📚 A-Level CIE Further Mathematics: Formula Summary Handbook | A-Level CIE 进阶数学：公式汇总手册
This handbook provides a comprehensive collection of essential formulas, identities, and standard results for the Cambridge International A-Level Further Mathematics syllabus. It is designed to help you quickly locate the tools you need for complex numbers, matrices, vectors, hyperbolic functions, differential equations, polar coordinates, series, mechanics, and statistics. Each section presents the key formulas in a clear, memorable format, paired with brief explanatory notes.
本手册汇集了剑桥国际 A-Level 进阶数学课程中必备的公式、恒等式和标准结果。旨在帮助你快速检索复数、矩阵、向量、双曲函数、微分方程、极坐标、级数、力学和统计等领域所需的工具。每个小节以清晰易记的方式呈现关键公式，并配有简要注释。
1. Complex Numbers | 复数
Let z = x + iy, where x = Re(z), y = Im(z), and i² = −1. The modulus-argument form is z = r(cos θ + i sin θ) = r e^(iθ), with r = √(x² + y²) and θ = arg(z). Multiplication and division in polar form follow r₁r₂ e^(i(θ₁+θ₂)) and (r₁/r₂) e^(i(θ₁−θ₂)). De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) for integer n. The nth roots of a complex number are given by z^(1/n) = r^(1/n) [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)], k = 0,1,…,n−1.
令 z = x + iy，其中 x = Re(z)，y = Im(z)，i² = −1。模–辐角形式为 z = r(cos θ + i sin θ) = r e^(iθ)，r = √(x² + y²)，θ = arg(z)。极坐标形式的乘法和除法分别遵循 r₁r₂ e^(i(θ₁+θ₂)) 和 (r₁/r₂) e^(i(θ₁−θ₂))。棣莫弗定理：对于整数 n，(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。复数的 n 次方根由 z^(1/n) = r^(1/n) [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)]，k = 0,1,…,n−1 给出。
2. Matrices and Linear Transformations | 矩阵与线性变换
The determinant of a 2×2 matrix A = [[a,b],[c,d]] is det(A) = ad − bc, and the inverse is A⁻¹ = (1/det(A)) [[d, −b],[−c, a]], provided det(A) ≠ 0. For a 3×3 matrix, the determinant can be computed by expansion along a row or column. A linear transformation T: v → Mv preserves the origin. Eigenvalues λ satisfy det(M − λI) = 0, and the corresponding eigenvectors x are non-zero solutions of (M − λI)x = 0.
二阶矩阵 A = [[a,b],[c,d]] 的行列式为 det(A) = ad − bc，其逆矩阵为 A⁻¹ = (1/det(A)) [[d, −b],[−c, a]]，前提是 det(A) ≠ 0。对于三阶矩阵，行列式可按行或列展开计算。线性变换 T: v → Mv 保持原点不变。特征值 λ 满足 det(M − λI) = 0，对应的特征向量 x 是 (M − λI)x = 0 的非零解。
3. Vectors | 向量
The scalar (dot) product: a·b = |a||b|cos θ = a₁b₁ + a₂b₂ + a₃b₃. The vector (cross) product: a×b = |a||b|sin θ n̂, with components (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁). The scalar triple product a·(b×c) gives the volume of the parallelepiped. The vector equation of a line: r = a + λd. The plane: r·n = a·n or in Cartesian form ax + by + cz = d.
标量积（点积）：a·b = |a||b|cos θ = a₁b₁ + a₂b₂ + a₃b₃。向量积（叉积）：a×b = |a||b|sin θ n̂，其分量为 (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁)。标量三重积 a·(b×c) 给出平行六面体的体积。直线的向量方程：r = a + λd。平面的方程：r·n = a·n 或笛卡尔形式 ax + by + cz = d。
4. Hyperbolic Functions | 双曲函数
The hyperbolic sine and cosine are defined as sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2. The fundamental identity is cosh² x − sinh² x = 1. The derivatives: d/dx(sinh x) = cosh x, d/dx(cosh x) = sinh x, d/dx(tanh x) = sech² x. Inverse hyperbolic functions: arsinh x = ln(x + √(x²+1)), arcosh x = ln(x + √(x²−1)) for x ≥ 1, artanh x = ½ ln((1+x)/(1−x)) for |x| < 1.
双曲正弦和双曲余弦定义为 sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2。基本恒等式为 cosh² x − sinh² 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：arsinh x = ln(x + √(x²+1))，arcosh x = ln(x + √(x²−1))（x ≥ 1），artanh x = ½ ln((1+x)/(1−x))（|x| < 1）。
5. Differential Equations | 微分方程
A first-order linear ODE of the form dy/dx + P(x)y = Q(x) is solved using the integrating factor I = e^(∫P dx). The solution is y I = ∫ Q I dx + C. For second-order linear ODEs with constant coefficients a d²y/dx² + b dy/dx + c y = f(x), the complementary function is obtained from the auxiliary equation am² + bm + c = 0. If the roots are real and distinct, y_cf = A e^(m₁x) + B e^(m₂x); if repeated, y_cf = (A + Bx)e^(mx); if complex m = α ± iβ, y_cf = e^(αx)(A cos βx + B sin βx). The particular integral depends on the form of f(x).
一阶线性常微分方程 dy/dx + P(x)y = Q(x) 使用积分因子 I = e^(∫P dx) 求解。通解为 y I = ∫ Q I dx + C。对于常系数二阶线性常微分方程 a d²y/dx² + b dy/dx + c y = f(x)，余函数由辅助方程 am² + bm + c = 0 求得。若根为相异实根，y_cf = A e^(m₁x) + B e^(m₂x)；若为重根，y_cf = (A + Bx)e^(mx)；若 m = α ± iβ 为复根，y_cf = e^(αx)(A cos βx + B sin βx)。特解的形式取决于 f(x) 的类型。
6. Polar Coordinates | 极坐标
In polar coordinates (r, θ), x = r cos θ, y = r sin θ. The area enclosed by a polar curve r = f(θ) between θ = α and θ = β is A = ½ ∫_α^β r² dθ. The arc length of the curve is s = ∫_α^β √(r² + (dr/dθ)²) dθ. Standard curves include cardioid r = a(1 + cos θ), circle r = a, and rose curves r = a sin(nθ) or r = a cos(nθ).
在极坐标 (r, θ) 中，x = r cos θ，y = r sin θ。由极坐标曲线 r = f(θ) 从 θ = α 到 θ = β 所围成的面积为 A = ½ ∫_α^β r² dθ。曲线的弧长为 s = ∫_α^β √(r² + (dr/dθ)²) dθ。标准曲线包括心脏线 r = a(1 + cos θ)、圆 r = a 以及玫瑰线 r = a sin(nθ) 或 r = a cos(nθ)。
7. Series and Summation | 级数与求和
The Maclaurin series expansion of a function f(x) is f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … + f⁽ⁿ⁾(0)xⁿ/n! + … . Standard expansions include eˣ = Σ xⁿ/n!, sin x = Σ (−1)ⁿ x²ⁿ⁺¹/(2n+1)!, cos x = Σ (−1)ⁿ x²ⁿ/(2n)!, ln(1+x) = Σ (−1)ⁿ⁻¹ xⁿ/n for −1 < x ≤ 1, and (1+x)^k = 1 + k x + k(k−1)x²/2! + … for |x| < 1. Also, Σ r = n(n+1)/2, Σ r² = n(n+1)(2n+1)/6, Σ r³ = n²(n+1)²/4.
函数 f(x) 的麦克劳林级数展开为 f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … + f⁽ⁿ⁾(0)xⁿ/n! + …。标准展开包括 eˣ = Σ xⁿ/n!，sin x = Σ (−1)ⁿ x²ⁿ⁺¹/(2n+1)!，cos x = Σ (−1)ⁿ x²ⁿ/(2n)!，ln(1+x) = Σ (−1)ⁿ⁻¹ xⁿ/n（−1 < x ≤ 1），以及 (1+x)^k = 1 + k x + k(k−1)x²/2! + …（|x| < 1）。此外还有 Σ r = n(n+1)/2，Σ r² = n(n+1)(2n+1)/6，Σ r³ = n²(n+1)²/4。
8. Further Mechanics | 进阶力学
In kinematics, for variable acceleration a = dv/dt = v dv/dx. The work done by a force is W = ∫ F dx. Kinetic energy = ½ m v², potential energy = mgh, and the principle of conservation of mechanical energy holds in the absence of non-conservative forces. Momentum p = mv; impulse = ∫ F dt = Δ(mv). For oblique elastic collisions, Newton’s experimental law gives e = (v₂ − v₁)/(u₁ − u₂) along the line of centres. Circular motion: radial acceleration a = v²/r = rω², tangential acceleration = rα.
运动学中，对于变加速度 a = dv/dt = v dv/dx。力做的功 W = ∫ F dx。动能 = ½ m v²，势能 = mgh，在没有非保守力的情况下机械能守恒定律成立。动量 p = mv；冲量 = ∫ F dt = Δ(mv)。对于斜弹性碰撞，牛顿实验定律沿中心线给出 e = (v₂ − v₁)/(u₁ − u₂)。圆周运动：径向加速度 a = v²/r = rω²，切向加速度 = rα。
9. Further Statistics: Discrete and Continuous Distributions | 进阶统计：离散与连续分布
For discrete random variables, E(X) = Σ x P(X=x), Var(X) = E(X²) − [E(X)]² = Σ (x−μ)² P(X=x). Common discrete distributions include the Poisson distribution P(X=r) = e^{−λ} λ^r / r!, with mean λ and variance λ. For continuous random variables, the cumulative distribution function (CDF) F(x) = P(X ≤ x) = ∫_{−∞}^x f(t) dt, and probability density function (PDF) f(x) = F'(x). The expectation E(X) = ∫ x f(x) dx. The normal distribution X ~ N(μ, σ²) has PDF f(x) = (1/σ√(2π)) e^{−(x−μ)²/(2σ²)}.
对于离散随机变量，E(X) = Σ x P(X=x)，Var(X) = E(X²) − [E(X)]² = Σ (x−μ)² P(X=x)。常见离散分布包括泊松分布 P(X=r) = e^{−λ} λ^r / r!，均值为 λ，方差为 λ。对于连续随机变量，累积分布函数（CDF）F(x) = P(X ≤ x) = ∫_{−∞}^x f(t) dt，概率密度函数（PDF）f(x) = F'(x)。期望 E(X) = ∫ x f(x) dx。正态分布 X ~ N(μ, σ²) 的 PDF 为 f(x) = (1/σ√(2π)) e^{−(x−μ)²/(2σ²)}。
10. Hypothesis Testing and Confidence Intervals | 假设检验与置信区间
For a normal sample with known variance, a z-test statistic is z = (x̄ − μ₀)/(σ/√n). The confidence interval for the population mean is x̄ ± z* σ/√n. For unknown variance, use the t-distribution: t = (x̄ − μ₀)/(s/√n) with n−1 degrees of freedom. For a binomial parameter p, the test statistic is based on the number of successes; a normal approximation can be used when np and nq are sufficiently large, with z = (p̂ − p)/√(p q/n) and continuity correction if appropriate.
对于已知方差的正态样本，z 检验统计量为 z = (x̄ − μ₀)/(σ/√n)。总体均值的置信区间为 x̄ ± z* σ/√n。对于未知方差，使用 t 分布：t = (x̄ − μ₀)/(s/√n)，自由度为 n−1。对于二项参数 p，检验统计量基于成功次数；当 np 和 nq 足够大时，可使用正态近似，z = (p̂ − p)/√(p q/n)，并视情况使用连续性校正。
11. Further Pure Techniques: Summation of Series and Method of Differences | 进阶纯数技巧：级数求和与差分法
The method of differences is used for series whose terms can be written as u_r = f(r) − f(r+1) or similar, so that Σ u_r telescopes. For example, Σ 1/(r(r+1)) = Σ (1/r − 1/(r+1)) = 1 − 1/(n+1). Standard summations for finite series of the form Σ r^n can be combined with algebraic manipulation to find sums of more complicated polynomials. The method of induction is often used to prove conjectured sum formulas.
差分法适用于项可表示为 u_r = f(r) − f(r+1) 等形式的级数，从而使 Σ u_r 产生相消效应。例如 Σ 1/(r(r+1)) = Σ (1/r − 1/(r+1)) = 1 − 1/(n+1)。形如 Σ r^n 的有限级数的标准求和公式可与代数操作结合，用于求更复杂的多项式之和。常用归纳法证明猜想的求和公式。
12. Vector Calculus and Further Applications | 向量微积分及进阶应用
The gradient vector ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z) gives the direction of the greatest rate of increase of f. The directional derivative in the direction of unit vector u is D_u f = ∇f · u. For a parametric surface r(u,v), the area element is |∂r/∂u × ∂r/∂v| du dv. In mechanics, vector methods simplify the analysis of moments and couples: the moment of a force F about a point A is r × F, where r is the position vector from A to any point on the line of action.
梯度向量 ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z) 给出 f 增加最快的方向。沿单位向量 u 的方向导数为 D_u f = ∇f · u。对于参数曲面 r(u,v)，面积元为 |∂r/∂u × ∂r/∂v| du dv。在力学中，向量方法简化了力矩和力偶的分析：力 F 关于点 A 的力矩为 r × F，其中 r 是从 A 到力作用线上任意点的位矢。
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