📚 Year 13 AQA Further Maths Formula and Theorem Quick Reference | AQA进阶数学公式定理速查手册
This rapid-reference handbook compiles essential formulae, theorems, and definitions for the Year 13 AQA Further Mathematics specification. It is designed as a revision aid for pure topics, covering complex numbers, matrices, hyperbolic functions, polar coordinates, advanced calculus, series, vectors, and proof techniques. Each section presents key results with concise explanations so you can quickly refresh your memory before assessments.
本速查手册汇编了AQA进阶数学Year 13阶段的核心公式、定理与定义，涵盖复数、矩阵、双曲函数、极坐标、进阶微积分、级数、向量与证明方法等纯数专题。每个要点先给出英文说明，紧跟中文解释，帮助高效复习与快速查阅。
1. Complex Numbers | 复数
The imaginary unit i satisfies i² = –1. A complex number can be written in Cartesian form as z = a + bi, where a, b ∈ ℝ.
虚数单位 i 满足 i² = –1。复数可写为笛卡儿形式 z = a + bi，其中 a, b 为实数。
The complex conjugate of z is z* = a – bi. The modulus is |z| = √(a² + b²). The argument arg z = θ is defined such that tan θ = b/a, with θ measured from the positive real axis.
共轭复数为 z* = a – bi。模为 |z| = √(a² + b²)。辐角 arg z = θ 满足 tan θ = b/a，θ 从正实轴开始度量。
Polar form: z = r(cos θ + i sin θ) = r cis θ, where r = |z| and θ = arg z.
极坐标形式：z = r(cos θ + i sin θ) = r cis θ，其中 r = |z|，θ = arg z。
Euler’s relation: e^(iθ) = cos θ + i sin θ.
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.
德莫弗定理：对整数 n，(cos θ + i sin θ)^n = cos nθ + i sin nθ。
Multiplication and division: |z₁z₂| = |z₁||z₂|, arg(z₁z₂) = arg z₁ + arg z₂; |z₁/z₂| = |z₁|/|z₂|, arg(z₁/z₂) = arg z₁ – arg z₂ (adding multiples of 2π as needed).
乘除法：|z₁z₂| = |z₁||z₂|，arg(z₁z₂) = arg z₁ + arg z₂；|z₁/z₂| = |z₁|/|z₂|，arg(z₁/z₂) = arg z₁ – arg z₂（必要时加上2π的整数倍）。
Roots of unity: The nth roots of unity are given by z = e^(2πik/n), k = 0, 1, …, n–1.
单位根：n次单位根由 z = e^(2πik/n) 给出，k = 0, 1, …, n–1。
Complex roots of real polynomials occur in conjugate pairs.
实系数多项式的复根成对共轭出现。
2. Matrices and Linear Transformations | 矩阵与线性变换
Matrix multiplication is defined only when the number of columns of the first equals the number of rows of the second. If A is m×n and B is n×p, then AB is m×p.
矩阵乘法仅在第一个矩阵的列数等于第二个矩阵的行数时有定义。若 A 为 m×n，B 为 n×p，则 AB 为 m×p。
The identity matrix I has 1s on the main diagonal and 0s elsewhere, and AI = IA = A for conformable matrices.
单位矩阵 I 的主对角线为1，其余为0，满足 AI = IA = A。
For a 2×2 matrix A = [a b; c d], the determinant is det A = ad – bc.
对于 2×2 矩阵 A = [a b; c d]，行列式为 det A = ad – bc。
The inverse exists if det A ≠ 0, and A⁻¹ = (1/det A)[d –b; –c a].
逆矩阵存在当 det A ≠ 0，且 A⁻¹ = (1/det A)[d –b; –c a]。
The determinant of a 3×3 matrix can be found by expansion or Sarrus’ rule.
3×3 矩阵的行列式可通过展开或萨吕法则求得。
Eigenvalues λ satisfy det(A – λI) = 0. An eigenvector x satisfies Ax = λx, x ≠ 0.
特征值 λ 满足特征方程 det(A – λI) = 0。特征向量 x 满足 Ax = λx，x ≠ 0。
The matrix of a linear transformation in ℝ² can represent reflections, rotations, enlargements, and shears. For rotation by θ anticlockwise: [cos θ –sin θ; sin θ cos θ].
ℝ² 中的线性变换矩阵可表示反射、旋转、放大和剪切。逆时针旋转 θ 的矩阵为 [cos θ –sin θ; sin θ cos θ]。
Reflection in the line y = x has matrix [0 1; 1 0]; reflection in the x-axis is [1 0; 0 –1].
关于直线 y = x 的反射矩阵为 [0 1; 1 0]；关于 x 轴的反射为 [1 0; 0 –1]。
3. Roots of Polynomials | 多项式根
For a quadratic equation ax² + bx + c = 0 with roots α, β: sum α + β = –b/a, product αβ = c/a.
二次方程 ax² + bx + c = 0 的两根为 α, β，有 根的和 α + β = –b/a，根的积 αβ = c/a。
For a cubic ax³ + bx² + cx + d = 0 with roots α, β, γ: Σα = –b/a, Σαβ = c/a, αβγ = –d/a.
三次方程 ax³ + bx² + cx + d = 0 的三根为 α, β, γ，有 Σα = –b/a，Σαβ = c/a，αβγ = –d/a。
For a quartic ax⁴ + bx³ + cx² + dx + e = 0: Σα = –b/a, Σαβ = c/a, Σαβγ = –d/a, αβγδ = e/a.
四次方程 ax⁴ + bx³ + cx² + dx + e = 0：Σα = –b/a，Σαβ = c/a，Σαβγ = –d/a，αβγδ = e/a。
Applications include finding unknown coefficients, forming new equations with transformed roots, and solving symmetric systems.
应用包括求未知系数、构造具有变换根的新方程以及求解对称方程组。
4. Sequences and Series | 数列与级数
Arithmetic sequence: n-th term uₙ = a + (n – 1)d. Sum to n terms: Sₙ = n/2 [2a + (n – 1)d] = n/2 (a + l).
等差数列：第n项 uₙ = a + (n – 1)d。前n项和 Sₙ = n/2 [2a + (n – 1)d] = n/2 (a + l)。
Geometric sequence: uₙ = ar^(n–1). Sum to n terms: Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1.
等比数列：uₙ = ar^(n–1)。前n项和 Sₙ = a(1 – rⁿ)/(1 – r)，r ≠ 1。
Infinite geometric series: if |r| < 1, S∞ = a/(1 – r).
无穷等比级数：若 |r| < 1，S∞ = a/(1 – r)。
The method of differences allows summation by cancelling terms when uₙ = f(n) – f(n–1).
差分法通过 uₙ = f(n) – f(n–1) 的抵消来求和。
Maclaurin series: f(x) = Σ [f⁽ⁿ⁾(0)/n!] xⁿ = f(0) + f'(0)x + f”(0)x²/2! + …
麦克劳林级数：f(x) = Σ [f⁽ⁿ⁾(0)/n!] xⁿ = f(0) + f'(0)x + f”(0)x²/2! + …
Standard expansions: 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 – … for |x| < 1.
标准展开：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 – …， |x| < 1。
5. 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。
Fundamental identity: cosh² x – sinh² x = 1.
基本恒等式：cosh² x – sinh² x = 1。
Double-argument formulae: sinh 2x = 2 sinh x cosh x, cosh 2x = cosh² x + sinh² x = 2 cosh² x – 1 = 1 + 2 sinh² x.
倍角公式：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)) for x ≥ 1, artanh x = ½ 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 = 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 where sech x = 1/cosh x.
导数：d/dx(sinh x) = cosh x，d/dx(cosh x) = sinh x，d/dx(tanh x) = sech² x，其中 sech x = 1/cosh x。
6. Polar Coordinates | 极坐标
Polar coordinates (r, θ) relate to Cartesian by x = r cos θ, y = r sin θ. Then r = √(x²+y²) and θ = arctan(y/x) with attention to quadrant.
极坐标 (r, θ) 与笛卡儿坐标转换：x = r cos θ，y = r sin θ；r = √(x²+y²)，θ = arctan(y/x) 并注意象限。
For a curve r = f(θ), the gradient is dy/dx = (f'(θ) sin θ + f(θ) cos θ) / (f'(θ) cos θ – f(θ) sin θ), where f'(θ) = dr/dθ.
对于曲线 r = f(θ)，切线斜率 dy/dx = (f'(θ) sin θ + f(θ) cos θ) / (f'(θ) cos θ – f(θ) sin θ)，其中 f'(θ) = dr/dθ。
Area enclosed by a polar curve between θ = α and θ = β is A = ½ ∫_α^β r² dθ.
极坐标曲线在 θ = α 到 θ = β 之间围成的面积为 A = ½ ∫_α^β r² dθ。
Common curves: circle r = a, cardioid r = a(1+cos θ), rose curves r = a cos nθ or r = a sin nθ.
常见曲线：圆 r = a，心形线 r = a(1+cos θ)，玫瑰线 r = a cos nθ 或 r = a sin nθ。
7. Further Differentiation | 进阶微分
Implicit differentiation: if f(x, y) = 0, differentiate both sides with respect to x, treating y as a function of x. Then solve for dy/dx.
隐函数微分：对方程 f(x, y) = 0 两边关于 x 求导，将 y 视作 x 的函数，然后解出 dy/dx。
Parametric differentiation: if x = x(t), y = y(t), then dy/dx = (dy/dt) / (dx/dt). The second derivative is d²y/dx² = d/dx(dy/dx) = (d/dt(dy/dx)) / (dx/dt).
参数微分：若 x = x(t), y = y(t)，则 dy/dx = (dy/dt) / (dx/dt)。二阶导数 d²y/dx² = d/dx(dy/dx) = (d/dt(dy/dx)) / (dx/dt)。
Derivatives of inverse trigonometric functions: d/dx(arcsin x) = 1/√(1–x²), d/dx(arccos x) = –1/√(1–x²), d/dx(arctan x) = 1
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