📚 Year 13 WJEC Further Maths: Formula & Theorem Quick Reference Handbook | 高三年级 WJEC 进阶数学公式定理速查手册
This handbook provides a concise summary of essential formulas and theorems for the Year 13 WJEC Further Mathematics course. It is designed for quick revision before exams, covering core pure topics such as complex numbers, matrices, vectors, hyperbolic functions, calculus, differential equations, polar coordinates, series, and proof. Each section presents key results in bite‑sized bilingual explanations to reinforce understanding.
本手册为高三年级 WJEC 进阶数学课程的重要公式和定理提供简明总结。专为考前快速复习设计，涵盖核心纯数学主题，如复数、矩阵、向量、双曲函数、微积分、微分方程、极坐标、级数与证明。每节均以中英双语要点解释，强化理解。
1. Complex Numbers | 复数
For a complex number z = x + iy, the modulus is r = √(x² + y²) and the argument θ = arctan(y/x), taking care to choose the correct quadrant.
对于复数 z = x + iy，模为 r = √(x² + y²)，辐角为 θ = arctan(y/x)，需注意选择正确的象限。
Euler’s formula connects exponentials with trigonometry: eⁱᶿ = cos θ + i sin θ. From this, De Moivre’s theorem follows: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ for any integer n.
欧拉公式将指数与三角函数联系起来：eⁱᶿ = cos θ + i sin θ。由此可得棣莫弗定理：对于任意整数 n，有 (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。
The n‑th roots of unity are given by z = e^(2πi k/n), where k = 0, 1, 2, …, n−1. They are equally spaced on the unit circle.
n 次单位根由 z = e^(2πi k/n) 给出，其中 k = 0, 1, 2, …, n−1。它们在单位圆上等距分布。
If z = x + iy, its complex conjugate is z* = x − iy. Important properties: z + z* = 2x and z z* = x² + y² = |z|².
若 z = x + iy，其共轭复数为 z* = x − iy。重要性质：z + z* = 2x 且 z z* = x² + y² = |z|²。
2. Matrices and Determinants | 矩阵与行列式
For a 2×2 matrix A = [a b; c d], the determinant is det A = ad − bc. The inverse, provided det A ≠ 0, is A⁻¹ = 1/(ad−bc) [d −b; −c a].
对于 2×2 矩阵 A = [a b; c d]，行列式为 det A = ad − bc。若 det A ≠ 0，其逆矩阵为 A⁻¹ = 1/(ad−bc) [d −b; −c a]。
A 3×3 determinant can be expanded along the first row: det A = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃, where Cᵢⱼ are cofactors.
3×3 行列式可按第一行展开：det A = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃，其中 Cᵢⱼ 是余子式。
For eigenvalues, solve det(A − λI) = 0 to obtain the characteristic equation. Eigenvectors satisfy Av = λv.
求特征值时，解特征方程 det(A − λI) = 0。特征向量满足 Av = λv。
In further work, matrices represent linear transformations. A rotation by θ anticlockwise is [cos θ −sin θ; sin θ cos θ], and a reflection in the x‑axis is [1 0; 0 −1].
在进阶内容中，矩阵表示线性变换。逆时针旋转 θ 的矩阵为 [cos θ −sin θ; sin θ cos θ]，关于 x 轴的反射为 [1 0; 0 −1]。
3. Vectors in 3D | 三维向量
The scalar (dot) product is a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃. Two vectors are perpendicular if and only if a·b = 0.
数量积（点积）为 a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。两向量垂直当且仅当 a·b = 0。
The vector (cross) product a×b yields a vector perpendicular to both a and b. Its magnitude |a×b| = |a||b| sin θ gives the area of the parallelogram. In components: a×b = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k.
向量积（叉积）a×b 给出一个垂直于 a 和 b 的向量。其模 |a×b| = |a||b| sin θ 为平行四边形的面积。分量形式：a×b = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k。
A line in 3D can be written as r = a + t b, where a is a point on the line and b is the direction vector. A plane has equation r·n = d or ax + by + cz = d, where n is a normal vector.
三维空间中的直线可写为 r = a + t b，其中 a 为线上一点，b 为方向向量。平面方程为 r·n = d 或 ax + by + cz = d，其中 n 是法向量。
The shortest distance from a point to a plane is |(r₀·n − d)| / |n|.
点到平面的最短距离为 |(r₀·n − d)| / |n|。
4. Hyperbolic Functions | 双曲函数
The definitions of the hyperbolic functions are: sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, and tanh x = sinh x / cosh x.
双曲函数的定义为：sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x。
Key identities parallel trigonometry: cosh² x − sinh² x = 1, sinh 2x = 2 sinh x cosh x, and cosh 2x = cosh² x + sinh² x.
关键恒等式与三角学对应：cosh² x − sinh² x = 1，sinh 2x = 2 sinh x cosh x，cosh 2x = cosh² x + sinh² x。
Derivatives: d/dx sinh x = cosh x, d/dx cosh x = sinh x, d/dx tanh x = sech² x. For inverse hyperbolic functions: d/dx arsinh x = 1/√(x²+1), d/dx arcosh x = 1/√(x²−1) (x>1).
导数：d/dx sinh x = cosh x，d/dx cosh x = sinh x，d/dx tanh x = sech² x。反双曲函数的导数：d/dx arsinh x = 1/√(x²+1)，d/dx arcosh x = 1/√(x²−1)（x>1）。
Integration often yields inverses: ∫ 1/√(a² + x²) dx = arsinh(x/a) + c, ∫ 1/√(x² − a²) dx = arcosh(x/a) + c (x>a).
积分常产生反双曲函数：∫ 1/√(a² + x²) dx = arsinh(x/a) + c，∫ 1/√(x² − a²) dx = arcosh(x/a) + c（x>a）。
5. Advanced Differentiation | 进阶微分
Implicit differentiation is used when y is not given explicitly. Differentiate both sides with respect to x, using the chain rule, and then solve for dy/dx.
当 y 未明确给出时使用隐函数微分。对等式两边关于 x 求导，运用链式法则，然后解出 dy/dx。
For parametric equations x = f(t), y = g(t), the derivative is dy/dx = (dy/dt) / (dx/dt). The second derivative is d²y/dx² = d/dx (dy/dt / dx/dt) = [d/dt (dy/dx)] / (dx/dt).
对于参数方程 x = f(t)，y = g(t)，导数为 dy/dx = (dy/dt) / (dx/dt)。二阶导数为 d²y/dx² = d/dx (dy/dt / dx/dt) = [d/dt (dy/dx)] / (dx/dt)。
Product rule for higher derivatives is given by Leibniz’s theorem: (uv)⁽ⁿ⁾ = Σⁿₖ₌₀ nCk u⁽ⁿ⁻ᵏ⁾ v⁽ᵏ⁾.
高阶乘积求导的莱布尼茨定理：(uv)⁽ⁿ⁾ = Σⁿₖ₌₀ nCk u⁽ⁿ⁻ᵏ⁾ v⁽ᵏ⁾。
Beyond standard results, the derivatives of inverse trig functions are vital: d/dx arcsin x = 1/√(1−x²), d/dx arctan x = 1/(1+x²).
除标准结果外，反三角函数的导数也很重要：d/dx arcsin x = 1/√(1−x²)，d/dx arctan x = 1/(1+x²)。
6. Advanced Integration | 进阶积分
Integration by parts is based on ∫ u dv = uv − ∫ v du. Choose u so that its differential simplifies the integrand.
分部积分法基于 ∫ u dv = uv − ∫ v du。选择 u 使得其微分能简化被积函数。
Standard integrals involving
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