: 'real part stays real – the bit without i'. The complex conjugate z* (or z̄) changes the sign of the imaginary component: a – bi. Conjugates are mirrors across the real axis.
虚数单位 i 定义为 i2 = –1。复数 z = a + bi 的实部 Re(z) = a，虚部 Im(z) = b（不是 bi）。记住：“实部就是不含 i 的那部分”。复共轭 z*（或 z̄）改变虚部的符号：a – bi。共轭就像是关于实轴的镜像。
The modulus of z, written |z|, is the distance from the origin in the Argand diagram: |z| = √a2 + b2 g(z), is the angle θ made with the positive real axis, usually given in radians and within –π < θ ≤ π. To avoid confusion, recall: 'modulus is length, argument is angle starting from the positive x-axis.' The polar form z = r(cos θ + i sin θ) and Euler's form z = reiθ both use r = |z| and θ = arg(z). De Moivre's theorem states (cos θ + i sin θ)n = cos nθ + i sin nθ, a powerful tool for powers and roots.
复数的模 |z| 是阿甘特图上到原点的距离：|z| = √a2 + b2。辐角 arg(z) 是与正实轴的夹角 θ，通常以弧度表示且满足 –π < θ ≤ π。避免混淆：“模是长度，辐角是从正 x 轴起始的角度”。极坐标形式 z = r(cos θ + i sin θ) 和欧拉形式 z = reiθ 均用 r = |z|, θ = arg(z)。棣莫弗定理指出 (cos θ + i sin θ)n = cos nθ + i sin nθ，是处理乘幂和方根的有力工具。
2. Matrices and Transformations | 矩阵与变换
ns. The identity matrix I acts like the number 1: AI = IA = A. A zero matrix contains only zeros. The transpose AT swaps rows and columns; a symmetric matrix satisfies AT = A. The determinant of a 2×2 matrix M = [[a, b], [c, d]] is det(M) = ad – bc. If det(M) = 0, the matrix is singular and has no inverse.
矩阵是矩形的元素阵列；其大小由行×列确定，称为阶。方阵具有相同的行数和列数。单位矩阵 I 就像数字 1：AI = IA = A。零矩阵仅包含零元素。转置 AT 交换行与列；对称矩阵满足 AT = A。2×2 矩阵 M = [[a, b], [c, d]] 的行列式为 det(M) = ad – bc。若 det(M) = 0，矩阵为奇异矩阵且没有逆矩阵。
The inverse A–1 satisfies AA–1 = I. For a 2×2 matrix, A–1 = (1/det A) [[d, –b], [– presented by 2×2 matrices: rotations (det = 1, preserving orientation), reflections (det = – s (scalar multiples), and shears (parallel lines remain parallel but angles change). Link: 'determinant tells you area scale factor and orientation sign.'
逆矩阵 A–1 满足 AA–1 = I。对于 2×2 矩阵，A–1 = (1/det A) [[d, –b], [–c, a]]。平面中的变换可用 2×2 矩阵表示：旋转（行列式 = 1，保持定向）、反射（行列式 = –1，反转定向）、缩放（标量倍数）和剪切（平行线保持平行但角度改变）。联系记忆：“行列式给出面积缩放因子和定向符号”。
3. Vectors and 3D Geometry | 向量与三维几何
Vectors are written in component form, e.g. 3i + 2j – k, or as column vectors. The magnitude of a = xi + yj + zk is |a| = √(x2 + y2 + z2). A unit vector has magnitude 1; the unit vector in direction a is â = a / |a| ation of a line: r = a + λb, where a is a point on the line and b is the direction vector.
向量以分量形式书写，如 3i + 2j – k，或列向量。向量 a = xi + yj + zk 的模为 |a| = √(x2 + y2 + z2)。单位向量的模为 1；沿 a 方向的单位向量是 â = a / |a|。点的位置向量是从原点到该点的位移。直线的向量方程为 r = a + λb，其中 a 是线上一点，b 是方向向量。
The scalar product (dot product) a · b = |a||b| duct (cross product) a × b produces a vector perpendicular to both a and b, with magnitude |a||b| sin θ and direction given by the right-hand rule. Remember: 'dot gives a number, cross gives a new perpendicular vector.'
标量积（点积）a · b = |a||b| cos θ，得出一个标量。两向量垂直当且仅当 a · b = 0。向量积（叉积）a × b 产生一个垂直于 a 和 b 的向量，大小为 |a||b| sin θ，方向由右手定则确定。记住：“点积得数，叉积得垂直新向量”。
4. Hyperbolic Functions | 双曲函数
Hyperbolic functions are defined via exponential functions: sinh x = (ex – e–x)/2, cosh x = (ex + e–x)/2, and tanh x = sinh x / cosh x. The key identity is cosh2 x – sinh2 x = 1, which resembles the trigonometric identity but with a minus sign. Osborn's rule helps convert trigonometric identities to hyperbolic ones by replacing sin2 with –sinh2 whenever a product of two sines appears.
双曲函数通过指数函数定义：sinh x = (ex – e–x)/2，cosh x = (ex + e–x)/2，tanh x = sinh x / cosh x。核心恒等式为 cosh2 x – sinh2 x = 1，与三角恒等式相似但含减号。奥士本法则帮助将三角恒等式转换为双曲恒等式：每当出现两个正弦的乘积时，将 sin2 替换为 –sinh2。
llow a clean pattern: d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech2 inh x, arcosh x and artanh x, can be expressed in logarithmic form, e.g. arsinh x = ln(x + √x2 + 1). Memory hook: 'hyperbolic sine and cosine swap derivatives without a sign change, unlike their circular cousins.'
导数具有简洁的模式：d/dx (sinh x) = cosh x，d/dx (cosh x) = sinh x，d/dx (tanh x) = sech2 x。反双曲函数 arsinh x、arcosh x 和 artanh x 可用对数形式表示，例如 arsinh x = ln(x + √x2 + 1)。记忆钩子：“双曲正弦和余弦的导数互相交换且无符号变化，这与圆函数不同”。
5. Polar Coordinates | 极坐标
measured from the initial line. Conversion to Cartesian: x = r cos θ, y = r sin θ, and r = √x2 + y2 cos θ), and roses. The area enclosed by a polar curve r = f(θ) between α and β is A = ½ ∫αβ r2 dθ. Think of it as a sum of tiny circular sectors.
点 P 由 (r, θ) 描述，其中 r 为到极点 O 的距离，θ 为从极轴起始的角度。转换为笛卡尔坐标：x = r cos θ, y = r sin θ，且 r = √x2 + y2。常见极坐标曲线包括圆、心形线 r = a(1 + cos θ) 和玫瑰线。由极曲线 r = f(θ) 在 α 与 β 之间围成的面积为 A = ½ ∫αβ r2 dθ。可将其视作许多微小的圆扇形之和。
Tangents to polar curves require a parametric approach. With r = r(θ), dy/dx = (r' sin θ + r cos θ) / (r' cos θ – r sin θ), where r' pendicular tangents are found by setting numerator or denominator to zero. A useful check: 'treat r and θ like a parametric pair; the formula pops out from dy/dθ over dx/dθ.'
求极曲线的切线需要参数化方法。对于 r = r(θ)，有 dy/dx = (r' sin θ + r cos θ) / (r' cos θ – r sin θ)，其中 r' = dr/dθ。平行或垂直切线通过令分子或分母为零来确定。实用的检查方法：“把 r 和 θ 当作一对参数；此公式就是从 dy/dθ 除以 dx/dθ 得出的”。
6. Sequences and Series | 数列与级数
An arithmetic sequence has a constant difference d between successive terms; its n-th term is un = a + (n–1)d. The sum of the first n terms is Sn = n/2 [2a + (n–1)d]. A geometric sequence has a constant ratio r; its n-th term is arn–1 and the sum Sn = a(1 – rn)/(1 – r) for r ≠ 1. When |r| < 1, an infinite geometric series converges to a/(1 – r).
等差数列的相邻项之差 d 为常数；第 n 项为 un = a + (n–1)d。前 n 项和为 Sn = n/2 [2a + (n–1)d]。等比数列具有常数公比 r；第 n 项为 arn–1，和为 Sn = a(1 – rn)/(1 – r)（r ≠ 1）。当 |r| < 1 时，无穷等比级数收敛于 a/(1–r)。
A Maclaurin series expands a function about x = 0: f(x) = f(0) + f'(0)x + f''(0)x2/2! + f'''(0)x3/3! + … . The series is a special case of the Taylor series. Important expansions include ex = 1 + x + x2/2! + …, sin x = x – x3/3! + …, and cos x = 1 – x2/2! + … . Memory tip: 'alternating signs for sine and cosine, all positive for exponential.' The term radius of convergence describes the interval in which the series is valid.
麦克劳林级数将函数在 x = 0 附近展开：f(x) = f(0) + f'(0)x + f''(0)x2/2! + f'''(0)x3/3! + … 。该级数是泰勒级数的特例。重要展开式包括 ex = 1 + x + x2/2! + …，sin x = x – x3/3! + …，cos x = 1 – x2/2! + …。记忆诀窍：“正弦和余弦的符号交替出现，而指数函数全为正”。收敛半径一词描述了级数有效的区间。
7. Proof and Polynomials | 证明与多项式
Proof by contradiction assumes the negation of the statement and derives an impossibility, e.g. the irrationality of √2 sume true for n = k), and an inductive step (show true for n = k+1). Think of induction as 'the domino effect: knock the first one, then the rest follow automatically.' A direct proof links hypotheses to conclusion via logical steps.
反证法假设命题的否定成立并推出矛盾，例如证明 √2 为无理数。归纳法证明包含基础步骤、归纳假设（假设 n = k 时成立）和归纳步骤（证明 n = k+1 时成立）。把归纳法想象成“多米诺效应：推倒第一块，其余自动跟进”。直接证明通过逻辑步骤将假设与结论联系起来。
the leading coefficient is the coefficient of that term. The factor theorem states that (x – if f(a) = 0. The remainder theorem gives f(a) as the remainder when f(x) is divided by (x – ys a degree-n polynomial has exactly n roots in the complex field (counting multiplicity).
多项式词汇：次数指 x 的最高次幂，首项系数是该次幂项的系数。因式定理指出 (x – a) 是 f(x) 的因式当且仅当 f(a) = 0。余式定理给出当 f(x) 除以 (x – a) 时的余数为 f(a)。对于实系数多项式，非实根以共轭对形式出现。代数基本定理指出 n 次多项式在复数域上恰有 n 个根（计重数）。
8. Calculus Techniques | 微积分技巧
= uv, then dy/dx = u dv/dx + v du/dx. The quotient rule: if y = u/v, then dy/dx = (v du/dx – u dv/dx)/v2. A common mnemonic for the quotient rule: 'low d-high minus high d-low, over low squared.' Implicit differentiation handles equations where y is not explicitly solved; differentiate both sides with respect to x, treating y as a function, and apply the chain rule to terms in y, e.g. d/dx (y2) = 2y dy/dx.
链式法则处理复合函数：dy/dx = dy/du × du/dx。乘积法则：若 y = uv，则 dy/dx = u dv/dx + v du/dx。商法则：若 y = u/v，则 dy/dx = (v du/dx – u dv/dx)/v2。商法则的常用记忆法：“下方乘上方导数减上方乘下方导数，除以下方平方”。隐函数微分处理 y 未显式解出的方程；两边同时对 x 求导，将 y 视为函数并对含 y 的项应用链式法则，例如 d/dx (y2) = 2y dy/dx。
Integration by parts reverses the product rule: ∫ u dv/dx dx = uv – r parametic equations x = f(t), y = g(t), the derivative is dy/dx = (dy/dt)/(dx/dt). Parametric integration for area uses ∫ y dx = ∫ y (dx/dt) dt. Keep the limits in t.
分部积分逆转乘积法则：∫ u dv/dx dx = uv – ∫ v du/dx dx。巧妙选取 u 是关键；首字母组合 LIATE（对数、反三角、代数、三角、指数）给出了 u 的优先顺序。对于参数方程 x = f(t), y = g(t)，导数为 dy/dx = (dy/dt)/(dx/dt)。求面积的参数积分为 ∫ y dx = ∫ y (dx/dt) dt，并将积分限保持在 t 中。
9. First-Order Differential Equations | 一阶微分方程
e by separating variables: ∫ 1/g(y) dy = ∫ f(x) dx, then add the constant of integration. The general solution contains an arbitrary constant; a particular solution is found by applying an initial condition, e.g. y(0) = 2.
一阶常微分方程 (ODE) 关联 dy/dx、x 和 y。若方程可写成 dy/dx = f(x)g(y)，则它是可分离变量的。通过分离变量求解：∫ 1/g(y) dy = ∫ f(x) dx，然后加上积分常数。通解包含任意常数；利用初始条件（例如 y(0) = 2）可确定特解。
ar first-order ODEs have the form dy/dx + P(x)y = Q(x). The integrating factor is I(x) = e∫ P(x) dx. Multiply the entire equation by I(x), which turns the left side into the exact derivative of I(x)y. Then integrate both sides to find y. Memory script: 'spot the linear pattern, compute the integrating factor, multiply through, integrate, and apply conditions.'
线性一阶常微分方程的形式为 dy/dx + P(x)y = Q(x)。积分因子为 I(x) = e∫ P(x) dx。将整个方程乘以 I(x)，可使左边变成 I(x)y 的精确导数。然后对两边积分求出 y。记忆流程：“识别线性模式，计算积分因子，乘以全式，积分并代入条件”。
10. Numerical Methods | 数值方法
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