Year 13 SQA Advanced Higher Mathematics: Quick Reference Formula & Theorem Handbook | SQA 进阶数学公式定理速查手册

📚 Year 13 SQA Advanced Higher Mathematics: Quick Reference Formula & Theorem Handbook | SQA 进阶数学公式定理速查手册

This handbook provides a concise yet comprehensive collection of essential formulae, theorems, and definitions required for the SQA Advanced Higher Mathematics course. Designed for Year 13 students, it covers all core topics: calculus, algebra, matrices, vectors, complex numbers, sequences and series, and proof techniques. Use this as your go‑to revision card to reinforce understanding and speed up problem‑solving in the exam.

本手册为 SQA 进阶数学课程整理了必备公式、定理与定义的速查集合。面向 Year 13 学生，内容涵盖所有核心主题：微积分、代数、矩阵、向量、复数、数列与级数以及证明方法。把它当作你的随身复习卡片，巩固理解，提高考试解题速度。

1. Essential Derivatives | 基本导数公式

The derivative of a function f(x) at a point gives the gradient of the tangent. The standard derivatives below must be memorised for the exam. Assume u and v are functions of x, and a, n are constants.

函数 f(x) 在某点的导数表示切线的斜率。以下标准导数公式必须在考试中熟记。假设 u 和 v 是 x 的函数，a、n 为常数。

d/dx (xⁿ) = n xⁿ⁻¹
d/dx (eˣ) = eˣ
d/dx (aˣ) = aˣ ln a
d/dx (ln x) = 1/x (x > 0)
d/dx (sin x) = cos x
d/dx (cos x) = −sin x
d/dx (tan x) = sec² x
d/dx (cot x) = −csc² x
d/dx (sec x) = sec x tan x
d/dx (csc x) = −csc x cot x
Product rule: d/dx (u v) = u’ v + u v’
Quotient rule: d/dx (u/v) = (u’ v − u v’) / v²
Chain rule: d/dx f(g(x)) = f'(g(x)) g'(x)

d/dx (xⁿ) = n xⁿ⁻¹
d/dx (eˣ) = eˣ
d/dx (aˣ) = aˣ ln a
d/dx (ln x) = 1/x (x > 0)
d/dx (sin x) = cos x
d/dx (cos x) = −sin x
d/dx (tan x) = sec² x
d/dx (cot x) = −csc² x
d/dx (sec x) = sec x tan x
d/dx (csc x) = −csc x cot x
乘积法则：d/dx (u v) = u’ v + u v’
商法则：d/dx (u/v) = (u’ v − u v’) / v²
链式法则：d/dx f(g(x)) = f'(g(x)) g'(x)

2. Key Integrals | 基本积分公式

Integration is the reverse process of differentiation. The constant of integration, C, must be included for indefinite integrals. Recognise standard forms to apply reverse chain rule or substitution efficiently.

积分是微分的逆运算。不定积分必须加上积分常数 C。识别标准形式能高效应用逆链式法则或换元法。

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
∫ 1/x dx = ln |x| + C
∫ eˣ dx = eˣ + C
∫ aˣ dx = aˣ / ln a + C
∫ sin x dx = −cos x + C
∫ cos x dx = sin x + C
∫ sec² x dx = tan x + C
∫ csc² x dx = −cot x + C
∫ sec x tan x dx = sec x + C
∫ csc x cot x dx = −csc x + C
∫ tan x dx = ln |sec x| + C
∫ cot x dx = ln |sin x| + C
∫ 1/(a² + x²) dx = (1/a) arctan(x/a) + C
∫ 1/√(a² − x²) dx = arcsin(x/a) + C

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
∫ 1/x dx = ln |x| + C
∫ eˣ dx = eˣ + C
∫ aˣ dx = aˣ / ln a + C
∫ sin x dx = −cos x + C
∫ cos x dx = sin x + C
∫ sec² x dx = tan x + C
∫ csc² x dx = −cot x + C
∫ sec x tan x dx = sec x + C
∫ csc x cot x dx = −csc x + C
∫ tan x dx = ln |sec x| + C
∫ cot x dx = ln |sin x| + C
∫ 1/(a² + x²) dx = (1/a) arctan(x/a) + C
∫ 1/√(a² − x²) dx = arcsin(x/a) + C

3. Techniques of Integration | 积分技巧

Advanced Higher expects fluency in substitution, integration by parts, and partial fractions. Choosing the correct method is critical for success.

进阶数学要求熟练运用换元法、分部积分法和部分分式。选择正确的方法是解题关键。

Integration by substitution: ∫ f(g(x)) g'(x) dx = ∫ f(u) du, where u = g(x). Always change limits for definite integrals.

换元积分法：∫ f(g(x)) g'(x) dx = ∫ f(u) du，其中 u = g(x)。定积分必须同步更换积分限。

Integration by parts: ∫ u dv = u v − ∫ v du. Choose u using the LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential).

分部积分法：∫ u dv = u v − ∫ v du。选择 u 时可使用 LIATE 优先顺序（对数、反三角、代数、三角、指数）。

Partial fractions: for rational functions, decompose into simpler fractions, e.g. 1/[(x−a)(x−b)] = A/(x−a) + B/(x−b). Use to integrate fractions with quadratic or linear factors.

部分分式：将有理函数分解为简单分式，例如 1/[(x−a)(x−b)] = A/(x−a) + B/(x−b)。用于含一次或二次因式的分式积分。

4. Definite Integrals and Applications | 定积分及其应用

Definite integrals are used to calculate areas, volumes, and other accumulated quantities. The Fundamental Theorem of Calculus links differentiation and integration.

定积分用于计算面积、体积及其他累积量。微积分基本定理将微分与积分联系起来。

Fundamental Theorem: If F'(x) = f(x), then ∫ₐᵇ f(x) dx = F(b) − F(a).

微积分基本定理：若 F'(x) = f(x)，则 ∫ₐᵇ f(x) dx = F(b) − F(a)。

Area under a curve between a and b: A = ∫ₐᵇ f(x) dx, taking absolute value if the curve goes below the x‑axis.

曲线下方面积 (a 到 b)：A = ∫ₐᵇ f(x) dx，若曲线在 x 轴下方则取绝对值。

Area between two curves: A = ∫ₐᵇ (f(x) − g(x)) dx, where f(x) ≥ g(x) on [a,b].

两曲线间面积：A = ∫ₐᵇ (f(x) − g(x)) dx，其中在 [a,b] 上 f(x) ≥ g(x)。

Volume of revolution about x‑axis: V = π ∫ₐᵇ [f(x)]² dx. About y‑axis: V = π ∫ₐᵇ [g(y)]² dy.

旋转体积 绕 x 轴：V = π ∫ₐᵇ [f(x)]² dx。绕 y 轴：V = π ∫ₐᵇ [g(y)]² dy。

5. Differential Equations | 微分方程

First‑order separable differential equations can be solved by separating variables and integrating. Second‑order linear homogeneous ODEs with constant coefficients require the characteristic equation.

一阶可分离变量微分方程可通过分离变量后积分求解。常系数二阶线性齐次常微分方程需使用特征方程。

Separable: dy/dx = g(x) h(y) → ∫ 1/h(y) dy = ∫ g(x) dx + C.

可分离变量型：dy/dx = g(x) h(y) → ∫ 1/h(y) dy = ∫ g(x) dx + C。

Second‑order linear homogeneous: a y” + b y’ + c y = 0. Form the characteristic equation a r² + b r + c = 0.

二阶线性齐次：a y” + b y’ + c y = 0。构成特征方程 a r² + b r + c = 0。

If two distinct real roots r₁, r₂: y = A e^(r₁ x) + B e^(r₂ x)
If repeated real root r: y = (A + B x) e^(r x)
If complex conjugate roots α ± iβ: y = e^(α x) (A cos βx + B sin βx)

两个相异实根 r₁, r₂：y = A e^(r₁ x) + B e^(r₂ x)
二重实根 r：y = (A + B x) e^(r x)
共轭复根 α ± iβ：y = e^(α x) (A cos βx + B sin βx)

6. Matrix Algebra | 矩阵代数

Matrices represent linear transformations. You must be confident with multiplication, determinants, inverses, and eigenvalues/eigenvectors. The order of multiplication matters: AB is generally not equal to BA.

矩阵表示线性变换。必须熟练掌握乘法、行列式、逆矩阵以及特征值与特征向量。乘法顺序重要：通常 AB ≠ BA。

Determinant of 2×2 matrix A = [[a, b], [c, d]] is det(A) = ad − bc. For 3×3, use Sarrus’ rule or cofactor expansion.

2×2 矩阵 A = [[a, b], [c, d]] 的行列式为 det(A) = ad − bc。3×3 矩阵可用萨鲁斯法则或余子式展开。

Inverse of 2×2: A⁻¹ = 1/det(A) [[d, −b], [−c, a]], provided det(A) ≠ 0.

2×2 逆矩阵：A⁻¹ = 1/det(A) [[d, −b], [−c, a]]，前提 det(A) ≠ 0。

Eigenvalues and eigenvectors: A v = λ v. Solve det(A − λ I) = 0 to find eigenvalues λ, then find non‑zero vectors v for each λ.

特征值与特征向量：A v = λ v。解 det(A − λ I) = 0 求特征值 λ，再对每个 λ 求非零向量 v。

Diagonalisation: If a matrix has n linearly independent eigenvectors, then A = P D P⁻¹, where D is diagonal matrix of eigenvalues and P contains corresponding eigenvectors as columns.

对角化：若矩阵有 n 个线性无关的特征向量，则 A = P D P⁻¹，其中 D 为特征值构成的对角矩阵，P 包含对应特征向量作为列。

7. Vectors in 3D | 三维向量

Vectors describe both magnitude and direction. Advanced Higher covers dot product, cross product, lines, and planes in three dimensions.

向量描述大小和方向。进阶数学涉及点积、叉积、三维空间中的直线与平面。

Dot product: a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃. Used to find angles and check orthogonality (a·b = 0).

点积：a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。用于求夹角和判断正交（a·b = 0）。

Cross product: a × b gives a vector perpendicular to both a and b, magnitude |a × b| = |a||b| sin θ. In component form, use the determinant method.

叉积：a × b 得到同时垂直于 a 和 b 的向量，模长 |a × b| = |a||b| sin θ。分量形式使用行列式计算。

Vector equation of a line: r = a + t d, where a is a point on the line and d is the direction vector.

直线向量方程：r = a + t d，其中 a 为线上一点，d 为方向向量。

Equation of a plane: r·n = p, where n is a normal vector, or r = a + s u + t v (parametric).

平面方程：r·n = p，其中 n 为法向量；或参数形式 r = a + s u + t v。

8. Complex Numbers | 复数

A complex number z = x + i y, where i² = −1. Polar and exponential forms simplify multiplication, division, and powers (De Moivre’s theorem).

复数 z = x + i y，其中 i² = −1。极坐标形式和指数形式能简化乘法、除法和乘方（棣莫弗定理）。

Modulus: |z| = √(x² + y²). Argument: Arg(z) = θ, where tan θ = y/x, taking care of quadrant.

模：|z| = √(x² + y²)。辐角：Arg(z) = θ，满足 tan θ = y/x，需注意象限。

Polar form: z = r (cos θ + i sin θ). Exponential form: z = r e^(iθ).

极坐标形式：z = r (cos θ + i sin θ)。指数形式：z = r e^(iθ)。

De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ), for integer n. Useful for finding roots: z^(1/n) = r^(1/n) [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)], k = 0,1,…,n−1.

棣莫弗定理：(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)，n 为整数。用于求根：z^(1/n) = r^(1/n) [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)]，k = 0,1,…,n−1。

Euler’s identity: e^(iπ) + 1 = 0 connects five fundamental constants.

欧拉恒等式：e^(iπ) + 1 = 0，连接五个基本常数。

9. Sequences and Series | 数列与级数

Arithmetic and geometric sequences recur throughout Advanced Higher, alongside binomial expansions and the method of differences.

等差与等比数列在进阶数学中反复出现，此外还有二项展开和差分法。

Arithmetic: uₙ = a + (n−1)d, sum Sₙ = n/2 [2a + (n−1)d].

等差：uₙ = a + (n−1)d，和 Sₙ = n/2 [2a + (n−1)d]。

Geometric: uₙ = a rⁿ⁻¹, sum Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. Infinite sum S∞ = a/(1 − r) for |r| < 1.

等比：uₙ = a rⁿ⁻¹，和 Sₙ = a(1 − rⁿ)/(1 − r)，r ≠ 1。无穷和 S∞ = a/(1 − r)，|r| < 1。

Binomial theorem: (1 + x)ⁿ = Σ (ⁿCₖ) xᵏ for |x| < 1, where ⁿCₖ = n!/(k!(n−k)!). For rational n, expansion is infinite.

二项式定理：(1 + x)ⁿ = Σ (ⁿCₖ) xᵏ，|x| < 1，其中 ⁿCₖ = n!/(k!(n−k)!)。有理数指数时展开为无穷级数。

Method of differences: for telescoping series, express terms as differences to cancel intermediate terms.

差分法：对于裂项级数，将各项表示为差式以消去中间项。

10. Proof and Logical Reasoning | 证明与逻辑推理

Proof technique questions assess your ability to construct rigorous arguments. Common methods include direct proof, contradiction, induction, and contrapositive.

证明题考查构建严谨论证的能力。常见方法包括直接证明、反证法、数学归纳法和逆否命题证明。

Mathematical induction: (1) Base case: verify P(1) true. (2) Inductive step: assume P(k) true, prove P(k+1) true. Conclude P(n) true for all positive integers n.

数学归纳法：(1) 基础步骤：验证 P(1) 为真。(2) 归纳步骤：假设 P(k) 为真，证明 P(k+1) 为真。得出结论对所有正整数 n，P(n) 成立。

Proof by contradiction: assume the negation of the statement, deduce an absurdity, therefore the original statement must hold.

反证法：假设命题的否定成立，推出矛盾，从而原命题成立。

Contrapositive: ‘if P then Q’ is logically equivalent to ‘if not Q then not P’. Useful for proving statements about squares or divisibility.

逆否命题：“若 P 则 Q” 等价于 “若非 Q 则非 P”。常用于平方数或整除性的证明。

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Year 13 SQA Advanced Higher Mathematics: Quick Reference Formula & Theorem Handbook | SQA 进阶数学公式定理速查手册