A-Level Further Mathematics: Formula Summary Handbook | A-Level 进阶数学：公式汇总手册

📚 A-Level Further Mathematics: Formula Summary Handbook | A-Level 进阶数学：公式汇总手册

This handbook is a comprehensive collection of the key formulas encountered in A-Level Further Mathematics (pure content). Each section includes essential results and identities, presented in a compact and exam-friendly format. Use it as a quick reference while practising past papers or revising core topics.

本手册全面汇总了 A-Level 进阶数学（纯数学部分）的核心公式。每一节都以紧凑、适合考试复习的形式列出了必备结论与恒等式。在刷真题或复习重点主题时，可将其用作快速查阅工具。

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1. Complex Numbers | 复数

The modulus of z = x + iy is |z| = √(x² + y²) and the argument θ satisfies tanθ = y/x. Complex numbers can be expressed in polar form z = r(cosθ + i sinθ) or exponential form z = re^iθ.

复数 z = x + iy 的模为 |z| = √(x² + y²)，辐角 θ 满足 tanθ = y/x。复数可表示为极坐标形式 z = r(cosθ + i sinθ) 或指数形式 z = re^iθ。

e^iθ = cosθ + i sinθ

Euler’s formula connects exponential and trigonometric functions. The conjugate is z* = x – iy = re^–iθ, with z z* = |z|².

欧拉公式联系指数与三角函数。共轭复数为 z* = x – iy = re^–iθ，且 z z* = |z|²。

Multiplication and division in polar form: z₁z₂ = r₁r₂ e^{i(θ₁+θ₂)}, z₁/z₂ = (r₁/r₂) e^{i(θ₁–θ₂)}. De Moivre’s theorem states (cosθ + i sinθ)ⁿ = cos nθ + i sin nθ, useful for finding powers and roots.

极坐标形式下的乘除法：z₁z₂ = r₁r₂ e^{i(θ₁+θ₂)}，z₁/z₂ = (r₁/r₂) e^{i(θ₁–θ₂)}。棣莫弗定理给出 (cosθ + i sinθ)ⁿ = cos nθ + i sin nθ，常用于计算幂与根。

The n-th roots of a complex number are given by z_k = r^1/n e^i(θ+2kπ)/n, k = 0,1,…,n−1, providing n equally spaced points on a circle of radius r^1/n.

复数的 n 次根由 z_k = r^1/n e^i(θ+2kπ)/n 给出，k = 0,1,…,n−1，在半径为 r^1/n 的圆上产生 n 个等距点。

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2. Matrices and Transformations | 矩阵与变换

For a 2×2 matrix A = [a b; c d], the determinant is det A = ad – bc. The matrix is invertible if and only if det A ≠ 0, and the inverse is A^–1 = (1/det A) [d –b; –c a].

对于 2×2 矩阵 A = [a b; c d]，行列式为 det A = ad – bc。矩阵可逆当且仅当 det A ≠ 0，逆矩阵为 A^–1 = (1/det A) [d –b; –c a]。

Matrix multiplication: the product AB requires the columns of A to match the rows of B. (AB)^–1 = B^–1 A^–1 and (AB)^T = B^T A^T.

矩阵乘法：乘积 AB 要求 A 的列数与 B 的行数相同。(AB)^–1 = B^–1 A^–1 且 (AB)^T = B^T A^T。

Eigenvalues λ satisfy the characteristic equation det(A – λI) = 0. For each λ, the eigenvector x is a non-zero solution to (A – λI)x = 0.

特征值 λ 满足特征方程 det(A – λI) = 0。对于每个 λ，特征向量 x 是 (A – λI)x = 0 的非零解。

Common linear transformations in the plane include rotation by θ through [cosθ –sinθ; sinθ cosθ], reflection in the line y = xtanα given by [cos2α sin2α; sin2α –cos2α], and scaling.

平面上常见的线性变换包括：旋转 θ 角的矩阵 [cosθ –sinθ; sinθ cosθ]；关于直线 y = xtanα 的反射矩阵为 [cos2α sin2α; sin2α –cos2α]；以及缩放变换。

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3. Vectors and Planes | 向量与平面

Dot product: a·b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃. Two vectors are perpendicular if a·b = 0. Cross product: a×b gives a vector perpendicular to both, with magnitude |a×b| = |a||b| sinθ.

点积：a·b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃。若 a·b = 0，则两向量垂直。叉积：a×b 得到垂直于 a 与 b 的向量，其大小为 |a×b| = |a||b| sinθ。

The vector equation of a line is r = a + λb, where a is a point on the line and b is the direction vector. The plane equation can be written as r·n = d, where n is the normal vector and d = a·n.

直线的向量方程为 r = a + λb，其中 a 是线上一点，b 是方向向量。平面方程可写为 r·n = d，n 是法向量，d = a·n。

Distance from a point P with position vector p to the plane r·n = d is |p·n – d| / |n|. The acute angle between two planes with normals n₁ and n₂ is given by cosθ = |n₁·n₂| / (|n₁||n₂|).

点 P（位置向量 p）到平面 r·n = d 的距离为 |p·n – d| / |n|。两平面（法向量 n₁, n₂）的夹角由 cosθ = |n₁·n₂| / (|n₁||n₂|) 给出。

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4. Hyperbolic Functions | 双曲函数

Definitions: sinh x = (e^x – e^–x)/2, cosh x = (e^x + e^–x)/2, tanh x = sinh x / cosh x. The fundamental identity is cosh²x – sinh²x = 1.

定义：sinh x = (e^x – e^–x)/2，cosh x = (e^x + e^–x)/2，tanh x = sinh x / cosh x。基本恒等式为 cosh²x – sinh²x = 1。

Derivatives: d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech²x. Inverse hyperbolic functions have logarithmic forms, e.g. arsinh x = ln(x + √(x²+1)).

导数：d/dx (sinh x) = cosh x，d/dx (cosh x) = sinh x，d/dx (tanh x) = sech²x。反双曲函数具有对数形式，例如 arsinh x = ln(x + √(x²+1))。

Osborn’s rule helps convert trigonometric identities to hyperbolic ones: change cos to cosh and sin to i sinh, then replace any product of two sines by – sinh².

奥斯本规则帮助将三角恒等式转化为双曲恒等式：将 cos 换为 cosh，sin 换为 i sinh，然后将两个正弦的乘积替换为 – sinh²。

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5. Polar Coordinates | 极坐标

Cartesian to polar: x = r cosθ, y = r sinθ; polar to Cartesian: r² = x² + y², θ = arctan(y/x) with care for quadrant. The curve is defined as r = f(θ).

直角坐标与极坐标互化：x = r cosθ，y = r sinθ；r² = x² + y²，θ = arctan(y/x)，须注意象限。曲线定义为 r = f(θ)。

The area enclosed by a polar curve from θ = α to θ = β is A = ½ ∫_α^β r² dθ. The arc length of a polar curve is s = ∫_α^β √(r² + (dr/dθ)²) dθ.

极坐标曲线从 θ = α 到 θ = β 所围面积为 A = ½ ∫_α^β r² dθ。弧长公式为 s = ∫_α^β √(r² + (dr/dθ)²) dθ。

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