AQA Year 13 Further Maths Quick-Reference Formula & Theorem Handbook | Year 13 AQA 进阶数学：公式定理速查手册

📚 AQA Year 13 Further Maths Quick-Reference Formula & Theorem Handbook | Year 13 AQA 进阶数学：公式定理速查手册

This concise handbook brings together all the essential formulas, theorems, and key results you need for the AQA Year 13 Further Mathematics course. It covers pure topics from complex numbers to vectors, serving as a rapid revision aid and a handy reference during practice. Each section is presented in both English and Chinese to support bilingual learners.

这本速查手册汇集了 AQA Year 13 进阶数学课程所需的全部核心公式、定理和重要结论。内容涵盖从复数到向量的纯数学主题，既可作为快速复习资料，又便于平时练习查阅。每个部分都以中英双语呈现，帮助双语学习者掌握知识。

1. Complex Numbers | 复数

Complex numbers can be expressed in Cartesian form z = x + iy or in polar form z = r(cosθ + i sinθ). By Euler’s formula, this is equivalent to z = r e^iθ. The modulus is |z| = √(x² + y²) and the argument, arg z, is arctan(y/x) adjusted by quadrant.

复数可用直角坐标形式 z = x + iy 或极坐标形式 z = r(cosθ + i sinθ) 表示。根据欧拉公式，这等价于 z = r e^iθ。模为 |z| = √(x² + y²)，辐角 arg z 为 arctan(y/x) 并按象限调整。

De Moivre’s theorem states that (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) for integer n. It can be used to find powers and roots of complex numbers. The n distinct n‑th roots of unity are given by e^2kπi/n for k = 0,1,…,n−1.

棣莫弗定理指出，对整数 n，有 (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)。该定理可用于计算复数的幂与方根。n 个不同的 n 次单位根为 e^2kπi/n，k = 0,1,…,n−1。

Multiplication and division in polar form: if z₁ = r₁e^iθ₁ and z₂ = r₂e^iθ₂, then z₁z₂ = r₁r₂ e^{i(θ₁+θ₂)} and z₁/z₂ = (r₁/r₂) e^{i(θ₁−θ₂)}. The conjugate of z = x + iy is z* = x − iy, and zz* = |z|².

极坐标形式的乘除运算：若 z₁ = r₁e^iθ₁，z₂ = r₂e^iθ₂，则 z₁z₂ = r₁r₂ e^{i(θ₁+θ₂)}，且 z₁/z₂ = (r₁/r₂) e^{i(θ₁−θ₂)}。z = x + iy 的共轭为 z* = x − iy，满足 zz* = |z|²。

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2. Matrices and Linear Transformations | 矩阵与线性变换

The inverse of a 2×2 matrix M = (a b; c d) is M⁻¹ = 1/(ad − bc) (d −b; −c a), provided the determinant det(M) = ad − bc is non‑zero. For a transformation represented by M, points are mapped by x’ = M x.

2×2 矩阵 M = (a b; c d) 的逆矩阵为 M⁻¹ = 1/(ad − bc) (d −b; −c a)，前提是行列式 det(M) = ad − bc ≠ 0。对于 M 表示的线性变换，点的映射为 x’ = M x。

Eigenvalues λ satisfy det(A − λI) = 0. Corresponding eigenvectors v are non‑zero vectors such that A v = λ v. For 2×2 matrices, the sum of eigenvalues equals the trace tr(A) = a + d, and the product equals det(A).

特征值 λ 满足 det(A − λI) = 0。相应的特征向量 v 是非零向量，满足 A v = λ v。对于 2×2 矩阵，特征值之和等于迹 tr(A) = a + d，特征值之积等于 det(A)。

Invariant lines: a line through the origin with direction vector v is invariant under transformation M if M v is parallel to v, i.e. v is an eigenvector. A line not through the origin may be invariant if it is mapped to itself; its image must satisfy the line equation.

不变直线：过原点且方向向量为 v 的直线在变换 M 下不变，当且仅当 M v 平行于 v，即 v 是特征向量。不过原点的直线若被映射到自身也可能是不变直线，其像必须满足同一直线方程。

Diagonalisation: if a 2×2 matrix A has two distinct eigenvalues λ₁, λ₂ with eigenvectors v₁, v₂, then A = PDP⁻¹, where P = (v₁ v₂) and D = diag(λ₁, λ₂). This allows easy computation of powers Aⁿ = PDⁿP⁻¹.

对角化：若 2×2 矩阵 A 有两个相异的特征值 λ₁, λ₂ 及对应的特征向量 v₁, v₂，则 A = PDP⁻¹，其中 P = (v₁ v₂)，D = diag(λ₁, λ₂)。由此可简便计算 A 的幂 Aⁿ = PDⁿP⁻¹。

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3. Polar Coordinates | 极坐标

In polar coordinates, a point is given by (r, θ) where x = r cosθ, y = r sinθ. The region is described by curves r = f(θ). The area bounded by r = f(θ) and the half‑lines θ = α, θ = β is A = ½ ∫_α^β r² dθ.

在极坐标中，点的坐标为 (r, θ)，其中 x = r cosθ，y = r sinθ。区域由曲线 r = f(θ) 描述。曲线 r = f(θ) 与射线 θ = α, θ = β 所围面积为 A = ½ ∫_α^β r² dθ。

The arc length of a polar curve from θ = α to θ = β is s = ∫_α^β √(r² + (dr/dθ)²) dθ. Symmetry about the initial line (θ = 0) or the pole can often simplify calculations.

极坐标曲线的弧长（从 θ = α 到 θ = β）为 s = ∫_α^β √(r² + (dr/dθ)²) dθ。利用关于初始线（θ = 0）或极点的对称性常能简化计算。

Common curves: r = a gives a circle; r = 2a cosθ gives a circle of radius a passing through the pole; r = a(1 ± cosθ) is a cardioid. The tangent at the pole occurs where r = 0.

常见曲线：r = a 表示圆；r = 2a cosθ 表示半径为 a 且过极点的圆；r = a(1 ± cosθ) 是心脏线。极点处的切线出现在 r = 0 的位置。

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4. Hyperbolic Functions | 双曲函数

Hyperbolic functions are defined in terms of exponentials: sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x. Their graphs are distinct from trig functions but share analogous identities.

双曲函数用指数函数定义：sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x。它们的图像与三角函数不同，但具有类似的恒等式。

Key identities: cosh²x − sinh²x = 1; sinh(2x) = 2 sinh x cosh x; cosh(2x) = cosh²x + sinh²x = 2cosh²x − 1 = 2sinh²x + 1. Also, tanh(2x) = 2 tanh x / (1 + tanh²x).

重要恒等式：cosh²x − sinh²x = 1；sinh(2x) = 2 sinh x cosh x；cosh(2x) = cosh²x + sinh²x = 2cosh²x − 1 = 2sinh²x + 1。此外，tanh(2x) = 2 tanh x / (1 + tanh²x)。

Inverse hyperbolic functions are logarithmic: arsinh x = ln(x + √(x²+1)), for all x; arcosh x = ln(x + √(x²−1)), for x ≥ 1; artanh x = ½ ln((1+x)/(1−x)), for |x| < 1. Differentiate these using the chain rule.

反双曲函数为对数形式：arsinh x = ln(x + √(x²+1))，定义域为全体实数；arcosh x = ln(x + √(x²−1))，x ≥ 1；artanh x = ½ ln((1+x)/(1−x))，|x| < 1。求导时可利用链式法则。

Derivatives: d/dx (sinh x) = cosh x; d/dx (cosh x) = sinh x; d/dx (tanh x) = sech²x. The reciprocals are sech x = 1/cosh x, cosech x = 1/sinh x, coth x = 1/tanh x.

导数：d/dx (sinh x) = cosh x；d/dx (cosh x) = sinh x；d/dx (tanh x) = sech²x。倒数双曲函数：sech x = 1/cosh x，cosech x = 1/sinh x，coth x = 1/tanh x。

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5. Differential Equations | 微分方程

First‑order linear ODEs have the form dy/dx + P(x)y = Q(x). The integrating factor is I = e^{∫P dx}. Multiply through by I to obtain d/dx (I y) = I Q, then integrate.

一阶线性常微分方程形如 dy/dx + P(x)y = Q(x)。积分因子为 I = e^{∫P dx}。方程两边乘以 I 得 d/dx (I y) = I Q，再积分即得解。

Second‑order linear ODEs with constant coefficients: a d²y/dx² + b dy/dx + c y = f(x). The auxiliary equation is am² + bm + c = 0. If roots m₁, m₂ are real and distinct, the complementary function is y_c = Ae^m₁x + Be^m₂x. For repeated root m, y_c = (A + Bx)e^mx. For complex roots α ± βi, y_c = e^αx(C cos βx + D sin βx).

二阶常系数线性常微分方程：a d²y/dx² + b dy/dx + c y = f(x)。辅助方程为 am² + bm + c = 0。若根 m₁, m₂ 为相异实数，补函数为 y_c = Ae^m₁x + Be^m₂x。重根 m 时 y_c = (A + Bx)e^mx。复根 α ± βi 时 y_c = e^αx(C cos βx + D sin βx)。

Particular integral y_p is found by undetermined coefficients: for f(x) = polynomial, try a polynomial of the same degree; for f(x) = ke^px, try Ce^px; for f(x) = k sin ωx or k cos ωx, try C sin ωx + D cos ωx. If the trial form appears in y_c, multiply by x.

特解 y_p 用待定系数法求得：若 f(x) 为多项式，则假设同次多项式；若 f(x) = ke^px，则设 Ce^px；若 f(x) = k sin ωx 或 k cos ωx，则设 C sin ωx + D cos ωx。若假设形式已出现在补函数中，需乘以 x。

The general solution is y = y_c + y_p. Initial or boundary conditions determine the arbitrary constants. For Euler equations of the type ax² d²y/dx² + bx dy/dx + c y = f(x), use the substitution x = eᵗ to reduce to a constant‑coefficient equation.

通解为 y = y_c + y_p。初始条件或边界条件用于确定任意常数。对于欧拉方程 ax² d²y/dx² + bx dy/dx + c y = f(x)，可使用代换 x = eᵗ 化为常系数方程。

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6. Maclaurin Series | 麦克劳林级数

Maclaurin’s expansion: f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … . It is valid for x within the interval of convergence.

麦克劳林展开式：f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … 。该展开式在收敛区间内成立。

Standard expansions: eˣ = 1 + x + x²/2! + x³/3! + …, for all x; sin x = x − x³/3! + x⁵/5! − …, for all x; cos x = 1 − x²/2! + x⁴/4! − …, for all x; ln(1 + x) = x − x²/2 + x³/3 − …, for −1 < x ≤ 1; (1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + …, for |x| < 1.

标准展开式：eˣ = 1 + x + x²/2! + x³/3! + …，对全体 x 成立；sin x = x − x³/3! + x⁵/5! − …，对全体 x 成立；cos x = 1 − x²/2! + x⁴/4! − …，对全体 x 成立；ln(1 + x) = x − x²/2 + x³/3 − …，收敛域 −1 < x ≤ 1；(1 + x)ⁿ = 1 + nx + n(n−1)x²/2! + …，|x| < 1。

Series for composite functions can be obtained by substituting into known expansions, multiplying series, or differentiating term‑by

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