📚 Pre-U CAIE Further Maths: Formula & Theorem Quick Reference Handbook | Pre-U CAIE 进阶数学：公式定理速查手册
This concise handbook assembles the most essential formulae, theorems, and key results required for the Cambridge Pre-U Further Mathematics syllabus (CAIE). Designed for quick revision and reference, it covers core topics from pure mathematics, mechanics, statistics, and discrete mathematics, ensuring you have a sturdy toolkit for problem solving and examination success. Each section pairs a topic overview with the critical equations and conditions you must know.
这份精编手册汇总了剑桥 Pre-U 进阶数学（CAIE）大纲中最关键的公式、定理和结论。它专为快速复习与查阅而设计，涵盖纯数学、力学、统计学和离散数学的核心内容，为你提供解决问题和应对考试的扎实工具。每个部分均以主题概览配以必须掌握的关键方程和条件。
1. Complex Numbers & Polar Form | 复数与极坐标形式
A complex number z = x + iy has modulus |z| = √(x² + y²) and argument arg(z) = θ where tan θ = y/x. The polar form is z = r(cos θ + i sin θ) and the exponential form is z = r e^(iθ). De Moivre’s theorem states that (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ). The roots of unity solve z^n = 1 and are given by z_k = e^(i·2πk/n) for k = 0, 1, …, n-1.
复数 z = x + iy 的模为 |z| = √(x² + y²)，幅角为 arg(z) = θ 满足 tan θ = y/x。极坐标形式为 z = r(cos θ + i sin θ)，指数形式为 z = r e^(iθ)。棣莫弗定理：(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)。单位根的方程 z^n = 1 的解为 z_k = e^(i·2πk/n)，k = 0, 1, …, n-1。
2. Matrices, Determinants & Linear Transformations | 矩阵、行列式与线性变换
For a 2×2 matrix M = [[a, b], [c, d]], the determinant is det(M) = ad – bc. The inverse exists if det(M) ≠ 0 and is M⁻¹ = (1/det(M)) [[d, -b], [-c, a]]. For a 3×3 matrix, expansion by cofactors is used. A linear transformation in the plane can be represented by a matrix; reflections, rotations by angle θ, and shear mappings have standard forms. The area scale factor of a transformation is |det(M)|.
对于 2×2 矩阵 M = [[a, b], [c, d]]，行列式为 det(M) = ad – bc。若 det(M) ≠ 0，则逆矩阵存在为 M⁻¹ = (1/det(M)) [[d, -b], [-c, a]]。3×3 矩阵用余子式展开。平面上的线性变换可由矩阵表示；反射、旋转 θ 角以及剪切映射均有标准形式。变换的面积缩放因子为 |det(M)|。
3. Vector Geometry & Scalar/Vector Products | 向量几何与数量/向量积
For vectors a and b, the scalar (dot) product is a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃. The vector (cross) product a×b yields a vector perpendicular to both, with magnitude |a×b| = |a||b| sin θ. The scalar triple product a·(b×c) gives the volume of the parallelepiped. Planes are defined by r·n = d, lines by r = r₀ + λ·v.
对于向量 a 和 b，数量积（点积）为 a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。向量积（叉积）a×b 产生垂直于两者的向量，大小 |a×b| = |a||b| sin θ。混合积 a·(b×c) 给出平行六面体的体积。平面由 r·n = d 定义，直线由 r = r₀ + λ·v 定义。
4. Further Calculus: Hyperbolics, Parametric & Polar | 进阶微积分：双曲函数、参数与极坐标
Hyperbolic functions: sinh x = (e^x – e^(-x))/2, cosh x = (e^x + e^(-x))/2, tanh x = sinh x / cosh x. Their derivatives: d/dx sinh x = cosh x, d/dx cosh x = sinh x. For parametric equations x = f(t), y = g(t), the gradient is dy/dx = (dy/dt)/(dx/dt). In polar coordinates, area = ∫(1/2) r² dθ and arc length = ∫ √(r² + (dr/dθ)²) dθ.
双曲函数：sinh x = (e^x – e^(-x))/2，cosh x = (e^x + e^(-x))/2，tanh x = sinh x / cosh x。导数：d/dx sinh x = cosh x，d/dx cosh x = sinh x。参数方程 x = f(t), y = g(t) 的斜率为 dy/dx = (dy/dt)/(dx/dt)。极坐标下，面积 = ∫(1/2) r² dθ，弧长 = ∫ √(r² + (dr/dθ)²) dθ。
5. Differential Equations: First & Second Order | 微分方程：一阶与二阶
First-order linear: dy/dx + P(x)y = Q(x) has integrating factor μ = e^(∫P dx). Separable ODEs: g(y) dy = h(x) dx are solved by integration. Second-order linear with constant coefficients: a y” + b y’ + c y = f(x). The complementary function (CF) solves the homogeneous case; for roots m₁, m₂, if real and distinct CF: Ae^(m₁x) + Be^(m₂x); if repeated: (A + Bx)e^(mx); if complex α ± iβ: e^(αx)(C cos βx + D sin βx). Particular integral (PI) is found by undetermined coefficients or variation of parameters.
一阶线性方程：dy/dx + P(x)y = Q(x)，积分因子为 μ = e^(∫P dx)。可分离的常微分方程：g(y) dy = h(x) dx 通过积分求解。二阶常系数线性方程：a y” + b y’ + c y = f(x)。余函数 (CF) 解齐次情形；特征根 m₁, m₂ 为相异实根时 CF: Ae^(m₁x) + Be^(m₂x)；重根时: (A + Bx)e^(mx)；共轭复根 α ± iβ 时: e^(αx)(C cos βx + D sin βx)。特解积分 (PI) 用待定系数法或参数变易法求得。
6. Further Sequences & Series: Summation & Maclaurin | 进阶数列与级数：求和与麦克劳林展开
Standard series: geometric ∑ ar^(n-1) = a/(1-r) for |r|<1. The Maclaurin series expansion f(x) = f(0) + f'(0)x + f''(0)/2! x² + ... . Key expansions: e^x = 1 + x + x²/2! + ...; sin x = x - x³/3! + x⁵/5! - ...; cos x = 1 - x²/2! + x⁴/4! - ...; ln(1+x) = x - x²/2 + x³/3 - ... (|x|<1). The method of differences can simplify series summation using partial fractions.
标准级数：等比数列 ∑ ar^(n-1) = a/(1-r)，|r|<1。麦克劳林级数展开：f(x) = f(0) + f'(0)x + f''(0)/2! x² + ... 。重要展开式：e^x = 1 + x + x²/2! + ...；sin x = x - x³/3! + x⁵/5! - ...；cos x = 1 - x²/2! + x⁴/4! - ...；ln(1+x) = x - x²/2 + x³/3 - ... (|x|<1)。差分法可利用部分分式简化级数求和。
7. Numerical Methods & Error Analysis | 数值方法与误差分析
Root-finding: Newton-Raphson iteration x_{n+1} = x_n – f(x_n)/f'(x_n). Numerical integration: Trapezium rule ∫_a^b f(x) dx ≈ (h/2)[y₀ + 2(y₁+y₂+…+y_{n-1}) + y_n] with h = (b-a)/n; Simpson’s rule (n even) ∫_a^b f(x) dx ≈ (h/3)[y₀ + 4(y₁+y₃+…) + 2(y₂+y₄+…) + y_n]. Error bounds involve derivatives: for trapezium E ≤ ((b-a)³/(12n²)) max |f”(x)|.
求根法：牛顿-拉弗森迭代 x_{n+1} = x_n – f(x_n)/f'(x_n)。数值积分：梯形法则 ∫_a^b f(x) dx ≈ (h/2)[y₀ + 2(y₁+y₂+…+y_{n-1}) + y_n]，其中 h = (b-a)/n；辛普森法则（n 为偶数）∫_a^b f(x) dx ≈ (h/3)[y₀ + 4(y₁+y₃+…) + 2(y₂+y₄+…) + y_n]。误差界涉及导数：梯形法则 E ≤ ((b-a)³/(12n²)) max |f”(x)|。
8. Mechanics: Kinematics, Forces & Energy | 力学：运动学、力与能量
Constant acceleration SUVAT: v = u + at, s = ut + ½at², v² = u² + 2as. Newton’s second law F = ma. Impulse = change in momentum = mv – mu. Work done by a force F over displacement s is Fs cos θ. Kinetic energy = ½mv², gravitational potential energy = mgh. Power = Fv. For projectiles, horizontal and vertical motions are independent; range = (u² sin 2θ)/g.
匀加速运动 SUVAT：v = u + at，s = ut + ½at²，v² = u² + 2as。牛顿第二定律 F = ma。冲量 = 动量变化 = mv – mu。力 F 在位移 s 上做功为 Fs cos θ。动能 = ½mv²，重力势能 = mgh。功率 = Fv。抛体运动中，水平和竖直运动独立；射程 = (u² sin 2θ)/g。
9. Further Mechanics: Circular Motion & SHM | 进阶力学：圆周运动与简谐运动
For horizontal circular motion, centripetal acceleration a = v²/r = rω², and centripetal force F = mrω² = mv²/r. Conical pendulum and banked tracks require resolving forces. Simple harmonic motion (SHM): defining equation ẍ = -ω² x. Solutions: x = A sin(ωt + φ) or x = A cos(ωt + φ). Period T = 2π/ω, speed v = ω√(A² – x²). Elastic potential energy = ½kx².
对于水平圆周运动，向心加速度 a = v²/r = rω²，向心力 F = mrω² = mv²/r。圆锥摆和倾斜轨道需分解力。简谐运动 (SHM)：定义方程 ẍ = -ω² x。解为 x = A sin(ωt + φ) 或 x = A cos(ωt + φ)。周期 T = 2π/ω，速度 v = ω√(A² – x²)。弹性势能 = ½kx²。
10. Statistics: Probability, Distributions & Hypothesis Testing | 统计学：概率、分布与假设检验
Probability: P(A|B) = P(A∩B)/P(B). Bayes’ theorem: P(A|B) = P(B|A)P(A)/P(B). Discrete random variables: expectation E(X) = ∑ x p(x), Var(X) = E(X²) – [E(X)]². Binomial: X ~ B(n, p), P(X=k) = C(n, k) p^k q^(n-k), mean = np, variance = npq. Poisson: X ~ Po(λ), P(X=k) = e^(-λ) λ^k / k!, mean = λ = variance. Normal: Z = (X – μ)/σ. Confidence intervals and hypothesis tests with critical values and p-values.
概率：P(A|B) = P(A∩B)/P(B)。贝叶斯定理：P(A|B) = P(B|A)P(A)/P(B)。离散随机变量：期望 E(X) = ∑ x p(x)，方差 Var(X) = E(X²) – [E(X)]²。二项分布：X ~ B(n, p)，P(X=k) = C(n, k) p^k q^(n-k)，均值 = np，方差 = npq。泊松分布：X ~ Po(λ)，P(X=k) = e^(-λ) λ^k / k!，均值 = λ = 方差。正态分布：Z = (X – μ)/σ。置信区间与假设检验需用临界值和 p 值。
11. Discrete Mathematics: Graphs, Networks & Algorithms | 离散数学：图、网络与算法
A graph G = (V, E) has vertices V and edges E. The sum of degrees = 2|E|. Eulerian trail exists if exactly 0 or 2 vertices have odd degree. Hamiltonian cycle visits every vertex once. Minimum spanning tree algorithms: Prim’s and Kruskal’s. Dijkstra’s algorithm finds shortest path from a source. Critical path analysis uses earliest and latest start times; float = LST – EST. Total float can be zero for critical activities.
图 G = (V, E) 有顶点集 V 和边集 E。度之和 = 2|E|。一条欧拉路径存在的充要条件是奇度顶点数为 0 或 2。哈密顿圈恰好经过每个顶点一次。最小生成树算法：普里姆算法和克鲁斯卡尔算法。迪杰斯特拉算法求源点到其他点的最短路径。关键路径分析使用最早和最晚开始时间；时差 = LST – EST。关键活动的总时差为零。
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