A-Level Edexcel Further Maths: Formula & Theorem Quick Reference Handbook | A-Level Edexcel 进阶数学：公式定理速查手册

📚 A-Level Edexcel Further Maths: Formula & Theorem Quick Reference Handbook | A-Level Edexcel 进阶数学：公式定理速查手册

This quick reference handbook collects the essential formulas, identities, and theorems required for the Edexcel A-Level Further Mathematics course. It is designed to help you consolidate core pure topics, including complex numbers, matrices, vectors, series, calculus, polar coordinates, hyperbolic functions, and differential equations. Each section presents key results in a clear layout with brief explanatory notes, making last-minute revision efficient and focused.

本速查手册汇总了 Edexcel A-Level 进阶数学课程所必需的核心公式、恒等式和定理。手册旨在帮助你巩固核心纯数内容，涵盖复数、矩阵、向量、级数、微积分、极坐标、双曲函数和微分方程。每节以清晰布局展示关键结论并附简要说明，让考前冲刺复习高效而专注。

1. Complex Numbers | 复数

The imaginary unit i is defined by i² = −1. A complex number is written as z = a + bi, where a, b ∈ ℝ. The real part is Re(z) = a and the imaginary part is Im(z) = b.

虚数单位 i 定义为 i² = −1。复数写作 z = a + bi，其中 a, b 为实数。实部为 Re(z) = a，虚部为 Im(z) = b。

The complex conjugate of z = a + bi is z* = a − bi. Modulus: |z| = √(a² + b²). Argument: arg(z) = θ, where tan θ = b/a, adjusted for the quadrant.

z = a + bi 的共轭复数为 z* = a − bi。模：|z| = √(a² + b²)。辐角：arg(z) = θ，其中 tan θ = b/a，需根据象限调整。

Polar form: z = r(cos θ + i sin θ) = re^(iθ), where r = |z| and θ = arg(z). Euler’s relation: e^(iθ) = cos θ + i sin θ.

极形式：z = r(cos θ + i sin θ) = re^(iθ)，其中 r = |z|，θ = arg(z)。欧拉公式：e^(iθ) = cos θ + i sin θ。

De Moivre’s theorem: (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ) for integer n. It is used to find powers and roots of complex numbers.

棣莫弗定理：对于整数 n，(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)。用于求复数的幂和根。

The n nth roots of unity are given by z = e^(2πik/n) for k = 0, 1, …, n−1. They form a regular n-gon on the unit circle.

n 次单位根由 z = e^(2πik/n)（k = 0, 1, …, n−1）给出。它们在单位圆上构成正 n 边形。

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2. Matrices | 矩阵

A matrix is a rectangular array of numbers. Order: m × n (rows × columns). The transpose Aᵀ swaps rows and columns: (Aᵀ)ᵢⱼ = Aⱼᵢ.

矩阵是数的矩形阵列。阶：m × n（行×列）。转置 Aᵀ 互换行与列：(Aᵀ)ᵢⱼ = Aⱼᵢ。

Matrix multiplication: if A is m × n and B is n × p, then C = AB is m × p with Cᵢⱼ = ∑ₖ Aᵢₖ Bₖⱼ. Multiplication is not commutative.

矩阵乘法：若 A 为 m×n，B 为 n×p，则 C = AB 为 m×p，其中 Cᵢⱼ = ∑ₖ Aᵢₖ Bₖⱼ。乘法不可交换。

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

二阶矩阵 A = [[a, b], [c, d]] 的行列式为 det(A) = ad − bc。三阶行列式使用萨鲁斯法则或余子式展开。

The inverse A⁻¹ exists iff det(A) ≠ 0. For 2×2: A⁻¹ = (1/det(A)) [[d, −b], [−c, a]]. Inverses satisfy AA⁻¹ = A⁻¹A = I.

若 det(A) ≠ 0，则逆阵 A⁻¹ 存在。二阶：A⁻¹ = (1/det(A)) [[d, −b], [−c, a]]。逆阵满足 AA⁻¹ = A⁻¹A = I。

Eigenvalue equation: Av = λv. The eigenvalues λ are roots of the characteristic equation det(A − λI) = 0. Corresponding eigenvectors v can then be found.

特征值方程：Av = λv。特征值 λ 是特征方程 det(A − λI) = 0 的根。然后可求出对应的特征向量 v。

For diagonalisation: if A has n linearly independent eigenvectors, then A = PDP⁻¹, where D is diagonal with eigenvalues and P is the matrix of eigenvectors.

对角化：若 A 有 n 个线性无关特征向量，则 A = PDP⁻¹，其中 D 为特征值对角阵，P 为特征向量矩阵。

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3. Vectors | 向量

A vector v has magnitude |v| and direction. In Cartesian coordinates v = x i + y j + z k. The scalar (dot) product: a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃.

向量 v 有大小 |v| 和方向。在直角坐标中 v = x i + y j + z k。数量积（点乘）：a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。

The vector (cross) product: a × b = |a||b| sin θ n̂, with direction given by the right-hand rule. In component form, a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k.

向量积（叉乘）：a × b = |a||b| sin θ n̂，方向由右手定则确定。分量形式：a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k。

Equation of a line: r = a + t d, where a is a point on the line and d is a direction vector. Cartesian form: (x − x₀)/d₁ = (y − y₀)/d₂ = (z − z₀)/d₃.

直线方程：r = a + t d，其中 a 为线上一点，d 为方向向量。直角坐标形式：(x − x₀)/d₁ = (y − y₀)/d₂ = (z − z₀)/d₃。

Equation of a plane: r · n = a · n = p (scalar product form), or r = a + λ u + μ v (parametric form). Normal vector n = u × v.

平面方程：r · n = a · n = p（点积形式），或 r = a + λ u + μ v（参数形式）。法向量 n = u × v。

Distance from a point to a plane |(AP · n)|/|n|. Intersection of two lines: solve r = a₁ + t d₁ = a₂ + s d₂. Check for skew lines (non-intersecting, non-parallel).

点到平面的距离 |(AP · n)|/|n|。两直线交点：解 r = a₁ + t d₁ = a₂ + s d₂。注意判断异面直线（不平行也不相交）。

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4. Series | 级数

The method of differences: sum a series of the form ∑ (uₙ₊₁ − uₙ) = uₙ₊₁ − u₁. Terms often need to be expressed as partial fractions.

差分法：对形如 ∑ (uₙ₊₁ − uₙ) 的级数求和得 uₙ₊₁ − u₁。通常需将项表示为部分分式。

Standard Maclaurin series: eˣ = ∑ xⁿ/n! (all x), sin x = ∑ (−1)ⁿ x²ⁿ⁺¹/(2n+1)! (all x), cos x = ∑ (−1)ⁿ x²ⁿ/(2n)! (all x), ln(1+x) = ∑ (−1)ⁿ⁻¹ xⁿ/n (|x| < 1).

标准麦克劳林级数：eˣ = ∑ xⁿ/n!（全体实数），sin x = ∑ (−1)ⁿ x²ⁿ⁺¹/(2n+1)!（全体实数），cos x = ∑ (−1)ⁿ x²ⁿ/(2n)!（全体实数），ln(1+x) = ∑ (−1)ⁿ⁻¹ xⁿ/n（|x| < 1）。

Binomial series: (1+x)ⁿ = 1 + nx + n(n−1)x²/2! + … for rational n, valid when |x| < 1.

二项式级数：(1+x)ⁿ = 1 + nx + n(n−1)x²/2! + …，其中 n 为有理数，当 |x| < 1 时收敛。

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5. Further Calculus | 进阶微积分

Reduction formulae: integrals of the form Iₙ = ∫ sinⁿ x dx or ∫ cosⁿ x dx, or Iₙ = ∫ xⁿ eᵏˣ dx, are evaluated by integration by parts, relating Iₙ to Iₙ₋₁ or Iₙ₋₂.

约化公式：形如 Iₙ = ∫ sinⁿ x dx 或 ∫ cosⁿ x dx 或 Iₙ = ∫ xⁿ eᵏˣ dx 的积分，用分部积分将 Iₙ 与 Iₙ₋₁ 或 Iₙ₋₂ 联系起来。

Arc length: for a curve y = f(x), s = ∫ √(1 + (dy/dx)²) dx. For parametric x(t), y(t): s = ∫ √((dx/dt)² + (dy/dt)²) dt.

弧长：曲线 y = f(x)，弧长 s = ∫ √(1 + (dy/dx)²) dx。用参数形式 x(t), y(t)：s = ∫ √((dx/dt)² + (dy/dt)²) dt。

Surface area of revolution: about x‑axis, S = ∫ 2π y ds, where ds is the arc length element; about y‑axis, S = ∫ 2π x ds.

旋转体表面积：绕 x 轴旋转，S = ∫ 2π y ds；绕 y 轴，S = ∫ 2π x ds，其中 ds 为弧长微元。

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6. Polar Coordinates | 极坐标

A point P is given by (r, θ), where r is the distance from the pole and θ is the angle from the initial line. Conversion: x = r cos θ, y = r sin θ, r² = x² + y².

点 P 用 (r, θ) 表示，r 为到极点的距离，θ 为从极轴转过的角度。转换：x = r cos θ, y = r sin θ, r² = x² + y²。

Area enclosed by a polar curve: A = ½ ∫ r² dθ between limits θ₁ and θ₂.

极曲线所围面积：A = ½ ∫ r² dθ，积分限为 θ₁ 和 θ₂。

Arc length in polar coordinates: s = ∫ √(r² + (dr/dθ)²) dθ.

极坐标下的弧长：s = ∫ √(r² + (dr/dθ)²) dθ。

Common polar curves: r = a (circle), r = a θ (Archimedean spiral), r = a cos nθ or r = a sin nθ (rose curves), r = a(1 ± cos θ) (cardioid).

常见极曲线：r = a（圆），r = a θ（阿基米德螺线），r = a cos nθ 或 r = a sin nθ（玫瑰线），r = a(1 ± cos θ)（心形线）。

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7. 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. sech x = 1/cosh x, cosech x = 1/sinh x, coth x = 1/tanh x.

定义：sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x。sech x = 1/cosh x，cosech x = 1/sinh x，coth x = 1/tanh x。

Osborn’s rule: identities for hyperbolic functions can be obtained from trigonometric identities by replacing sin → i sinh, cos → cosh, but changing the sign of any product (or implied product) of two sines.

奥斯本法则：双曲恒等式可由三角恒等式通过 sin → i sinh, cos → cosh 获得，但每出现两个正弦乘积（或隐含乘积）的项需变号。

Key identity: cosh² x − sinh² x = 1. Others: sinh 2x = 2 sinh x cosh x, cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x.

核心恒等式：cosh² x − sinh² x = 1。其他：sinh 2x = 2 sinh x cosh x，cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x。

Inverse hyperbolic functions: arsinh x = ln(x + √(x² + 1)), arcosh x = ln(x + √(x² − 1)) (x ≥ 1), artanh x = ½ ln((1+x)/(1−x)) (|x| < 1).

反双曲函数：arsinh x = ln(x + √(x² + 1))，arcosh x = ln(x + √(x² − 1))（x ≥ 1），artanh x = ½ ln((1+x)/(1−x))（|x| < 1）。

Derivatives: d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech² x, d/dx (arsinh x) = 1/√(x²+1), d/dx (arcosh x) = 1/√(x²−1), d/dx (artanh x) = 1/(1−x²).

导数：d/dx (sinh x) = cosh x，d/dx (cosh x) = sinh x，d/dx (tanh x) = sech² x，d/dx (arsinh x) = 1/√(x²+1)，d/dx (arcosh x) = 1/√(x²−1)，d/dx (artanh x) = 1/(1−x²)。

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8. Differential Equations | 微分方程

First-order separable: dy/dx = g(x)h(y) → ∫ 1/h(y) dy = ∫ g(x) dx. Integrating factor: for dy/dx + P(x)y = Q(x), μ = e^(∫ P dx), then d/dx (μ y) = μ Q.

一阶可分离变量：dy/dx = g(x)h(y) → ∫ 1/h(y) dy = ∫ g(x) dx。积分因子：对于 dy/dx + P(x)y = Q(x)，μ = e^(∫ P dx)，然后 d/dx (μ y) = μ Q。

Second-order homogeneous with constant coefficients: ay” + by’ + cy = 0. Auxiliary equation: am² + bm + c = 0. If roots real distinct m₁, m₂: y = A e^(m₁x) + B e^(m₂x). Repeated root m: y = (A + Bx) e^(mx). Complex roots α ± iβ: y = e^(αx) (A cos βx + B sin βx).

常系数二阶齐次方程：ay” + by’ + cy = 0。辅助方程：am² + bm + c = 0。相异实根 m₁, m₂ → y = A e^(m₁x) + B e^(m₂x)。重根 m → y = (A + Bx) e^(mx)。复根 α ± iβ → y = e^(αx)(A cos βx + B sin βx)。

Inhomogeneous case: ay” + by’ + cy = f(x). General solution = complementary function + particular integral. Particular integral trial forms: f(x) = polynomial → polynomial of same degree; f(x) = p e^(kx) → C e^(kx) (modify if k is a root of auxiliary eqn); f(x) = cos ωx or sin ωx → C cos ωx + D sin ωx.

非齐次情形：ay” + by’ + cy = f(x)。通解 = 补函数 + 特解。特解试函数：f(x) 为多项式 → 同次多项式；f(x) = p e^(kx) → C e^(kx)（若 k 是辅助方程的根则需修正）；f(x) = cos ωx 或 sin ωx → C cos ωx + D sin ωx。

Simple harmonic motion: x” + ω² x = 0 → x = A cos ωt + B sin ωt = R cos(ωt − φ). Damped oscillations: x” + 2k x’ + ω² x = 0, with under/critical/over-damping.

简谐运动：x” + ω² x = 0 → x = A cos ωt + B sin ωt = R cos(ωt − φ)。阻尼振动：x” + 2k x’ + ω² x = 0，分为欠阻尼、临界阻尼和过阻尼。

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9. Numerical Methods | 数值方法

Interval bisection: if f(a) and f(b) have opposite signs, the root lies in (a,b). Repeatedly halve the interval to narrow the approximation.

二分法：若 f(a) 与 f(b) 异号，则根在 (a,b) 内。反复将区间二分以逼近根。

Newton-Raphson method: xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ). Requires a good initial guess and f'(xₙ) ≠ 0. Quadratic convergence near simple roots.

牛顿-拉弗森法：xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ)。需要较好的初值且 f'(xₙ) ≠ 0。在单根附近具有二次收敛速度。

Euler’s method for 1st-order ODE y’ = f(x,y): yₙ₊₁ = yₙ + h f(xₙ, yₙ), where h is step size. Improved Euler (Heun’s method) uses an average slope.

欧拉法解一阶常微分方程 y’ = f(x,y)：yₙ₊₁ = yₙ + h f(xₙ, yₙ)，h 为步长。改进欧拉法（休恩法）使用平均斜率。

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10. Proof by Induction | 数学归纳法

Structure: 1) Base case: show truth for n = 1 (or smallest value). 2) Inductive step: assume true for n = k, then prove true for n = k+1. 3) Conclusion: by induction, statement true for all positive integers n.

结构：1) 基础情形：验证 n = 1（或最小整数）时命题成立。2) 归纳步骤：假设 n = k 时成立，证明 n = k+1 时成立。3) 结论：由归纳法，命题对所有正整数 n 成立。

Common applications: summation of series, divisibility proofs (e.g., 3²ⁿ − 1 is divisible by 8), matrix powers (Aⁿ formula), inequalities, and recurrence relations.

常见应用：级数求和、整除性证明（如 3²ⁿ − 1 能被 8 整除）、矩阵幂次 (Aⁿ 公式)、不等式以及递推关系。

The inductive hypothesis must be used explicitly in the proof for k+1. Look to extract the k-case from the k+1 expression.

归纳假设必须在证明 k+1 时显式使用。注意从 k+1 的表达式中提取出 k 的情形。

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