📚 Pre-U OCR Further Mathematics: Quick Reference Handbook of Formulas and Theorems | Pre-U OCR 进阶数学：公式定理速查手册
This article provides a concise yet comprehensive summary of the essential formulas and theorems required for the OCR Pre-U Further Mathematics syllabus. Each section pairs key results with brief explanations, enabling efficient revision and quick reference. Topics covered include complex numbers, matrices, vectors, hyperbolic functions, differentiation, integration, differential equations, polar coordinates, Maclaurin series, inequalities, and an introduction to group theory.
本文为 OCR Pre-U 进阶数学大纲中的核心公式与定理提供了一份简明全面的速查手册。每个小节将关键结果与简要解释配对呈现，便于高效复习和快速查阅。涵盖的专题包括复数、矩阵、向量、双曲函数、微分、积分、微分方程、极坐标、麦克劳林级数、不等式以及群论入门。
1. Complex Numbers | 复数
A complex number is expressed as z = a + bi, where a, b ∈ ℝ and i² = −1. The real part is a and the imaginary part is b.
复数表示为 z = a + bi，其中 a, b ∈ ℝ 且 i² = −1。实部为 a，虚部为 b。
The modulus of z is |z| = √a² + b² and the argument (principal value) is θ = arg(z), measured from the positive real axis.
z 的模为 |z| = √a² + b²，幅角（主值）为 θ = arg(z)，从正实轴开始测量。
Euler's formula: eⁱθ = cos θ + i sin θ, allowing the polar form z = r eⁱθ, where r = |z|.
欧拉公式：eⁱθ = cos θ + i sin θ，由此可得极坐标形式 z = r eⁱθ，其中 r = |z|。
De Moivre's theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) for any integer n.
棣莫弗定理：对任意整数 n，(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。
The n nth roots of unity are given by e2πik/n for k = 0, 1, …, n−1.
n 次单位根为 e2πik/n，其中 k = 0, 1, …, n−1。
- Complex conjugate: z̄ = a − bi. Properties: z + z̄ = 2a, zz̄ = |z|².
- 共轭复数：z̄ = a − bi。性质：z + z̄ = 2a，zz̄ = |z|²。
2. Matrices and Determinants | 矩阵与行列式
2×2 matrix A = [[a, b], [c, d]], det(A) = ad − bc and A⁻¹ = (1/det(A)) [[d, −b], [−c, a]].
矩阵 A 可逆当且仅当 det(A) ≠ 0。对于 2×2 矩阵 A = [[a, b], [c, d]]，det(A) = ad − bc，且 A⁻¹ = (1/det(A)) [[d, −b], [−c, a]]。
The transpose of A, denoted Aᵀ, satisfies (AB)ᵀ = Bᵀ Aᵀ.
A 的转置记为 Aᵀ，满足 (AB)ᵀ = Bᵀ Aᵀ。
The determinant of a product: det(AB) = det(A) det(B).
乘积的行列式：det(AB) = det(A) det(B)。
satisfy det(A − λI) = 0. Corresponding eigenvectors v are nonzero vectors with A v = λ v.
方阵 A 的特征值 λ 满足 det(A − λI) = 0。对应的特征向量 v 为非零向量，满足 A v = λ v。
AP = D, where D is a diagonal matrix of eigenvalues and columns of P are the eigenvectors.
对角化：若 A 有 n 个线性无关的特征向量，则 P⁻¹AP = D，其中 D 为特征值构成的对角矩阵，P 的列为特征向量。
| Matrix type | Definition |
| Symmetric | Aᵀ = A |
| Orthogonal | Aᵀ = A⁻¹ |
3. Vectors | 向量
a₁, a₂, a₃) and b = (b₁, b₂, b₃), the scalar (dot) product is a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ.
对于向量 a = (a₁, a₂, a₃) 与 b = (b₁, b₂, b₃)，数量积（点积）为 a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ。
a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). Its magnitude gives the area of the parallelogram: |a × b| = |a||b| sin θ.
向量积（叉积）为 a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)。其大小给出平行四边形的面积：|a × b| = |a||b| sin θ。
Scalar triple product: [a, b, c] = a · (b × c) gives the volume of a parallelepiped.
标量三重积：[a, b, c] = a · (b × c) 给出平行六面体的体积。
Vector equation of a line: r = a + t d, t ∈ ℝ. Equation of a plane: r · n = a · n, or equivalently ax + by + cz = d.
直线的向量方程：r = a + t d，t ∈ ℝ。平面的方程：r · n = a · n，或等价地 ax + by + cz = d。
4. Hyperbolic Functions | 双曲函数
Definitions: sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x.
定义：sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x。
Key identity: cosh² x − sinh² x = 1.
核心恒等式：cosh² x − sinh² x = 1。
Derivatives: d/dx(sinh x) = cosh x, d/dx(cosh x) = sinh x, d/dx(tanh x) = sech² x.
导数：d/dx(sinh x) = cosh x，d/dx(cosh x) = sinh x，d/dx(tanh x) = sech² x。
Inverse hyperbolic functions: arsinh x = ln(x + √x²+1), arcosh x = ln(x + √x²−1) for x ≥ 1, artanh x = (1/2)ln((1+x)/(1−x)) for |x| < 1.
反双曲函数：arsinh x = ln(x + √x²+1)，arcosh x = ln(x + √x²−1)（x ≥ 1），artanh x = (1/2)ln((1+x)/(1−x))（|x| < 1）。
5. Differentiation | 微分
Standard derivatives:
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 (arcsin x) = 1/√1−x², d/dx (arctan x) = 1/(1+x²)
Product rule: d/dx (uv) = u′ v + u v′.
乘法法则：d/dx (uv) = u′ v + u v′。
Quotient rule: d/dx (u/v) = (u′ v − u v′)/v².
除法法则：d/dx (u/v) = (u′ v − u v′)/v²。
Chain rule: dy/dx = (dy/du)(du/dx).
链式法则：dy/dx = (dy/du)(du/dx)。
Implicit differentiation: differentiate both sides of an equation with respect to x, treating y as a function of x.
隐函数求导：对方程两边关于 x 求导，视 y 为 x 的函数。
Parametric differentiation: if x = f(t), y = g(t), then dy/dx = (dy/dt) / (dx/dt).
参数方程求导：若 x = f(t)，y = g(t)，则 dy/dx = (dy/dt) / (dx/dt)。
6. Integration | 积分
Standard integrals:
标准积分：
- ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
- ∫ 1/x dx = ln|x| + C
- ∫ eˣ dx = eˣ + C
- ∫ cos x dx = sin x + C, ∫ sin x dx = −cos x + C
- ∫ sec² x dx = tan x + C
- ∫ 1/√a² − x² dx = arcsin(x/a) + C, ∫ 1/(a² + x²) dx = (1/a) arctan(x/a) + C
Integration by parts: ∫ u dv = uv − ∫ v du.
分部积分：∫ u dv = uv − ∫ v du。
he variable and the differential to simplify the integrand, e.g., u = g(x), du = g′(x) dx.
换元积分：替换变量与微分以简化被积函数，例如 u = g(x)，du = g′(x) dx。
Integrals involving hyperbolic functions work analogously, e.g., ∫ cosh x dx = sinh x + C.
涉及双曲函数的积分类似，如 ∫ cosh x dx = sinh x + C。
For partial fractions, express a rational function as a sum of simpler terms before integrating.
对于部分分式积分，先将有理函数分解为简单分式之和再积分。
7. First-Order Differential Equations | 一阶微分方程
Separable equations: if dy/dx = f(x) g(y), rearrange as ∫ (1/g(y)) dy = ∫ f(x) dx.
可分离变量方程：若 dy/dx = f(x) g(y)，重排为 ∫ (1/g(y)) dy = ∫ f(x) dx。
he integrating factor is μ(x) = e^(∫ P(x) dx). Multiply through, then d/dx (μ y) = μ Q(x).
积分因子法：对于 dy/dx + P(x) y = Q(x)，积分因子为 μ(x) = e^(∫ P(x) dx)。乘开，则 d/dx (μ y) = μ Q(x)。
omogeneous equations of the form dy/dx = F(y/x) are solved using the substitution y = v x.
形如 dy/dx = F(y/x) 的齐次方程用代换 y = v x 求解。
8. Second-Order Differential Equations | 二阶微分方程
For a linear homogeneous equation a d²y/dx² + b dy/dx + c y = 0, assume y = eλx; the auxiliary equation is a λ² + b λ + c = 0.
对于线性齐次方程 a d²y/dx² + b dy/dx + c y = 0，设 y = eλx；辅助方程为 a λ² + b λ + c = 0。
If roots λ₁, λ₂ are real and distinct, general solution: y = A eλ₁x + B eλ₂x.
若根 λ₁、λ₂ 为不等实根，通解为 y = A eλ₁x + B eλ₂x。
If repeated real root λ: y = (A + B x) eλx.
若为重实根 λ：y = (A + B x) eλx。
If complex roots λ = α ± iβ: y = eαx (A cos βx + B sin βx).
若为复根 λ = α ± iβ：y = eαx (A cos βx + B sin βx)。
For a non-homogeneous equation, find the complementary function (CF) from the homogeneous part, then a particular integral (PI). Trial forms for PI depend on the forcing term: for polynomial, try polynomial; for ekx, try C ekx; for sin/cos, try C sin ωx + D cos ωx, etc. Modify if it overlaps with CF.
对于非齐次方程，先求齐次部分的余函数 (CF)，再求特解 (PI)。PI 的试解形式取决于外力项：多项式则试多项式；ekx 试 C ekx；正/余弦试 C sin ωx + D cos ωx 等。若与 CF 重叠则需修改。
9. Polar Coordinates | 极坐标
Conversion: x = r cos θ, y = r sin θ; r = √x² + y², θ = arctan(y/x).
坐标转换：x = r cos θ，y = r sin θ；r = √x² + y²，θ = arctan(y/x)。
Area enclosed by a polar curve r = f(θ) from θ = α to θ = β: A = (1/2) ∫[α,β] r² dθ.
极坐标曲线 r = f(θ) 在 θ 从 α 到 β 所围面积：A = (1/2) ∫[α,β] r² dθ。
Arc length of a polar curve: s = ∫[α,β] √(r² + (dr/dθ)²) dθ.
极坐标曲线的弧长：s = ∫[α,β] √(r² + (dr/dθ)²) dθ。
slope at a point on a polar curve is dy/dx = [r′ sin θ + r cos θ] / [r′ cos θ − r sin θ].
极坐标曲线上一点处的切线斜率为 dy/dx = [r′ sin θ + r cos θ] / [r′ cos θ − r sin θ]。
10. Maclaurin Series | 麦克劳林级数
The Maclaurin series expansion of a function f(x) is f(x) = ∑[n=0,∞] (f⁽ⁿ⁾(0)/n!) xⁿ.
函数 f(x) 的麦克劳林级数展开为 f(x) = ∑[n=0,∞] (f⁽ⁿ⁾(0)/n!) xⁿ。
Key series:
关键展开式：
- 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 − … (valid for −1 < x ≤ 1)
- (1+x)ⁿ = 1 + n x + n(n−1)/2! x² + … (Binomial series, |x| < 1)
Compositions: series can be used to approximate functions, evaluate limits, and find integrals in a power series form.
组合运算：级数可用于函数逼近、计算极限以及将积分表示为幂级数形式。
11. Inequalities | 不等式
Basic rules: addition preserves inequality; multiplying by a positive number preserves inequality, multiplying by a negative number reverses it.
基本规则：加法保持不等号方向；乘以正数保持不等号方向，乘以负数则反号。
Solving polynomial inequalities: bring all terms to one side, factorise, determine sign intervals using a sign table or critical points.
求解多项式不等式：将所有项移到一边，因式分解，通过符号表或临界点确定符号区间。
Cauchy–Schwarz inequality: |a·b| ≤ |a| |b| for vectors a and b.
柯西-施瓦茨不等式：对于向量 a 和 b，|a·b| ≤ |a| |b|。
Triangle inequality: |a + b| ≤ |a| + |b|.
三角不等式：|a + b| ≤ |a| + |b|。
AM–GM inequality: for non-negative numbers, (x₁+x₂+…+xₙ)/n ≥ (x₁x₂…xₙ)1/n, equality when all xᵢ are equal.
算术–几何平均不等式：对于非负数，(x₁+x₂+…+xₙ)/n ≥ (x₁x₂…xₙ)1/n，等号当所有 xᵢ 相等时成立。
12. Introduction to Groups | 群论入门
A group (G, ∘) is a set G with a binary operation ∘ satisfying closure, associativity, existence of an identity element e, and existence of inverses for each element.
群 (G, ∘) 是由集合 G 与二元运算 ∘ 组成，满足封闭性、结合律、存在单位元 e，以及每个元素存在逆元。
Order of a group is the number of elements |G|. The order of an element a is the smallest positive integer n such that aⁿ = e.
群的阶为元素个数 |G|。元素 a 的阶为使得 aⁿ = e 的最小正整数 n。
Cyclic groups: a group generated by a single element, Cₙ = {e, a, a², …, aⁿ⁻¹} where aⁿ = e.
循环群：由单一元素生成的群，Cₙ = {e, a, a², …, aⁿ⁻¹}，其中 aⁿ = e。
Subgroup test: a non-empty subset H of a group G is a subgroup if for all a, b ∈ H, ab⁻¹ ∈ H.
子群检验：群 G 的非空子集 H 是子群当且仅当对所有 a, b ∈ H，有 ab⁻¹ ∈ H。
Lagrange's theorem: for a finite group, the order of any subgroup divides the order of the group.
拉格朗日定理：对于有限群，任意子群的阶整除群的阶。
Isomorphism: two groups are isomorphic if there exists a bijective homomorphism between them, preserving the group structure.
同构：若两个群之间存在双射同态并保持群结构，则称它们同构。
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