📚 Quick Reference: Formulae and Theorems for Pre-U AQA Further Mathematics | Pre-U AQA 进阶数学公式定理速查手册
This concise handbook compiles essential formulae and theorems required for the Pre-U AQA Further Mathematics course. Each section presents key results in a bilingual format, ensuring that you can quickly reference both the English statements and their Chinese equivalents during revision.
本简明手册汇编了 Pre-U AQA 进阶数学课程所需的核心公式与定理。每个部分以中英双语呈现关键结论，确保您在复习时能快速查阅英文表述及其对应的中文释义。
1. Algebra and Roots of Polynomials | 代数与多项式根
a, αβ+βγ+γα = c/a, αβγ = −d/a. For a quartic, similar symmetric sums extend to four roots.
对于三次方程 ax³ + bx² + cx + d = 0，根为 α, β, γ，根的和为 α+β+γ = −b/a，两两积之和为 αβ+βγ+γα = c/a，三根积为 αβγ = −d/a。对于四次方程，类似对称和推广至四个根。
The substitution x = u − b/(na) eliminates the xⁿ⁻¹ term in a polynomial of degree n.
代换 x = u − b/(na) 可消去 n 次多项式中的 xⁿ⁻¹ 项。
A polynomial identity holds for all values of the variable; equate coefficients to find unknowns.
多项式恒等式对于变量的所有值成立；可通过比较系数求未知量。
2. Complex Numbers | 复数
A complex number is z = x + iy, where i² = −1. Its conjugate is z* = x − iy, modulus |z| = √x² + y², argument arg(z) = θ such that tanθ = y/x.
复数表示为 z = x + iy，其中 i² = −1。共轭复数为 z* = x − iy，模 |z| = √x² + y²，辐角 arg(z) = θ 满足 tanθ = y/x。
In polar form, z = r(cosθ + i sinθ) = r eiθ. De Moivre's theorem: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ) for integer n.
极坐标形式为 z = r(cosθ + i sinθ) = r eiθ。棣莫弗定理：对于整数 n，(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)。
The n distinct nth roots of unity are e2kπi/n for k = 0,1,…,n−1, and their sum is zero.
n 个不同的 n 次单位根为 e2kπi/n，k = 0,1,…,n−1，它们的和为零。
Loci in the complex plane include circles |z − a| = r, perpendicular bisectors |z − a| = |z − b|, and half-lines arg(z − a) = θ.
复平面上的轨迹包括圆 |z − a| = r，垂直平分线 |z − a| = |z − b|，以及射线 arg(z − a) = θ。
3. Matrices and Linear Transformations | 矩阵与线性变换
minant is det(M) = ad − bc. The inverse M⁻¹ = (1/det(M))[d −b; −c a] exists if det(M) ≠ 0.
对于 2×2 矩阵 M = [a b; c d]，行列式 det(M) = ad − bc。若 det(M) ≠ 0，其逆矩阵为 M⁻¹ = (1/det(M))[d −b; −c a]。
Eigenvalues λ satisfy det(M − λI) = 0. The corresponding eigenvectors v fulfil Mv = λv.
特征值 λ 满足 det(M − λI) = 0。对应的特征向量 v 满足 Mv = λv。
in line y = (tanθ)x is [cos2θ sin2θ; sin2θ −cos2θ]; shear parallel to x-axis: [1 k; 0 1].
标准变换：旋转 θ 角的矩阵为 [cosθ −sinθ; sinθ cosθ]；关于直线 y = (tanθ)x 的反射为 [cos2θ sin2θ; sin2θ −cos2θ]；平行于 x 轴的剪切为 [1 k; 0 1]。
A matrix M is singular if det(M) = 0; for a non-singular matrix, area scale factor = |det(M)|.
若 det(M) = 0，则矩阵 M 奇异；对于非奇异矩阵，面积缩放因子 = |det(M)|。
4. Vectors | 向量
The scalar product a·b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃. Vectors are perpendicular if a·b = 0.
数量积 a·b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃。若 a·b = 0，则两向量垂直。
The vector product a×b = |a||b|sinθ n̂ s area of parallelogram. In component form, a×b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁).
向量积 a×b = |a||b|sinθ n̂，其大小等于平行四边形的面积。分量形式为 a×b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)。
of a line: r = a + λb. Equation of a plane: r·n = p or r = a + λu + μv, where n is normal.
直线方程：r = a + λb。平面方程：r·n = p 或 r = a + λu + μv，其中 n 为法向量。
Distance from a point with position vector p to plane r·n = d is |p·n − d|/|n|.
位置向量为 p 的点到平面 r·n = d 的距离为 |p·n − d|/|n|。
5. Calculus and Series Expansions | 微积分与级数展开
Maclaurin series: f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + … . Standard expansions: eˣ = Σᵢ₌₀∞ xⁿ/n!, sin x = Σᵢ₌₀∞ (−1)ⁿ x²ⁿ⁺¹/(2n+1)!, cos x = Σᵢ₌₀∞ (−1)ⁿ x²ⁿ/(2n)!, ln(1+x) = Σᵢ₌₁∞ (−1)ⁿ⁻¹ xⁿ/n (|x| < 1).
麦克劳林级数：f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + …。标准展开：eˣ = Σₙ₌₀∞ xⁿ/n!，sin x = Σₙ₌₀∞ (−1)ⁿ x²ⁿ⁺¹/(2n+1)!，cos x = Σₙ₌₀∞ (−1)ⁿ x²ⁿ/(2n)!，ln(1+x) = Σₙ₌₁∞ (−1)ⁿ⁻¹ xⁿ/n（|x| < 1）。
The chain rule, product rule and quotient rule are essential: d/dx [f(g(x))] = f'(g(x))g'(x); d/dx (uv) = u'v + uv'; d/dx (u/v) = (u'v − uv')/v².
链式法则、乘积法则和商法则是基础：d/dx [f(g(x))] = f'(g(x))g'(x)；d/dx (uv) = u'v + uv'；d/dx (u/v) = (u'v − uv')/v²。
fractions. Volumes of revolution: about x-axis, V = π∫ y² dx; about y-axis, V = π∫ x² dy.
积分技巧：代换法，分部积分 ∫ u dv = uv − ∫ v du，部分分式法。旋转体体积：绕 x 轴，V = π∫ y² dx；绕 y 轴，V = π∫ x² dy。
6. 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. The identity cosh² x − sinh² x = 1 holds.
定义：sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x。恒等式 cosh² x − sinh² x = 1 成立。
x) = sech² x. Inverse hyperbolic functions can be expressed in log form: arsinh x = ln(x + √x²+1), arcosh x = ln(x + √x²−1) (x ≥ 1), artanh x = ½ ln((1+x)/(1−x)) (|x| < 1).
导数：d/dx (sinh x) = cosh x，d/dx (cosh x) = sinh x，d/dx (tanh x) = sech² x。反双曲函数可用对数表示：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: to convert a trigonometric identity into a hyperbolic one, replace every product of two sines by minus the product of two sinhs (if using sin² directly, replace with −sinh²).
奥斯本法则：将三角恒等式转换为双曲恒等式时，每个包含两个正弦的乘积替换为两个双曲正弦乘积的相反数（若直接使用 sin²，则替换为 −sinh²）。
7. Polar Coordinates | 极坐标
re r ≥ 0 and θ is the angle from the initial line. Conversion: x = r cosθ, y = r sinθ, r = √x² + y², θ = arctan(y/x).
点 P 表示为 (r, θ)，其中 r ≥ 0，θ 为从极轴量起的角度。转换：x = r cosθ，y = r sinθ，r = √x² + y²，θ = arctan(y/x)。
sector: A = ½ ∫ r² dθ between limits. Tangents to polar curves are found using dy/dx = (r' sinθ + r cosθ)/(r' cosθ − r sinθ).
扇形面积：A = ½ ∫ r² dθ（在界限之间）。极坐标曲线的切线使用 dy/dx = (r' sinθ + r cosθ)/(r' cosθ − r sinθ) 求解。
dioid r = a(1 + cosθ), limacon r = a + b cosθ, rose curves r = a sin(nθ) or r = a cos(nθ).
常见曲线：心形线 r = a(1 + cosθ)，蜗线 r = a + b cosθ，玫瑰线 r = a sin(nθ) 或 r = a cos(nθ)。
8. Summation of Series | 级数求和
Standard sums: Σ₁ⁿ r = n(n+1)/2, Σ₁ⁿ r² = n(n+1)(2n+1)/6, Σ₁ⁿ r³ = n²(n+1)²/4.
标准求和：Σ₁ⁿ r = n(n+1)/2，Σ₁ⁿ r² = n(n+1)(2n+1)/6，Σ₁ⁿ r³ = n²(n+1)²/4。
series, e.g. Σ 1/(r(r+1)) = 1 − 1/(n+1). General technique: express term as f(r) − f(r+1).
裂项法处理伸缩级数，例如 Σ 1/(r(r+1)) = 1 − 1/(n+1)。一般技巧：将项表为 f(r) − f(r+1)。
ometric series, nth term = arⁿ⁻¹, sum = a(1−rⁿ)/(1−r) (r≠1), sum to infinity = a/(1−r) for |r|<1.
等差数列第 n 项 = a + (n−1)d，和 = n/2 (2a + (n−1)d)。等比数列第 n 项 = arⁿ⁻¹，和 = a(1−rⁿ)/(1−r)（r≠1），无穷和 = a/(1−r)（|r|<1）。
9. Differential Equations | 微分方程
First order separable: dy/dx = f(x)g(y) ⇒ ∫ 1/g(y) dy = ∫ f(x) dx. First order linear: dy/dx + P(x)y = Q(x) has integrating factor e∫ P dx.
一阶可分离：dy/dx = f(x)g(y) ⇒ ∫ 1/g(y) dy = ∫ f(x) dx。一阶线性：dy/dx + P(x)y = Q(x) 的积分因子为 e∫ P dx。
y = 0. Auxiliary equation am² + bm + c = 0; if real distinct roots m₁, m₂, solution y = Aem₁x + Bem₂x; repeated root m: y = (A+Bx)emx; complex roots α±iβ: y = eαx(Acosβx + B sinβx).
常系数二阶线性齐次方程：a d²y/dx² + b dy/dx + c y = 0。辅助方程 am² + bm + c = 0；若为不等实根 m₁, m₂，解为 y = Aem₁x + Bem₂x；重根 m：y = (A+Bx)emx；复根 α±iβ：y = eαx(Acosβx + B sinβx)。
For non-homogeneous equations, the particular integral is found by trial functions: polynomial, exponential, trigonometric, or combinations.
对于非齐次方程，特解通过试探函数求得：多项式、指数、三角函数或它们的组合。
Simple harmonic motion: d²x/dt² = −ω²x has solution x = A cosωt + B sinωt = R sin(ωt + φ).
简谐运动：d²x/dt² = −ω²x 的解为 x = A cosωt + B sinωt = R sin(ωt + φ)。
10. Mechanics: Kinematics and Dynamics | 力学：运动学与动力学
ension with constant acceleration: v = u + at, s = ut + ½at², v² = u² + 2as, s = (u+v)t/2.
匀加速直线运动：v = u + at，s = ut + ½at²，v² = u² + 2as，s = (u+v)t/2。
rce × distance moved in direction of force. Kinetic energy = ½mv², potential energy = mgh.
牛顿第二定律：F = ma。冲量 = 动量变化 = mv − mu。功 = 力 × 沿力方向的位移。动能 = ½mv²，势能 = mgh。
ernal work done. Coefficient of restitution e = (speed of separation)/(speed of approach).
功率 P = Fv = dW/dt。能量守恒：若无外力做功，总机械能保持不变。恢复系数 e =（分离速度）/（接近速度）。
ange R = (u² sin2θ)/g, maximum height H = (u² sin²θ)/(2g), time of flight T = (2u sinθ)/g.
抛体运动：水平射程 R = (u² sin2θ)/g，最大高度 H = (u² sin²θ)/(2g)，飞行时间 T = (2u sinθ)/g。
11. Statistics and Probability | 统计与概率
Var(X) = E(X²) − [E(X)]². For a linear function, E(aX+b) = aE(X)+b, Var(aX+b) = a² Var(X).
离散随机变量 X：期望 E(X) = Σ x P(X=x)，方差 Var(X) = E(X²) − [E(X)]²。对于线性函数，E(aX+b) = aE(X)+b，Var(aX+b) = a² Var(X)。
): mean = λ, variance = λ. Geometric distribution Geo(p): mean = 1/p, variance = (1−p)/p².
二项分布 B(n, p)：均值 = np，方差 = np(1−p)。泊松分布 Po(λ)：均值 = λ，方差 = λ。几何分布 Geo(p)：均值 = 1/p，方差 = (1−p)/p²。
nction f(x) ≥ 0, ∫ f(x) dx = 1 over domain. E(X) = ∫ x f(x) dx, Var(X) = ∫ (x−μ)² f(x) dx.
连续随机变量：概率密度函数 f(x) ≥ 0，在定义域上 ∫ f(x) dx = 1。E(X) = ∫ x f(x) dx，Var(X) = ∫ (x−μ)² f(x) dx。
Normal distribution N(μ, σ²): if X ~ N(μ, σ²), then Z = (X−μ)/σ ~ N(0,1). Use tables for Φ(z). Sum of independent normal variables is also normal.
正态分布 N(μ, σ²)：若 X ~ N(μ, σ²)，则 Z = (X−μ)/σ ~ N(0,1)。使用标准正态分布表求 Φ(z)。独立正态变量之和仍服从正态分布。
12. Numerical Methods | 数值方法
f roots: if f(a) and f(b) have opposite signs and f is continuous on [a,b], there exists c ∈ (a,b) with f(c)=0. The Newton-Raphson iteration: xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ).
根的定位：若 f(a) 与 f(b) 异号且 f 在 [a,b] 上连续，则存在 c ∈ (a,b) 使得 f(c)=0。牛顿-拉夫森迭代：xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ)。
Numerical integration: the trapezium rule approximates ∫ₐᵇ f(x) dx ≈ h/2 [y₀ + 2(y₁+ y₂+ … + yₙ₋₁) + yₙ] where h = (b−a)/n. Simpson's rule (n even): ≈ h/3 [y₀ + 4y₁ + 2y₂ + 4y₃ + … + 4yₙ₋₁ + yₙ].
数值积分：梯形法则近似 ∫ₐᵇ f(x) dx ≈ h/2 [y₀ + 2(y₁+ y₂+ … + yₙ₋₁) + yₙ]，其中 h = (b−a)/n。辛普森法则（n 为偶数）：≈ h/3 [y₀ + 4y₁ + 2y₂ + 4y₃ + … + 4yₙ₋₁ + yₙ]。
f(x,y): yₙ₊₁ = yₙ + h f(xₙ, yₙ). Improved Euler (Heun’s) method uses predictor-corrector.
欧拉法求解 dy/dx = f(x,y)：yₙ₊₁ = yₙ + h f(xₙ, yₙ)。改进欧拉法（休恩法）使用预测-校正公式。
Fixed-point iteration: rearrange equation to x = g(x); iteration xₙ₊₁ = g(xₙ) converges if |g'(x)| < 1 near the root.
不动点迭代：将方程改写为 x = g(x)；迭代 xₙ₊₁ = g(xₙ) 在根附近若 |g'(x)| < 1 则收敛。
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