A-Level Edexcel Further Maths Formula Handbook | 爱德思进阶数学公式汇总手册

📚 A-Level Edexcel Further Maths Formula Handbook | 爱德思进阶数学公式汇总手册

This comprehensive handbook brings together every essential formula from the Edexcel A-Level Further Mathematics specification. Whether you are tackling Core Pure 1, Core Pure 2, or the optional applied modules, this revision tool organises the key results in one clear, bilingual reference. Use it to check your recall, structure your note‑making, and build fluency in the symbolic language of advanced mathematics.

本手册汇集了爱德思 A-Level 进阶数学大纲中的全部核心公式。无论你正在学习纯数核心 1、核心 2，还是应用选修模块，这份中英对照的复习工具都能帮你把关键结论整理得井井有条。你可以用它来检验记忆、梳理笔记，并提升运用数学符号语言的熟练度。

1. Complex Numbers | 复数

A complex number is written as z = x + iy, where x, y ∈ ℝ and i² = −1. The real part is Re(z) = x, the imaginary part is Im(z) = y.

复数写作 z = x + iy，其中 x, y ∈ ℝ，i² = −1。实部为 Re(z) = x，虚部为 Im(z) = y。

The complex conjugate is z* = x − iy. Modulus: |z| = √(x² + y²). Argument: arg(z) = θ, where tan θ = y/x, with θ ∈ (−π, π].

共轭复数为 z* = x − iy。模：|z| = √(x² + y²)。辐角：arg(z) = θ，其中 tan θ = y/x，θ ∈ (−π, π]。

Multiplication and division in modulus‑argument form: if z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂), then z₁z₂ = r₁r₂[cos(θ₁+θ₂) + i sin(θ₁+θ₂)], z₁/z₂ = (r₁/r₂)[cos(θ₁−θ₂) + i sin(θ₁−θ₂)].

模－辐角形式的乘除法：若 z₁ = r₁(cos θ₁ + i sin θ₁)，z₂ = r₂(cos θ₂ + i sin θ₂)，则 z₁z₂ = r₁r₂[cos(θ₁+θ₂) + i sin(θ₁+θ₂)]，z₁/z₂ = (r₁/r₂)[cos(θ₁−θ₂) + i sin(θ₁−θ₂)]。

De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) for any integer n. Roots of unity: the nᵗʰ roots of unity are zₖ = cos(2kπ/n) + i sin(2kπ/n), k = 0, 1, …, n−1.

棣莫弗定理：(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) 对任意整数 n 成立。单位根：1 的 n 次方根为 zₖ = cos(2kπ/n) + i sin(2kπ/n)，k = 0, 1, …, n−1。

Euler’s relation: e^(iθ) = cos θ + i sin θ. Hence z = re^(iθ).

欧拉公式：e^(iθ) = cos θ + i sin θ。因此 z = re^(iθ)。

2. Matrices | 矩阵

The determinant of a 2×2 matrix M = [[a, b], [c, d]] is det M = ad − bc. For a 3×3 matrix, use Sarrus’ rule or expansion by minors.

2×2 矩阵 M = [[a, b], [c, d]] 的行列式为 det M = ad − bc。3×3 矩阵的行列式可通过沙路法则或按子式展开计算。

Inverse of a 2×2 matrix: M⁻¹ = (1/det M) [[d, −b], [−c, a]], provided det M ≠ 0.

2×2 矩阵的逆：M⁻¹ = (1/det M) [[d, −b], [−c, a]]，前提是 det M ≠ 0。

For a 3×3 matrix A, A⁻¹ = (1/det A) adj A, where adj A is the transpose of the cofactor matrix.

3×3 矩阵 A 的逆为 A⁻¹ = (1/det A) adj A，其中 adj A 是余子式矩阵的转置。

Eigenvalues satisfy det(A − λI) = 0. Eigenvectors x are non‑zero solutions of (A − λI)x = 0.

特征值满足 det(A − λI) = 0。特征向量 x 为 (A − λI)x = 0 的非零解。

Diagonalisation: if A has n linearly independent eigenvectors, then P⁻¹AP = D, where columns of P are eigenvectors and D is the diagonal matrix of eigenvalues.

对角化：若 A 有 n 个线性无关的特征向量，则 P⁻¹AP = D，其中 P 的列为特征向量，D 为由特征值构成的对角矩阵。

Transformations in the plane: reflection, rotation, enlargement and shear are represented by 2×2 matrices. A rotation through angle θ (anticlockwise) is given by [[cos θ, −sin θ], [sin θ, cos θ]].

平面上的变换：反射、旋转、放大和剪切可用 2×2 矩阵表示。绕原点逆时针旋转 θ 角的矩阵为 [[cos θ, −sin θ], [sin θ, cos θ]]。

3. Vectors in 3D | 三维向量

For vectors a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, the scalar (dot) product is a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ.

对于向量 a = a₁i + a₂j + a₃k 与 b = b₁i + b₂j + b₃k，数量积（点积）为 a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ。

The vector (cross) product: a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k. Its magnitude |a × b| = |a||b| sin θ gives the area of the parallelogram spanned by a and b.

向量积（叉积）：a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k。其模 |a × b| = |a||b| sin θ 是以 a 和 b 为邻边的平行四边形的面积。

Scalar triple product: a·(b × c) = det([a₁ a₂ a₃; b₁ b₂ b₃; c₁ c₂ c₃]). It gives the volume of the parallelepiped.

标量三重积：a·(b × c) = det([a₁ a₂ a₃; b₁ b₂ b₃; c₁ c₂ c₃])，其绝对值为平行六面体的体积。

Equation of a line: r = a + λd, where a is a point on the line and d is a direction vector.

直线方程：r = a + λd，其中 a 为直线上一点，d 为方向向量。

Equation of a plane: r·n = p, or r = a + λu + μv, where n is a normal vector and u, v are direction vectors lying in the plane.

平面方程：r·n = p，或 r = a + λu + μv，其中 n 为法向量，u, v 为平面内的方向向量。

Distance from a point to a plane: |(AP·n)|/|n|, where A lies on the plane, P is the point and n is a normal vector to the plane.

点到平面的距离：|(AP·n)|/|n|，其中 A 在平面上，P 为给定点，n 为平面的法向量。

4. 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。

Osborne’s rule: To convert a trigonometric identity into a hyperbolic one, replace each trigonometric function by its hyperbolic counterpart and change the sign of any term that contains a product of two sines.

奥斯本法则：将三角恒等式转换为双曲恒等式时，把三角函数替换为相应的双曲函数，并改变任何含有两个正弦乘积项的符号。

Key identity: cosh² x − sinh² x = 1. Other identities: sinh 2x = 2 sinh x cosh x, cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x.

核心恒等式：cosh² x − sinh² x = 1。其他：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)) (x ≥ 1), artanh x = ½ ln((1+x)/(1−x)) (|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. 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) = sech² x。d/dx (arsinh x) = 1/√(x²+1)，d/dx (arcosh x) = 1/√(x²−1)，d/dx (artanh x) = 1/(1−x²)。

5. Polar Coordinates | 极坐标

A curve can be defined by r = f(θ), with x = r cos θ, y = r sin θ. The area of a sector is A = ½ ∫_α^β r² dθ.

曲线可用极坐标方程 r = f(θ) 表示，其中 x = r cos θ, y = r sin θ。曲线与射线 θ = α, θ = β 围成的扇形面积为 A = ½ ∫_α^β r² dθ。

The arc length of a polar curve between θ = α and θ = β is s = ∫_α^β √(r² + (dr/dθ)²) dθ.

极坐标曲线在 θ = α 与 θ = β 之间的弧长为 s = ∫_α^β √(r² + (dr/dθ)²) dθ。

Tangents to polar curves: The tangent is parallel to the initial line when d/dθ (r sin θ) = 0, and perpendicular to the initial line when d/dθ (r cos θ) = 0.

极坐标曲线的切线：当 d/dθ (r sin θ) = 0 时，切线平行于极轴；当 d/dθ (r cos θ) = 0 时，切线垂直于极轴。

Common curves: cardioid r = a(1 + cos θ), circle r = 2a cos θ (centre (a,0)), rose curves r = a cos(nθ) or r = a sin(nθ).

常见曲线：心形线 r = a(1 + cos θ)，圆 r = 2a cos θ（圆心在 (a,0)），玫瑰线 r = a cos(nθ) 或 r = a sin(nθ)。

6. Series | 级数

Method of differences: if a general term uᵣ = f(r) − f(r+1), then ∑_{r=1}^{n} uᵣ = f(1) − f(n+1). Use this for rational functions and logarithmic telescoping sums.

裂项相消法：若通项 uᵣ = f(r) − f(r+1)，则 ∑_{r=1}^{n} uᵣ = f(1) − f(n+1)。此法适用于有理函数及对数型裂项求和。

Maclaurin series: f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + …

麦克劳林级数：f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + …

Key expansions: eˣ = 1 + x + x²/2! + x³/3! + …, valid for all x. ln(1+x) = x − x²/2 + x³/3 − x⁴/4 + … (−1 < x ≤ 1).

重要展开式：eˣ = 1 + x + x²/2! + x³/3! + …，对所有 x 成立。ln(1+x) = x − x²/2 + x³/3 − x⁴/4 + … (−1 < x ≤ 1)。

sin x = x − x³/3! + x⁵/5! − …; cos x = 1 − x²/2! + x⁴/4! − …; (1+x)ᵅ = 1 + αx + α(α−1)x²/2! + … for |x| < 1.

sin x = x − x³/3! + x⁵/5! − …；cos x = 1 − x²/2! + x⁴/4! − …；(1+x)ᵅ = 1 + αx + α(α−1)x²/2! + … (|x| < 1)。

7. Integration Techniques | 积分方法

Reduction formulae: integrals of the form Iₙ = ∫ sinⁿ x dx or ∫ cosⁿ x dx satisfy Iₙ = −(1/n) sinⁿ⁻¹ x cos x + (n−1)/n Iₙ₋₂. Similar recurrence relations apply for ∫ tanⁿ x dx, ∫ secⁿ x dx and ∫ xⁿeᵏˣ dx.

递推公式：形如 Iₙ = ∫ sinⁿ x dx 或 ∫ cosⁿ x dx 的积分满足 Iₙ = −(1/n) sinⁿ⁻¹ x cos x + (n−1)/n Iₙ₋₂。类似的递推关系也适用于 ∫ tanⁿ x dx、∫ secⁿ x dx 和 ∫ xⁿeᵏˣ dx。

Arc length of a curve y = f(x) from x=a to x=b: s = ∫_a^b √(1 + (dy/dx)²) dx. For parametric curves (x(t), y(t)): s = ∫ √((dx/dt)² + (dy/dt)²) dt.

曲线弧长：对于 y = f(x) 从 x=a 到 x=b，s = ∫_a^b √(1 + (dy/dx)²) dx。对于参数曲线 (x(t), y(t))，s = ∫ √((dx/dt)² + (dy/dt)²) dt。

Surface area of revolution: about the x‑axis, S = 2π ∫ y ds, where ds = √(1 + (dy/dx)²) dx (or parametric form). About the y‑axis, S = 2π ∫ x ds.

旋转体表面积：绕 x 轴旋转时，S = 2π ∫ y ds，其中 ds = √(1 + (dy/dx)²) dx（或参数形式）。绕 y 轴旋转时，S = 2π ∫ x ds。

Mean value of a function: (1/(b−a)) ∫_a^b f(x) dx. Root mean square (RMS): √( (1/(b−a)) ∫_a^b [f(x)]² dx ).

函数的平均值：(1/(b−a)) ∫_a^b f(x) dx。均方根 (RMS)：√( (1/(b−a)) ∫_a^b [f(x)]² dx )。

8. Differential Equations | 微分方程

First order separable: dy/dx = g(x)h(y) ⇒ ∫ 1/h(y) dy = ∫ g(x) dx. Don’t forget the constant of integration.

一阶可分离变量方程：dy/dx = g(x)h(y) ⇒ ∫ 1/h(y) dy = ∫ g(x) dx。不要遗漏积分常数。

First order linear: dy/dx + P(x)y = Q(x). Integrating factor I(x) = e^(∫ P(x) dx). Solution: y I = ∫ Q I dx + C.

一阶线性方程：dy/dx + P(x)y = Q(x)。积分因子 I(x) = e^(∫ P(x) dx)。通解为 y I = ∫ Q I dx + C。

Second order homogeneous with constant coefficients: a d²y/dx² + b dy/dx + cy = 0. Auxiliary equation: am² + bm + c = 0. Real distinct roots m₁, m₂: y = Ae^(m₁x) + Be^(m₂x). Repeated root m: y = (A + Bx)e^(mx). Complex roots α ± iβ: y = e^(αx)(A cos βx + B sin βx).

二阶常系数齐次方程：a d²y/dx² + b dy/dx + cy = 0。辅助方程：am² + bm + c = 0。不等实根 m₁, m₂：y = Ae^(m₁x) + Be^(m₂x)。重根 m：y = (A + Bx)e^(mx)。共轭复根 α ± iβ：y = e^(αx)(A cos βx + B sin βx)。

Second order non‑homogeneous: find particular integral by trying a form similar to the RHS. If the RHS is e^(kx), try yₚ = Ce^(kx); if RHS is a polynomial, try a general polynomial of the same degree. If overlap with complementary function occurs, multiply by x.

二阶非齐次方程：根据右边形式试探特解。若右边为 e^(kx)，试取 yₚ = Ce^(kx)；若为多项式，试取同次多项式。若与补函数重复，则乘以 x。

Simple harmonic motion: d²x/dt² + ω² x = 0 ⇒ x = A cos ωt + B sin ωt = R cos(ωt − φ), where R = √(A²+B²).

简谐运动：d²x/dt² + ω² x = 0 ⇒ x = A cos ωt + B sin ωt = R cos(ωt − φ)，其中 R = √(A²+B²)。

9. Numerical Methods | 数值方法

Newton‑Raphson method for f(x)=0: xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ). Iterate until convergence.

牛顿－拉弗森法求 f(x)=0：xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ)。迭代直至收敛。

Euler’s method for dy/dx = f(x,y): yₙ₊₁ = yₙ + h f(xₙ, yₙ), with step length h.

欧拉法用于 dy/dx = f(x,y)：yₙ₊₁ = yₙ + h f(xₙ, yₙ)，步长为 h。

Improved Euler method (Heun’s method): predictor yₚ = yₙ + h f(xₙ, yₙ), corrector yₙ₊₁ = yₙ + h/2 [f(xₙ, yₙ) + f(xₙ₊₁, yₚ)].

改进欧拉法（豪恩法）：预测 yₚ = yₙ + h f(xₙ, yₙ)，校正 yₙ₊₁ = yₙ + h/2 [f(xₙ, yₙ) + f(xₙ₊₁, yₚ)]。

Midpoint method: yₙ₊₁ = yₙ + h f(xₙ + h/2, yₙ + (h/2) f(xₙ, yₙ)).

中点法：yₙ₊₁ = yₙ + h f(xₙ + h/2, yₙ + (h/2) f(xₙ, yₙ))。

10. Statistics Formulas (FS1) | 统计公式（进阶统计 1）

Discrete random variables: E(X) = Σ x P(X=x). Var(X) = E(X²) − [E(X)]² = Σ (x − µ)² P(X=x).

离散随机变量：E(X) = Σ x P(X=x)。Var(X) = E(X²) − [E(X)]² = Σ (x − µ)² P(X=x)。

Poisson distribution: X ~ Po(λ), P(X=r) = e^(−λ) λʳ / r!. Mean = λ, variance = λ. Sum of independent Poissons: X+Y ~ Po(λ₁+λ₂).

泊松分布：X ~ Po(λ)，P(X=r) = e^(−λ) λʳ / r!。均值 = λ，方差 = λ。独立泊松变量的和：X+Y ~ Po(λ₁+λ₂)。

Geometric distribution: X ~ Geo(p), P(X=r) = p(1−p)⁰⁻¹? Wait, geometric: P(X=r) = (1−p)⁰⁻¹ p, r=1,2,… Mean = 1/p, variance = (1−p)/p².

几何分布：X ~ Geo(p)，P(X=r) = (1−p)⁰⁻¹ p，r=1,2,… 均值 = 1/p，方差 = (1−p)/p²。

Negative binomial: number of trials to get r successes, X ~ NB(r,p), P(X=x) = C(x−1, r−1) p⁰ (1−p)⁰⁻⁰? Actually P(X=x) = C(x−1, r−1) pʳ (1−p)^(x−r). Mean = r/p, variance = r(1−p)/p².

负二项分布：达到 r 次成功所需试验次数 X ~ NB(r,p)，P(X=x) = C(x−1, r−1) pʳ (1−p)^(x−r)。均值 = r/p，方差 = r(1−p)/p²。

Chi‑squared tests: test statistic χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ. Degrees of freedom: for goodness of fit, ν = number of classes − 1 − number of estimated parameters. For contingency tables, ν = (rows−1)(columns−1).

卡方检验：检验统计量 χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ。自由度：拟合优度检验中，ν = 类别数 − 1 − 估计参数个数；列联表中，ν = (行数−1)(列数−1)。

11. Mechanics Formulae (FM1) | 力学公式（进阶力学 1）

Kinematics with constant acceleration: v = u + at, s = ut + ½ at², s = ½ (u+v)t, v² = u² + 2as. These apply only when acceleration is constant.

匀加速直线运动：v = u + at，s = ut + ½ at²，s = ½ (u+v)t，v² = u² + 2as。仅适用于加速度恒定情形。

Newton’s second law: F = ma. Impulse‑momentum principle: Impulse = change in momentum = mv − mu.

牛顿第二定律：F = ma。冲量－动量定理：冲量 = 动量变化 = mv − mu。

Work done by a constant force: W = Fd cos θ. Work‑energy principle: total work done by all forces = change in kinetic energy = ½ m (v² − u²).

恒力做功：W = Fd cos θ。功能原理：所有力做功的总和 = 动能变化 = ½ m (v² − u²)。

Power = dW/dt = Fv for a force acting at speed v. Mechanical energy: kinetic energy ½ mv², gravitational potential energy mgh.

功率 = dW/dt = Fv（力作用在速度为 v 的物体上）。机械能：动能 ½ mv²，重力势能 mgh。

Elastic strings and springs: Hooke’s law T = kx, where k is the stiffness, x is extension. Elastic potential energy = ½ k x².

弹性弦与弹簧：胡克定律 T = kx，其中 k 为劲度系数，x 为伸长量。弹性势能 = ½ k x²。

12. Key Trigonometric and Calculus Results | 常用三角与微积分结果

Compound angle formulae: sin(A±B) = sin A cos B ± cos A sin B, cos(A±B) = cos A cos B ∓ sin A sin B, tan(A±B) = (tan A ± tan B)/(1 ∓ tan A tan B).

和角公式：sin(A±B) = sin A cos B ± cos A sin B，cos(A±B) = cos A cos B ∓ sin A sin B，tan(A±B) = (tan A ± tan B)/(1 ∓ tan A tan B)。

Double angle: sin 2A = 2 sin A cos A, cos 2A = cos²A − sin²A = 2 cos²A − 1 = 1 − 2 sin²A, tan 2A = 2 tan A/(1 − tan²A).

倍角公式：sin 2A = 2 sin A cos A，cos 2A = cos²A − sin²A = 2 cos²A − 1 = 1 − 2 sin²A，tan 2A = 2 tan A/(1 − tan²A)。

Factoring forms: sin P + sin Q = 2 sin((P+Q)/2) cos((P−Q)/2), sin P − sin Q = 2 cos((P+Q)/2) sin((P−Q)/2). Similar for sum/difference of cosines.

和差化积：sin P + sin Q = 2 sin((P+Q)/2) cos((P−Q)/2)，sin P − sin Q = 2 cos((P+Q)/2) sin((P−Q)/2)。余弦类似。

Volume of revolution: about x‑axis, V = π ∫ y² dx; about y‑axis, V = π ∫ x² dy.

旋转体体积：绕 x 轴旋转，V = π ∫ y² dx；绕 y 轴旋转，V = π ∫ x² dy。

Integration by parts: ∫ u dv/dx dx = uv − ∫ v du/dx dx. For definite integrals, apply limits to the uv term.

分部积分法：∫ u dv/dx dx = uv − ∫ v du/dx dx。对定积分，uv 项需要代入上下限。

Standard derivatives: d/dx (aˣ) = aˣ ln a; d/dx (arcsin x) = 1/√(1−x²); d/dx (arccos x) = −1/√(1−x²); d/dx (arctan x) = 1/(1+x²).

标准导数：d/dx (aˣ) = aˣ ln a；d/dx (arcsin x) = 1/√(1−x²)；d/dx (arccos x) = −1/√(1−x²)；d/dx (arctan x) = 1/(1+x²)。

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A-Level Edexcel Further Maths Formula Handbook | 爱德思进阶数学公式汇总手册