📚 Year 12 SQA Advanced Higher Maths: Formula & Theorem Quick Reference Handbook | SQA 进阶数学：公式定理速查手册
This quick-reference handbook brings together all the essential formulas, theorems and standard results needed for the SQA Advanced Higher Mathematics course. It is designed as a fast revision aid, helping you recall key facts for differentiation, integration, complex numbers, matrices, vectors, sequences, differential equations and proof techniques. Every entry is accompanied by a concise explanation in both English and Chinese, ensuring clarity for bilingual learners.
本速查手册汇集了 SQA 进阶数学课程所需的所有核心公式、定理和标准结果。它旨在作为快速复习工具，帮助你回忆微分、积分、复数、矩阵、向量、数列、微分方程和证明技巧的关键知识点。每个条目都配有简洁的中英文解释，确保双语学习者能清晰理解。
1. Algebraic Manipulation and Partial Fractions | 代数运算与部分分式
The expansion of (a+b)ⁿ for rational n is given by the general binomial theorem, while partial fractions decompose rational functions into simpler sums. These techniques are foundational for integration and series work.
有理指数 (a+b)ⁿ 的二项式展开由广义二项式定理给出，而部分分式将有理函数拆分为更简单的分式之和。这些方法是积分和级数运算的基础。
(1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + … , |x| < 1
General binomial series valid for |x|<1 and any rational n. For a pure linear denominator (ax+b), use 1/(ax+b) = (1/a)·1/(1 - (-bx/a)) to apply the expansion.
广义二项式级数在 |x|<1 且 n 为任意有理数时成立。对于纯线性分母 (ax+b)，可化为 1/(ax+b) = (1/a)·1/(1 - (-bx/a)) 后套用展开式。
Partial fraction types:
部分分式类型：
- Linear factors: 1/((x-a)(x-b)) = A/(x-a) + B/(x-b)
- 线性因式： 1/((x-a)(x-b)) = A/(x-a) + B/(x-b)
- Repeated linear: 1/((x-a)²) = A/(x-a) + B/(x-a)²
- 重复线性因式： 1/((x-a)²) = A/(x-a) + B/(x-a)²
- Irreducible quadratic: 1/((x²+a)(x+b)) = (Ax+B)/(x²+a) + C/(x+b)
- 不可约二次因式： 1/((x²+a)(x+b)) = (Ax+B)/(x²+a) + C/(x+b)
Solving for constants: multiply through by the denominator, equate coefficients or substitute convenient x-values.
求解常数：两边同乘分母，比较系数或代入方便计算的 x 值。
2. Differentiation Rules and Standard Derivatives | 微分法则与标准导数
Advanced Higher extends differentiation to products, quotients, composites, inverses and parametric forms. Second derivatives and implicit differentiation are also essential.
进阶数学将微分扩展到乘积、商、复合、反函数和参数形式。二阶导数和隐函数微分同样至关重要。
| Rule | Formula | 中文 |
|---|---|---|
| Product Rule | d/dx [u v] = u’ v + u v’ | 乘积法则 |
| Quotient Rule | d/dx [u/v] = (u’ v – u v’) / v² | 商法则 |
| Chain Rule | dy/dx = dy/du × du/dx | 链式法则 |
| Parametric | dy/dx = (dy/dt) / (dx/dt) | 参数微分 |
| Implicit | Differentiate both sides w.r.t. x, treating y as a function of x. | 隐函数微分：两边对 x 求导，视 y 为 x 的函数。 |
| Second derivative parametric | d²y/dx² = d/dx (dy/dx) = d/dt (dy/dx) / (dx/dt) | 参数方程的二阶导数 |
Standard derivatives to memorise:
需要记住的标准导数：
d/dx (xⁿ) = n xⁿ⁻¹, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x
d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec² x
d/dx (sin⁻¹ x) = 1/√(1−x²), d/dx (cos⁻¹ x) = −1/√(1−x²), d/dx (tan⁻¹ x) = 1/(1+x²)
Note the sign difference for inverse cosine.
注意反余弦的符号差异。
3. Integration Techniques | 积分技巧
Advanced Higher requires fluency with substitution, integration by parts, partial fractions, and standard integrals. Be prepared to handle definite integrals, improper integrals and use of inverse trig results.
进阶数学要求熟练运用换元法、分部积分法、部分分式积分以及标准积分形式。还需要处理定积分、瑕积分以及反三角函数的积分结果。
∫ 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
Integration by parts formula:
分部积分公式：
∫ u dv = u v − ∫ v du
Tip: choose u using ‘LIATE’ (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential).
提示：用“LIATE”规则选择 u（对数、反三角、代数、三角、指数）。
Substitution: For ∫ f(g(x))·g'(x) dx, let u=g(x) so du=g'(x)dx. Also be able to recognise standard forms like ∫ f'(x)/f(x) dx = ln|f(x)|+C.
换元积分：若 ∫ f(g(x))·g'(x) dx，令 u=g(x)，则 du=g'(x)dx。同时要能识别标准形如 ∫ f'(x)/f(x) dx = ln|f(x)|+C。
Important inverse trig integrals (often in formula sheet, but need to recognise):
重要的反三角函数积分（常出现在公式表，但要能识别）：
∫ 1/√(a²−x²) dx = sin⁻¹(x/a) + C, ∫ 1/(a²+x²) dx = (1/a) tan⁻¹(x/a) + C
4. Differential Equations | 微分方程
First-order separable equations and second-order linear homogeneous ODEs with constant coefficients form the core. You must be able to find general and particular solutions, and apply initial/boundary conditions.
一阶可分离变量方程和二阶常系数线性齐次常微分方程是核心内容。必须能求出通解和特解，并能代入初始条件或边界条件。
Separable type: dy/dx = g(x)h(y) → ∫ 1/h(y) dy = ∫ g(x) dx.
可分离变量型：dy/dx = g(x)h(y) → ∫ 1/h(y) dy = ∫ g(x) dx。
Second-order homogeneous: a d²y/dx² + b dy/dx + c y = 0.
二阶齐次方程：a d²y/dx² + b dy/dx + c y = 0。
Auxiliary equation: a m² + b m + c = 0.
辅助方程：a m² + b m + c = 0。
| Roots of auxiliary eq. | General solution | 根的情况 |
|---|---|---|
| Real distinct m₁, m₂ | y = A e^(m₁ x) + B e^(m₂ x) | 相异实根 |
| Repeated real root m | y = (A + Bx) e^(m x) | 重实根 |
| Complex conjugate p ± iq | y = e^(p x) (A cos qx + B sin qx) | 共轭复根 p ± iq |
To find a particular solution given initial conditions, differentiate the general solution and substitute the given x, y, dy/dx values.
给定初始条件求特解时，先将通解求导，然后代入给定的 x、y 和 dy/dx 值。
5. Matrices and Determinants | 矩阵与行列式
Matrix algebra, determinants, inverses, and the ability to solve systems of linear equations using Gaussian elimination or inverse matrices are essential. Transformations in 2D and 3D rely on matrix multiplication.
矩阵运算、行列式、逆矩阵以及利用高斯消元法或逆矩阵求解线性方程组是关键技能。二维和三维变换依赖于矩阵乘法。
For a 3×3 matrix A, the determinant is:
对于 3×3 矩阵 A，行列式为：
det A = a(ei − fh) − b(di − fg) + c(dh − eg)
The inverse A⁻¹ exists iff det A ≠ 0, and is given by (1/det A) adj(A) where adj(A) is the adjugate (transpose of cofactor matrix).
当且仅当 det A ≠ 0 时逆矩阵存在，且 A⁻¹ = (1/det A) adj(A)，其中 adj(A) 为伴随矩阵（余子式矩阵的转置）。
Solving a system AX = B: if A is invertible, X = A⁻¹B. Elementary row operations (Gaussian elimination) reduce the augmented matrix to row-echelon form.
求解方程组 AX = B：若 A 可逆，则 X = A⁻¹B。初等行变换（高斯消元法）将增广矩阵化为行阶梯形。
Properties of determinants:
行列式性质：
- det (AB) = det A × det B
- det (Aᵀ) = det A
- Swapping two rows changes the sign of det.
- 交换两行，行列式变号。
6. Vectors in 3D | 三维向量
Vectors are used to describe points, lines, and planes in three dimensions. The scalar (dot) product and vector (cross) product are fundamental for angles, areas, and perpendicularity conditions.
向量用于描述三维空间中的点、线和面。数量积（点乘）和向量积（叉乘）是处理角度、面积和垂直条件的基础工具。
Vector equation of a line: r = a + t d , where a is a point on the line, d is direction vector, t scalar.
直线向量方程：r = a + t d，其中 a 为直线上一点，d 为方向向量，t 为标量。
Scalar product: a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃.
数量积：a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。
If a·b = 0, vectors are perpendicular.
若 a·b = 0，则向量垂直。
Cross product: a×b = (a₂b₃ − a₃b₂) i − (a₁b₃ − a₃b₁) j + (a₁b₂ − a₂b₁) k. The result is perpendicular to both a and b, and its magnitude gives the area of the parallelogram spanned by them.
向量积：a×b = (a₂b₃ − a₃b₂) i − (a₁b₃ − a₃b₁) j + (a₁b₂ − a₂b₁) k。结果同时垂直于 a 和 b，其模长等于以它们为边的平行四边形面积。
Equation of a plane: r·n = a·n = d, where n is normal vector. Or parametric form: r = a + su + tv.
平面方程：r·n = a·n = d，其中 n 为法向量。或参数形式：r = a + su + tv。
The angle between two planes is the angle between their normals, found using dot product.
两平面的夹角即其法向量的夹角，可用点积求得。
7. Complex Numbers | 复数
Complex numbers are expressed in Cartesian form a+ib, polar form r(cosθ + i sinθ), and exponential form re^(iθ). De Moivre’s theorem and roots of unity are key topics.
复数可用代数形式 a+ib、极坐标形式 r(cosθ + i sinθ) 和指数形式 re^(iθ) 表示。棣莫弗定理和单位根是重点内容。
i² = −1, |z| = √(a²+b²), arg(z) = θ, z = a+bi
z₁z₂ = r₁r₂ (cos(θ₁+θ₂) + i sin(θ₁+θ₂))
De Moivre: [r(cos θ + i sin θ)]ⁿ = rⁿ (cos nθ + i sin nθ)
To find the n nth roots of a complex number w: write w = s(cos φ + i sin φ), then roots are s^(1/n) [cos((φ+2kπ)/n) + i sin((φ+2kπ)/n)], k=0,1,…,n−1.
求复数 w 的 n 次方根：将 w 写成 s(cos φ + i sin φ)，则 n 个根为 s^(1/n) [cos((φ+2kπ)/n) + i sin((φ+2kπ)/n)]，k=0,1,…,n−1。
Conjugate z̄ = a−bi. Useful property: z × z̄ = |z|².
共轭复数 z̄ = a−bi，常用性质：z × z̄ = |z|²。
Solving polynomial equations with real coefficients: non-real roots occur in conjugate pairs.
解实系数多项式方程：非实根共轭成对出现。
8. Sequences and Series | 数列与级数
Arithmetic and geometric sequences, sum to infinity for convergent geometric series, and the method of differences for telescoping sums appear frequently. The Maclaurin series and binomial expansion complete the series toolkit.
等差数列与等比数列、收敛等比级数的无穷和，以及裂项求和的差分法经常出现。麦克劳林级数和二项式展开完善了级数工具包。
Arithmetic: uₙ = a + (n−1)d, Sₙ = n/2 [2a + (n−1)d].
等差数列：uₙ = a + (n−1)d，Sₙ = n/2 [2a + (n−1)d]。
Geometric: uₙ = a rⁿ⁻¹, Sₙ = a(1−rⁿ)/(1−r) for r≠1. Sum to infinity: S∞ = a/(1−r) if |r| < 1.
等比数列：uₙ = a rⁿ⁻¹，Sₙ = a(1−rⁿ)/(1−r)（r≠1）。若 |r| < 1，无穷和为 S∞ = a/(1−r)。
Method of differences: express the general term as f(r) − f(r+1) so that intermediate terms cancel. For example, ∑ 1/(r(r+1)) = 1 − 1/(n+1).
差分法：将通项写成 f(r) − f(r+1) 的形式，使中间项抵消。例如 ∑ 1/(r(r+1)) = 1 − 1/(n+1)。
Maclaurin series: f(x) = f(0) + f ‘(0)x + f ”(0)x²/2! + f ”'(0)x³/3! + …
麦克劳林级数：f(x) = f(0) + f ‘(0)x + f ”(0)x²/2! + f ”'(0)x³/3! + …
Standard Maclaurin expansions:
标准麦克劳林展开式：
eˣ = 1 + x + x²/2! + x³/3! + …
sin x = x − x³/3! + x⁵/5! − … , cos x = 1 − x²/2! + x⁴/4! − …
ln(1+x) = x − x²/2 + x³/3 − x⁴/4 + … , −1 < x ≤ 1
9. Further Calculus: Volumes of Revolution and Improper Integrals | 进阶微积分：旋转体体积与瑕积分
A solid of revolution is formed by rotating a curve about the x- or y-axis. Its volume is calculated using definite integrals. Improper integrals handle infinite limits or infinite integrands.
旋转体由曲线绕 x 轴或 y 轴旋转而成，其体积用定积分计算。瑕积分处理无穷积分限或被积函数无穷的情形。
Volume of revolution about x-axis: V = π ∫ₐᵇ y² dx.
绕 x 轴旋转体积：V = π ∫ₐᵇ y² dx。
About y-axis: V = π ∫ₐᵇ x² dy, where limits are y-values.
绕 y 轴旋转：V = π ∫ₐᵇ x² dy，积分限为 y 值。
Improper integrals: replace ∞ with a large number b and take limit b→∞. If the limit exists and is finite, the integral converges; otherwise it diverges.
瑕积分：用大数 b 代替 ∞ 并取 b→∞ 的极限。若极限存在且有限，则积分收敛；否则发散。
Example: ∫₁^∞ 1/x² dx = lim_(b→∞) [−1/x]₁ᵇ = 1, converges.
示例：∫₁^∞ 1/x² dx = lim_(b→∞) [−1/x]₁ᵇ = 1，收敛。
Watch for infinite integrands within the interval; split the interval and treat each improper limit separately.
注意区间内被积函数无界的情况，需拆分区间并分别处理每个瑕积分。
10. Proof by Induction | 归纳法证明
Proof by induction is a standard method for establishing results that depend on a positive integer n. You must show a base case, the inductive hypothesis, and the inductive step.
归纳法证明是一种用于证明依赖正整数 n 的结论的标准方法。必须展示基础步骤、归纳假设和归纳步骤。
Steps:
步骤：
- Base case: Verify the statement holds for n=1 (or other starting value).
- 基础步骤： 验证当 n=1（或其他起始值）时命题成立。
- Inductive hypothesis: Assume true for n=k.
- 归纳假设： 假设 n=k 时命题为真。
- Inductive step: Prove it holds for n=k+1 using the assumption.
- 归纳步骤： 利用该假设证明 n=k+1 时命题也成立。
- Conclusion: By the principle of mathematical induction, the statement is true for all n ≥ 1.
- 结论： 由数学归纳法原理，命题对所有 n ≥ 1 成立。
Common induction proofs: summation formulas (e.g., ∑ r = n(n+1)/2), divisibility (e.g., 3^(2n)−1 is divisible by 8), matrix powers, and inequalities.
常见的归纳法证明：求和公式（如 ∑ r = n(n+1)/2）、整除性（如 3^(2n)−1 可被 8 整除）、矩阵幂次和不等式。
For summation, in the inductive step write S(k+1) = S(k) + the (k+1)th term, substitute the closed form of S(k) from the hypothesis, and simplify to the required expression for n=k+1.
对于求和证明，在归纳步骤中写出 S(k+1) = S(k) + 第 (k+1) 项，代入归纳假设中 S(k) 的闭式，化简得到 n=k+1 时所需的形式。
11. Further Techniques: L’Hôpital’s Rule and Limits | 进阶技巧：洛必达法则与极限
When evaluating limits that result in indeterminate forms 0/0 or ∞/∞, l’Hôpital’s rule allows differentiation of numerator and denominator separately. This topic often appears alongside Maclaurin series and improper integrals.
当求极限遇到 0/0 或 ∞/∞ 等未定式时，洛必达法则允许对分子分母分别求导。该知识点常与麦克劳林级数和瑕积分结合考查。
If lim f(x) = 0 and lim g(x) = 0, then lim f(x)/g(x) = lim f ‘(x)/g'(x), provided the latter limit exists.
若 lim f(x) = 0 且 lim g(x) = 0，则 lim f(x)/g(x) = lim f ‘(x)/g'(x)，前提是后者极限存在。
For repeated applications, continue differentiating until an indeterminate form no longer occurs. Always check that the conditions are satisfied before applying.
可重复应用法则，直到不再出现未定式为止。应用前务必检查条件是否满足。
The rule also works for ∞/∞ forms. Transform products or differences into quotients when necessary.
该法则同样适用于 ∞/∞ 型。必要时可将乘积或差式转化为商式。
12. Standard Results for Quick Recall | 常用结论速记
Memorising these facts can save valuable time in the exam:
熟记以下结论可在考试中节省宝贵时间：
- Double-angle: sin 2θ = 2 sinθ cosθ, cos 2θ = cos²θ − sin²θ = 2cos²θ−1 = 1−2sin²θ
- 倍角公式： sin 2θ = 2 sinθ cosθ，cos 2θ = cos²θ − sin²θ = 2cos²θ−1 = 1−2sin²θ
- Integral of odd/even functions: ∫₋ₐᵃ f(x) dx = 0 if f is odd; = 2∫₀ᵃ f(x) dx if f is even.
- 奇偶函数积分： 若 f 为奇函数，∫₋ₐᵃ f(x) dx = 0；若为偶函数，则等于 2∫₀ᵃ f(x) dx。
- Scalar triple product: a·(b×c) gives the volume of parallelepiped; zero implies coplanarity.
- 标量三重积： a·(b×c) 等于平行六面体的体积；为零则三向量共面。
- Sum of cubes and differences: a³ + b³ = (a+b)(a²−ab+b²), a³ − b³ = (a−b)(a²+ab+b²)
- 立方和与立方差： a³ + b³ = (a+b)(a²−ab+b²)，a³ − b³ = (a−b)(a²+ab+b²)
Keep these relationships at your fingertips — they often appear as intermediate steps in larger problems.
把这些关系式熟记于心——它们常常作为大题中的中间步骤出现。
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