📚 Year 13 Edexcel Further Maths: Essential Formulas & Theorems Quick Reference | Year 13 爱德思进阶数学：公式定理速查手册
This concise yet comprehensive quick-reference guide covers the core formulas, theorems and techniques needed for the Year 13 Edexcel Further Mathematics A Level (9FM0). It brings together material from Core Pure Mathematics 2 and common optional units such as Further Pure 1 & 2, ensuring you have a one‑stop toolkit for revision, homework and exam preparation. Each section pairs key English statements with their Chinese equivalents, so that bilingual learners can strengthen understanding in both languages while mastering rigorous A‑level content.
这本简明而全面的速查手册涵盖了 Year 13 爱德思进阶数学 A Level (9FM0) 所需的核心公式、定理与方法。它整合了核心纯数学 2 以及常见的选修单元（如进阶纯数 1&2）的内容，为你提供复习、作业和备考的一站式工具箱。每个要点均以中英双语配对呈现，帮助双语学习者在掌握严谨的 A‑level 知识的同时加深两种语言的理解。
1. Complex Numbers – Euler & De Moivre | 复数 – 欧拉公式与棣莫弗定理
Euler’s formula links exponential and trigonometric forms: eiθ = cos θ + i sin θ. Hence any complex number can be written as z = r eiθ with r = |z| and θ = arg(z), where the principal argument satisfies −π < θ ≤ π.
欧拉公式将指数形式与三角形式联系起来：eiθ = cos θ + i sin θ。因此任何复数都可写作 z = r eiθ，其中 r = |z|，θ = arg(z)，主辐角满足 −π < θ ≤ π。
De Moivre’s theorem: (cos θ + i sin θ)n = cos(nθ) + i sin(nθ) for integer n. Together with the modulus, [r(cos θ + i sin θ)]n = rn(cos nθ + i sin nθ). It is used to find powers, multiple‑angle identities and n‑th roots.
棣莫弗定理：(cos θ + i sin θ)n = cos(nθ) + i sin(nθ)（n 为整数）。考虑模长则 [r(cos θ + i sin θ)]n = rn(cos nθ + i sin nθ)。该定理用于求幂、多倍角恒等式以及 n 次方根。
n‑th roots of unity: the solutions of zn = 1 are z = e2πik/n, k = 0, 1, 2, …, n−1. They lie evenly spaced on the unit circle. The sum of all n‑th roots of unity is zero.
单位的 n 次方根：zn = 1 的解为 z = e2πik/n, k = 0, 1, 2, …, n−1，它们在单位圆上等距分布。所有 n 次方根之和为零。
2. Series & Maclaurin Expansions | 级数与麦克劳林展开
The method of differences simplifies sums by expressing a term as f(r) − f(r+1). Standard Maclaurin series (valid for |x| < R):
差分法通过将通项写作 f(r) − f(r+1) 以化简和式。标准麦克劳林级数（在收敛域内）：
ex = 1 + x + x²/2! + x³/3! + …
ex = 1 + x + x²/2! + x³/3! + …
sin x = x − x³/3! + x⁵/5! − …
sin x = x − x³/3! + x⁵/5! − …
cos x = 1 − x²/2! + x⁴/4! − …
cos x = 1 − x²/2! + x⁴/4! − …
ln(1 + x) = x − x²/2 + x³/3 − x⁴/4 + …
ln(1 + x) = x − x²/2 + x³/3 − x⁴/4 + …
(1 + x)n = 1 + nx + n(n−1)x²/2! + …
(1 + x)n = 1 + nx + n(n−1)x²/2! + …
To sum series using known expansions, manipulate the general term to match a standard series, possibly differentiating or integrating.
利用已知展开求级数和时，需将通项变形以匹配标准级数，有时需通过微分或积分处理。
3. Further Calculus – Volumes, Improper Integrals & Mean Value | 进阶微积分 – 体积、瑕积分与均值
Volume of revolution about the x‑axis: V = π ∫ab y² dx (for y = f(x)). About the y‑axis: V = π ∫cd x² dy. For parametric curves (x(t), y(t)), volume about the x‑axis is V = π ∫t₁t₂ y² (dx/dt) dt.
绕 x 轴旋转体积：V = π ∫ab y² dx（y = f(x)）。绕 y 轴：V = π ∫cd x² dy。对于参数曲线 (x(t), y(t))，绕 x 轴体积为 V = π ∫t₁t₂ y² (dx/dt) dt。
Improper integrals: an integral with infinite limit(s) or an unbounded integrand. Evaluate by replacing the infinite limit with a variable, then take the limit. Convergence occurs if the limit is finite.
瑕积分：积分限含无穷大或被积函数无界。计算时用变量替代无穷限再取极限，若极限有限则积分收敛。
The mean value of a function f(x) on [a, b] is (1/(b−a)) ∫ab f(x) dx. For volumes of revolution, the mean square value appears.
函数 f(x) 在 [a, b] 上的平均值为 (1/(b−a)) ∫ab f(x) dx。旋转体问题中会出现均方值。
4. Polar Coordinates – Curves, Area & Tangents | 极坐标 – 曲线、面积与切线
A polar curve is given by r = f(θ). The area swept out from θ = α to θ = β is ½ ∫αβ r² dθ. Often you will find the area of a loop or between two curves.
极坐标曲线由 r = f(θ) 给出。从 θ = α 到 θ = β 扫过的面积为 ½ ∫αβ r² dθ。常需求出一个环的面积或两条曲线之间的面积。
To find the gradient of a tangent, use the chain rule: dy/dx = (dy/dθ) / (dx/dθ), where x = r cos θ, y = r sin θ. Substitute r and dr/dθ to find tangents parallel to the initial line or perpendicular.
求切线斜率时，利用链式法则：dy/dx = (dy/dθ) / (dx/dθ)，其中 x = r cos θ, y = r sin θ。代入 r 与 dr/dθ 即可找出平行或垂直于极轴的切线。
Common curves: cardioid r = a(1 + cos θ), limacon, rose curves r = a sin(nθ). Sketching requires knowledge of maxima, minima and symmetries.
常见曲线：心形线 r = a(1 + cos θ)、利马松、玫瑰线 r = a sin(nθ)。绘图需掌握最值点与对称性。
5. Hyperbolic Functions – Definitions, Identities & Calculus | 双曲函数 – 定义、恒等式与微积分
Definitions: sinh x = (ex − e−x)/2, cosh x = (ex + e−x)/2, tanh x = sinh x / cosh x. Their inverses are arsinh, arcosh, artanh, with logarithmic forms: arsinh x = ln(x + √(x² + 1)); arcosh x = ln(x + √(x² − 1)) (x ≥ 1); artanh x = ½ ln((1+x)/(1−x)) (|x| < 1).
定义：sinh x = (ex − e−x)/2，cosh x = (ex + e−x)/2，tanh x = sinh x / cosh x。其反函数为 arsinh, arcosh, artanh，具有对数形式：arsinh x = ln(x + √(x² + 1))；arcosh x = ln(x + √(x² − 1)) (x ≥ 1)；artanh x = ½ ln((1+x)/(1−x)) (|x| < 1)。
Osborn’s rule: any trigonometric identity can be converted to a hyperbolic identity by replacing cos ↔ cosh and sin ↔ i sinh, then whenever a product of two sines occurs, change the sign. Key identity: cosh²x − sinh²x = 1.
奥斯本法则：将三角恒等式中的 cos 换成 cosh，sin 换成 i sinh，每当出现两个正弦的乘积时改变符号，即可得到双曲恒等式。核心恒等式：cosh²x − sinh²x = 1。
Derivatives: d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech²x. Integrals: ∫ sinh x dx = cosh x + C, ∫ cosh x dx = sinh x + C. For inverse hyperbolic functions: 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) = sech²x。积分：∫ sinh x dx = cosh x + C，∫ cosh x dx = sinh x + C。反双曲函数的导数：d/dx (arsinh x) = 1/√(x²+1)，d/dx (arcosh x) = 1/√(x²−1)，d/dx (artanh x) = 1/(1−x²)。
6. First-Order Differential Equations – Integrating Factor | 一阶微分方程 – 积分因子
A linear first-order ODE is of the form dy/dx + P(x) y = Q(x). The integrating factor (IF) is μ(x) = e∫P(x) dx. Multiply both sides by μ to obtain d/dx (μ y) = μ Q(x), then integrate.
一阶线性常微分方程的标准形式为 dy/dx + P(x) y = Q(x)。积分因子为 μ(x) = e∫P(x) dx。两边同乘 μ 可得 d/dx (μ y) = μ Q(x)，再积分求解。
For separable equations g(y) dy = f(x) dx, integrate directly. For homogeneous type dy/dx = f(y/x), substitute v = y/x. Always state the general solution and find particular solutions using initial conditions.
对于可分离方程 g(y) dy = f(x) dx，直接积分。对于齐次型 dy/dx = f(y/x)，设 v = y/x 进行代换。始终给出通解，并利用初值条件求特解。
7. Second-Order Differential Equations – Constant Coefficients | 二阶微分方程 – 常系数
Homogeneous equation: a d²y/dx² + b dy/dx + c y = 0. Auxiliary equation: a m² + b m + c = 0. Roots real and distinct (m₁, m₂): y = A em₁x + B em₂x. Repeated root m: y = (A + Bx) emx. Complex roots α ± iβ: y = eαx (A cos βx + B sin βx).
齐次方程：a d²y/dx² + b dy/dx + c y = 0。特征方程：a m² + b m + c = 0。实根相异时 y = A em₁x + B em₂x；重根 m 时 y = (A + Bx) emx；共轭复根 α ± iβ 时 y = eαx (A cos βx + B sin βx)。
Inhomogeneous equation: a d²y/dx² + b dy/dx + c y = f(x). General solution = complementary function (CF) + particular integral (PI). Find PI by trying a form similar to f(x) (polynomial, exponential, sin/cos, or a combination) with undetermined coefficients, substituting into the ODE. If f(x) overlaps with CF, multiply by x.
非齐次方程：a d²y/dx² + b dy/dx + c y = f(x)。通解 = 余函数 (CF) + 特解 (PI)。根据 f(x) 的形式（多项式、指数、正弦/余弦或组合）设定待定系数形式的试探解，代入方程确定。若 f(x) 与 CF 项重叠，需乘以 x。
8. Vectors in 3D – Cross Product, Planes & Distances | 三维向量 – 向量积、平面与距离
Vector (cross) product: a × b is a vector perpendicular to both a and b, magnitude |a||b| sin θ. In components, if a = a₁i + a₂j + a₃k, b = b₁i + b₂j + b₃k, then a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k. Area of triangle = ½ |a × b|.
向量积（叉积）：a × b 垂直于 a 与 b，其模为 |a||b| sin θ。分量形式下，若 a = a₁i + a₂j + a₃k，b = b₁i + b₂j + b₃k，则 a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k。三角形的面积 = ½ |a × b|。
Equation of a line: r = a + t b (a is a point on the line, b is the direction vector). Equation of a plane: r·n = d (n is a normal vector) or r = a + s u + t v. The scalar triple product a·(b × c) gives the volume of a parallelepiped; it is zero if vectors are coplanar.
直线方程：r = a + t b（a 为线上一点，b 为方向向量）。平面方程：r·n = d（n 为法向量）或 r = a + s u + t v。标量三重积 a·(b × c) 给出平行六面体的体积；若三向量共面则该积为零。
Shortest distance from a point P with position vector p to the line r = a + t b: d = |(p − a) × b| / |b|. Distance from point to plane r·n = d: |p·n − d| / |n|. Intersections: solve parametric equations simultaneously.
点 P（位矢 p）到直线 r = a + t b 的最短距离：d = |(p − a) × b| / |b|。点到平面 r·n = d 的距离：|p·n − d| / |n|。求交
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