📚 SQA Advanced Higher Mathematics: Vocabulary & Terminology Quick Memory Guide | SQA 进阶数学：词汇术语速记指南
Mastering Advanced Higher Mathematics begins with owning the precise language of the subject. This guide presents key terms from the SQA course alongside memory hooks using word roots, analogies and visual patterns, so you can recall definitions, properties and common exam traps with confidence.
学好 SQA 进阶数学，首先要掌握这门学科精准的语言。本指南精选课程核心术语，结合词根联想、类比和视觉模式等记忆术，帮你牢记定义、性质与常见失分点，在考场上从容应对。
1. Functions and Graphs | 函数与图像
A function assigns exactly one output (y) to each input (x). The vertical line test checks this: if any vertical line cuts the graph more than once, y is not a function of x.
函数为每一个输入 (x) 指派唯一一个输出 (y)。判定方法为垂直线检验：若任何一条垂直线与图像相交多于一次，则 y 不是 x 的函数。
Think of an inverse function f⁻¹(x) as ‘undoing’ f. Graphically, the inverse reflects the original in the line y = x. Memory hook: the superscript ⁻¹ behaves like a reciprocal exponent only in notation — it never means 1/f.
把反函数 f⁻¹(x) 想象成 f 的“撤销键”。几何上，反函数的图像是原函数关于直线 y = x 的对称。记忆提示：上标 ⁻¹ 此处仅是符号，不表示倒数 1/f。
Odd and even functions describe symmetry. An even function satisfies f(−x) = f(x) (mirror across y‑axis); an odd function satisfies f(−x) = −f(x) (180° rotation about origin). Connect ‘even’ with ‘symmetric like the letter E’s vertical mirror’.
奇函数与偶函数描述对称性。偶函数满足 f(−x) = f(x)（关于 y 轴对称）；奇函数满足 f(−x) = −f(x)（关于原点旋转 180°）。联想：even（偶）与字母 E 的垂直镜像是同一类对称。
2. Differentiation Techniques | 微分技巧
The chain rule differentiates a function of a function: dy/dx = dy/du × du/dx. Picture peeling an onion — you differentiate the outer layer first, then multiply by the derivative of the inner layer.
链式法则用于复合函数求导：dy/dx = dy/du × du/dx。把它想象成剥洋葱——先对外层求导，再乘以内层的导数。
Product rule: if y = uv, then dy/dx = u dv/dx + v du/dx. The mnemonic ‘u-dee-v plus v-dee-u’ is widely used. Always check if the product can be simplified before working.
乘法法则：若 y = uv，则 dy/dx = u dv/dx + v du/dx。常用口诀“u 乘 v 的导数加上 v 乘 u 的导数”。动手之前先看看乘积能否先化简。
Quotient rule: for y = u/v, dy/dx = (v du/dx − u dv/dx) / v². Remember the rhyme: ‘low dee-high minus high dee-low, over the square of what’s below’.
除法法则：y = u/v 时，dy/dx = (v du/dx − u dv/dx) / v²。记住顺口溜：“分母乘分子导数减去分子乘分母导数，除以分母的平方”。
Implicit differentiation handles equations where y is not isolated. Differentiate term‑by‑term, treating y as a function of x and multiplying by dy/dx whenever a y‑term is differentiated.
隐函数微分处理无法显式写出 y 的方程。逐项求导，把 y 看作 x 的函数，每次对含 y 的项求导都要补乘 dy/dx。
3. Integration and Its Applications | 积分及其应用
Integration reverses differentiation. The indefinite integral ∫ f(x) dx yields a family of functions differing by a constant, + C. The definite integral ∫ₐᵇ f(x) dx computes the signed area under a curve.
积分是微分的逆运算。不定积分 ∫ f(x) dx 得到一族只差一个常数的函数，+ C。定积分 ∫ₐᵇ f(x) dx 计算曲线下的带符号面积。
Visualise the Fundamental Theorem of Calculus as a two‑way bridge: the derivative of the integral gives back the original function, and the definite integral is the change in the antiderivative between a and b.
把微积分基本定理想象成双向桥：积分的导数还原原函数，而定积分等于原函数在 b 和 a 处的差值。
Integration by substitution chooses u = g(x) so that du = g'(x) dx appears. Label it ‘reverse chain rule’ and always change limits when evaluating a definite integral.
换元积分法选取 u = g(x) 使 du = g'(x) dx 出现。把它当作“倒过来的链式法则”，计算定积分时务必同步更换积分上下限。
Integration by parts comes from the product rule: ∫ u dv = uv − ∫ v du. Use the LIATE priority (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) to choose u.
分部积分法源自乘法法则：∫ u dv = uv − ∫ v du。用 LIATE 顺序（对数、反三角、代数、三角、指数）选取 u。
4. Complex Numbers | 复数
A complex number z = a + bi combines a real part a and an imaginary part b, where i² = −1. The imaginary unit i is engineered to answer ‘What is √(−1)?’
复数 z = a + bi 包含实部 a 和虚部 b，其中 i² = −1。虚数单位 i 就是为了回答“√(−1) 是什么”而创造的。
Modulus |z| = √(a² + b²) measures distance from the origin; argument arg(z) is the angle measured anticlockwise from the positive real axis. Think of a radar: modulus = range, argument = bearing.
模 |z| = √(a² + b²) 表示到原点的距离；辐角 arg(z) 是从正实轴逆时针度量的角度。像雷达一样联想：模是距离，辐角是方位。
Polar form z = r(cos θ + i sin θ) and Euler’s form z = r e^(iθ) simplify multiplication: multiply moduli, add arguments. The exponential form is a powerful shorthand for de Moivre’s theorem.
极坐标形式 z = r(cos θ + i sin θ) 和欧拉形式 z = r e^(iθ) 使乘法简化：模相乘，辐角相加。欧拉形式是棣莫弗定理的强力缩写。
De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). Use it to find multiple‑angle trig identities or to solve zⁿ = w by taking the nth root of the modulus and dividing the argument by n, then adding 2πk/n.
棣莫弗定理：(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。可用于推导多倍角三角恒等式，或解 zⁿ = w：模开 n 次方，辐角除以 n 后依次加上 2πk/n。
5. Vectors in 3D | 三维向量
A position vector points from the origin to a point. The vector equation of a line is r = a + λb, where a is a point on the line and b is the direction vector.
位置向量从原点指向某一点。直线的向量方程为 r = a + λb，其中 a 是直线上一点的位置向量，b 是方向向量。
The scalar product a·b = |a||b| cos θ is used to test perpendicularity (a·b = 0) and to find the angle between vectors. It yields a scalar; think ‘dot gives a number (dot = digit)’.
数量积 a·b = |a||b| cos θ 可检验垂直（a·b = 0）并求向量夹角。结果是一个标量；联想“点乘得到数字（点 = 小数点）”。
Vector product a × b produces a vector perpendicular to both a and b, with magnitude |a||b| sin θ. Direction is given by the right‑hand rule. This product is zero when vectors are parallel.
向量积 a × b 得到一个同时垂直于 a 和 b 的向量，大小为 |a||b| sin θ。方向由右手定则决定。两向量平行时，向量积为零向量。
The equation of a plane can be given in vector form r·n = a·n, where n is the normal vector, or in Cartesian form ax + by + cz = d. The normal vector is perpendicular to every line in the plane.
平面的方程可用向量形式 r·n = a·n（n 为法向量），或用笛卡儿形式 ax + by + cz = d。法向量垂直于平面内所有直线。
6. Matrices and Transformations | 矩阵与变换
A matrix is a rectangular array of numbers. The order is rows × columns. Two matrices multiply only when the inside dimensions match: (m × n) × (n × p) = (m × p).
矩阵是数字的矩形阵列。矩阵的阶记为 行数 × 列数。两矩阵相乘要求内部维度相等：(m × n) × (n × p) = (m × p)。
The identity matrix I behaves like the number 1: AI = IA = A. In 2×2, I = [1 0; 0 1]. A matrix has an inverse A⁻¹ if AA⁻¹ = A⁻¹A = I. The determinant det(A) must be non‑zero for an inverse to exist.
单位矩阵 I 的功能如同数字 1：AI = IA = A。2×2 单位阵为 [1 0; 0 1]。若 AA⁻¹ = A⁻¹A = I，则 A 有逆矩阵。逆矩阵存在的前提是行列式 det(A) ≠ 0。
A 2×2 matrix represents a linear transformation of the plane. The columns are the images of (1,0) and (0,1). The determinant gives the area scale factor; a negative determinant signals reflection.
2×2 矩阵表示平面的线性变换。矩阵的列就是基向量 (1,0) 和 (0,1) 的像。行列式给出面积缩放因子；行列式为负表明含有反射。
An eigenvalue λ and eigenvector v satisfy Av = λv. The eigenvector’s direction is preserved during the transformation. To find eigenvalues, solve det(A − λI) = 0, the characteristic equation.
特征值 λ 和特征向量 v 满足 Av = λv。特征向量的方向在变换中保持不变。求特征值需解特征方程 det(A − λI) = 0。
7. Sequences and Series | 序列与级数
A sequence is an ordered list; a series is the sum of terms. The limit of a sequence is the value terms approach as n → ∞. If a limit exists, the sequence converges; otherwise it diverges.
序列是有序列表；级数是各项之和。序列的极限是当 n → ∞ 时项趋近的值。极限存在则序列收敛，否则发散。
Arithmetic sequences have a common difference d: uₙ = a + (n−1)d. Sum: Sₙ = n/2 (2a + (n−1)d). Geometric sequences have a common ratio r: uₙ = arⁿ⁻¹. Sum: Sₙ = a(1−rⁿ)/(1−r) for r ≠ 1.
等差数列有公差 d：uₙ = a + (n−1)d。求和公式 Sₙ = n/2 (2a + (n−1)d)。等比数列有公比 r：uₙ = arⁿ⁻¹。求和 Sₙ = a(1−rⁿ)/(1−r)（r ≠ 1）。
An infinite geometric series converges if |r| < 1, with sum to infinity S∞ = a/(1−r). The word 'infinite' is safe only when the common ratio shrinks the terms.
无穷等比级数当 |r| < 1 时收敛，无穷和为 S∞ = a/(1−r)。只有当公比让项越来越小时，才能安全地谈论“无穷”。
The Maclaurin series approximates a function near 0: f(x) = f(0) + f'(0)x + f”(0)x²/2! + … . Memory cue: it builds the function from its derivatives at zero, like a polynomial sculpture.
麦克劳林级数在 x=0 附近逼近函数：f(x) = f(0) + f'(0)x + f”(0)x²/2! + … 。记忆线索：它用函数在零点的各阶导数塑造一个多项式雕像。
8. Proof by Induction | 归纳法证明
Mathematical induction proves statements for all positive integers. It works like dominoes: prove the first falls (base case), then show if one falls the next falls (inductive step).
数学归纳法证明命题对所有正整数成立。原理如同多米诺骨牌：证明第一块倒下（奠基步），再证明若前一块倒下则后一块必定倒下（递推步）。
Typical structure: Step 1 – verify P(1) is true. Step 2 – assume P(k) true, and use it to prove P(k+1) true. Step 3 – conclude by induction that P(n) holds for all n ∈ ℕ.
典型结构：第一步——验证 P(1) 真。第二步——假设 P(k) 真，并以此证明 P(k+1) 真。第三步——由归纳原理断定 P(n) 对所有正整数 n 成立。
Common pitfalls: forgetting the base case, or manipulating the assumption without linking to P(k+1). Always state the induction hypothesis clearly and label where you use it.
常见失分点：遗漏奠基步，或机械变换假设而未真正联系到 P(k+1)。务必清晰写出归纳假设，并标明在何处使用。
9. Differential Equations | 微分方程
A differential equation involves an unknown function and its derivatives. The order is the highest derivative present. A general solution contains arbitrary constants; a particular solution fits given initial conditions.
微分方程包含未知函数及其导数。阶数由方程中出现的最高阶导数决定。通解含有任意常数；特解由初始条件确定常数。
Separation of variables works for first‑order equations dy/dx = g(x)h(y). Rearrange so all y‑terms are with dy and all x‑terms with dx, then integrate both sides. Memory: ‘split the tea’ — x side, y side.
分离变量法适用于一阶方程 dy/dx = g(x)h(y)。重排使所有含 y 的项与 dy 同行，所有含 x 的项与 dx 同行，然后两边积分。联想：“分开茶”——x 一边，y 一边。
A first‑order linear equation has the form dy/dx + P(x)y = Q(x). Use an integrating factor I.F. = e^(∫ P dx). Multiply through by I.F. so the left side becomes d/dx (y × I.F.).
一阶线性方程形式为 dy/dx + P(x)y = Q(x)。使用积分因子 I.F. = e^(∫ P dx)。方程两边同乘积分因子后，左侧变成 d/dx (y × I.F.)。
The second‑order homogeneous equation a d²y/dx² + b dy/dx + c y = 0 yields an auxiliary equation am² + bm + c = 0. Real distinct roots give exponentials; repeated roots give (A + Bx)e^(mx); complex conjugate roots give e^(αx)(A cos βx + B sin βx).
二阶齐次方程 a d²y/dx² + b dy/dx + c y = 0 产生辅助方程 am² + bm + c = 0。相异实根给出指数函数；重根给出 (A + Bx)e^(mx)；共轭复根给出 e^(αx)(A cos βx + B sin βx)。
10. Partial Fractions | 部分分式
Partial fractions decompose a rational expression into simpler fractions, making integration or series expansion possible. The denominator must be factored into linear and irreducible quadratic factors.
部分分式将有型分式拆分为更简单的分式，便于积分或级数展开。使用前必须先将分母因式分解为一次和不可约二次因式的乘积。
For distinct linear factors (ax + b)(cx + d), write A/(ax + b) + B/(cx + d). For a repeated factor (ax + b)ⁿ, include terms A₁/(ax + b) + A₂/(ax + b)² + … + Aₙ/(ax + b)ⁿ. For an irreducible quadratic, use (Ax + B)/(quadratic).
对于不同一次因式 (ax + b)(cx + d)，设为 A/(ax + b) + B/(cx + d)。重因式 (ax + b)ⁿ 则需包含 A₁/(ax + b) + A₂/(ax + b)² + … + Aₙ/(ax + b)ⁿ。不可约二次因式使用 (Ax + B)/(二次式)。
Always check the degree of the numerator. If it is greater than or equal to the denominator, perform polynomial long division first. Then apply partial fractions to the remainder term.
务必检查分子与分母的次数。若分子次数 ≥ 分母次数，先作多项式长除法，再对余项施用部分分式。
11. Probability & Statistics | 概率与统计
Conditional probability P(A|B) = P(A ∩ B) / P(B) updates the likelihood of A after knowing B has occurred. Think of shrinking the sample space to B.
条件概率 P(A|B) = P(A ∩ B) / P(B) 在已知 B 发生后更新 A 的可能性。想象样本空间缩小到了 B。
Independence means A and B do not affect each other: P(A ∩ B) = P(A)P(B). Equivalently, P(A|B) = P(A). Test independence; do not assume it.
独立事件意味着 A 与 B 互不影响：P(A ∩ B) = P(A)P(B)。等价地，P(A|B) = P(A)。独立性需要检验，切勿默认。
The binomial distribution B(n, p) models the number of successes in n independent trials, each with probability p. The probability of exactly r successes is C(n, r) p^r (1−p)^(n−r).
二项分布 B(n, p) 描述 n 次独立试验中成功的次数，每次成功概率为 p。恰有 r 次成功的概率为 C(n, r) p^r (1−p)^(n−r)。
Expected value E(X) is the long‑run average; for binomial, E(X) = np. Variance Var(X) measures spread; for binomial, Var(X) = np(1−p).
期望值 E(X) 是长期平均数；二项分布的期望 E(X) = np。方差 Var(X) 衡量离散程度；二项分布 Var(X) = np(1−p)。
The Poisson distribution Po(λ) counts events in a fixed interval with average rate λ. P(X = r) = e^(−λ) λ^r / r!. It is a limit of the binomial when n is large and p is small.
泊松分布 Po(λ) 计算固定区间内事件的发生次数，平均发生率为 λ。P(X = r) = e^(−λ) λ^r / r!。它是二项分布在 n 大、p 小时的极限。
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