📚 A-Level Cambridge Further Mathematics: Formula and Theorem Quick Reference Handbook | A-Level Cambridge 进阶数学：公式定理速查手册
This quick-reference guide collects the essential formulas, theorems, and identities required for the Cambridge International A-Level Further Mathematics syllabus (9231). It covers Further Pure Mathematics topics from complex numbers and matrices to differential equations and series expansions, presenting each concept with a concise statement in English followed by its Chinese equivalent. Whether you are consolidating your revision or seeking a rapid reminder before the exam, this handbook is designed to provide clarity and speed.
本手册汇总了剑桥国际 A-Level 进阶数学（9231）课程所需的核心公式、定理和恒等式。内容涵盖进阶纯数学的复数、矩阵、微分方程、级数展开等专题，每个概念先用英文简述，再配以中文对应说明。无论是巩固复习还是考前速查，这本手册都旨在帮助你快速理清要点，高效备考。
1. Complex Numbers – Key Forms and Operations | 复数 – 基本形式与运算
A complex number can be expressed in Cartesian form as z = x + iy, where x, y are real numbers and i² = −1. The modulus is |z| = √(x² + y²) and the argument is arg(z) = θ, where tan θ = y/x, with θ usually chosen in (−π, π]. The complex conjugate is z* = x − iy.
复数可表示为笛卡尔形式 z = x + iy，其中 x, y 为实数，且 i² = −1。模为 |z| = √(x² + y²)，辐角为 arg(z) = θ，满足 tan θ = y/x，θ 通常取在 (−π, π] 范围内。共轭复数为 z* = x − iy。
The polar form is z = r(cos θ + i sin θ) with r = |z|. Euler’s formula gives the exponential form: z = r e^(iθ). Multiplication and division in polar form: if z₁ = r₁e^(iθ₁) and z₂ = r₂e^(iθ₂), then z₁z₂ = r₁r₂e^(i(θ₁+θ₂)) and z₁/z₂ = (r₁/r₂)e^(i(θ₁−θ₂)). De Moivre’s theorem states (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ) for integer n.
极坐标形式为 z = r(cos θ + i sin θ)，其中 r = |z|。利用欧拉公式可得指数形式：z = r e^(iθ)。极坐标下的乘除运算：若 z₁ = r₁e^(iθ₁)，z₂ = r₂e^(iθ₂)，则 z₁z₂ = r₁r₂e^(i(θ₁+θ₂))，z₁/z₂ = (r₁/r₂)e^(i(θ₁−θ₂))。棣莫弗定理指出，对于整数 n，(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)。
2. Roots of Polynomials and Relationships | 多项式根与系数关系
For a cubic equation ax³ + bx² + cx + d = 0 with roots α, β, γ, the sum of roots Σα = −b/a, the sum of pairwise products Σαβ = c/a, and the product αβγ = −d/a. For a quartic ax⁴ + bx³ + cx² + dx + e = 0, Σα = −b/a, Σαβ = c/a, Σαβγ = −d/a, αβγδ = e/a.
对于三次方程 ax³ + bx² + cx + d = 0，其根为 α, β, γ，则有根之和 Σα = −b/a，两两根积之和 Σαβ = c/a，根之积 αβγ = −d/a。对于四次方程 ax⁴ + bx³ + cx² + dx + e = 0，Σα = −b/a，Σαβ = c/a，Σαβγ = −d/a，αβγδ = e/a。
These symmetric sums can be used to find expressions like Σα² = (Σα)² − 2Σαβ. They also help in forming new equations whose roots are transformations of the original roots, for example using substitution y = f(x).
这些对称和可用于计算如 Σα² = (Σα)² − 2Σαβ 等表达式，也可用于构造以原根变换为新根的新方程，例如通过替换 y = f(x) 来实现。
3. Matrices – Determinants, Inverses and Transformations | 矩阵 – 行列式、逆矩阵与变换
For a 2×2 matrix M = [[a, b], [c, d]], the determinant is det(M) = ad − bc. The inverse is M⁻¹ = (1/det(M)) [[d, −b], [−c, a]], provided det(M) ≠ 0. For a 3×3 matrix, the determinant can be evaluated by expansion along a row or column.
对于 2×2 矩阵 M = [[a, b], [c, d]]，行列式为 det(M) = ad − bc。若 det(M) ≠ 0，逆矩阵为 M⁻¹ = (1/det(M)) [[d, −b], [−c, a]]。对于 3×3 矩阵，行列式可按某一行或某一列展开计算。
A matrix represents a linear transformation. The image of a vector v under transformation M is Mv. The characteristic equation is det(M − λI) = 0, whose roots are eigenvalues. Eigenvectors satisfy (M − λI)v = 0. Diagonalisation is possible if there are enough linearly independent eigenvectors: M = PDP⁻¹, where D is diagonal and contains eigenvalues.
矩阵代表线性变换。向量 v 在变换 M 下的像为 Mv。特征方程为 det(M − λI) = 0，其根为特征值。特征向量满足 (M − λI)v = 0。若存在足够多线性无关的特征向量，则可对角化：M = PDP⁻¹，其中 D 为对角矩阵且对角线元素为特征值。
4. Hyperbolic Functions – Definitions and Identities | 双曲函数 – 定义与恒等式
The hyperbolic functions are defined by: sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x. Their derivatives are d/dx(sinh x) = cosh x, d/dx(cosh x) = sinh x, d/dx(tanh x) = sech² x. Key identity: cosh² x − sinh² x = 1.
双曲函数定义为：sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x。其导数为：d/dx(sinh x) = cosh x，d/dx(cosh x) = sinh x，d/dx(tanh x) = sech² x。核心恒等式：cosh² x − sinh² x = 1。
Inverse hyperbolic functions are expressible in logarithms: 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. These are useful for integration and solving equations.
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）。这些形式在积分和方程求解中非常实用。
5. Polar Coordinates – Curves and Area | 极坐标 – 曲线与面积
In polar coordinates, a point is given by (r, θ), where r is the distance from the origin and θ the angle from the initial line. Cartesian conversion: x = r cos θ, y = r sin θ. The area enclosed by a polar curve r = f(θ) from θ = α to θ = β is A = ½ ∫[α,β] r² dθ.
在极坐标中，点用 (r, θ) 表示，r 为到原点的距离，θ 为从极轴起量的角度。与直角坐标的转换：x = r cos θ，y = r sin θ。极坐标曲线 r = f(θ) 在 θ = α 到 θ = β 之间围成的面积为 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. Common curves include cardioids (r = a(1 + cos θ)) and circles (r = 2a cos θ). Tangents are parallel to the initial line when d/dθ(r sin θ) = 0, and perpendicular when d/dθ(r cos θ) = 0.
极坐标曲线在 θ = α 到 θ = β 之间的弧长为 s = ∫[α,β] √(r² + (dr/dθ)²) dθ。常见曲线包括心形线 r = a(1 + cos θ) 和圆 r = 2a cos θ。切线平行于极轴当 d/dθ(r sin θ) = 0，垂直于极轴当 d/dθ(r cos θ) = 0。
6. Differential Equations – First and Second Order | 微分方程 – 一阶与二阶
First-order separable equations: if dy/dx = g(x)h(y), rewrite as ∫(1/h(y)) dy = ∫g(x) dx. The integrating factor method applies to linear equations dy/dx + P(x)y = Q(x); the integrating factor is IF = e^(∫P(x) dx). The solution is y·IF = ∫ Q(x)·IF dx.
一阶可分离变量方程：若 dy/dx = g(x)h(y)，可改写为 ∫(1/h(y)) dy = ∫g(x) dx。积分因子法适用于线性方程 dy/dx + P(x)y = Q(x)；积分因子为 IF = e^(∫P(x) dx)。通解为 y·IF = ∫ Q(x)·IF dx。
Second-order linear homogeneous differential equations with constant coefficients: a d²y/dx² + b dy/dx + c y = 0. The auxiliary equation is am² + bm + c = 0. If roots m₁, m₂ are real and distinct, y = Ae^(m₁x) + Be^(m₂x). If repeated root m, y = (A + Bx)e^(mx). If complex conjugate roots α ± iβ, y = e^(αx)(A cos βx + B sin βx). For non-homogeneous equations, find a particular integral using trial functions (polynomial, exponential, trigonometric) and add to the complementary function.
常系数二阶线性齐次微分方程：a d²y/dx² + b dy/dx + c y = 0。辅助方程为 am² + bm + c = 0。若两实根 m₁, m₂ 不等，则 y = Ae^(m₁x) + Be^(m₂x)；若为重根 m，则 y = (A + Bx)e^(mx)；若为共轭复根 α ± iβ，则 y = e^(αx)(A cos βx + B sin βx)。对于非齐次方程，使用多项式、指数函数或三角函数形式的试探解求特积分，再与补函数相加。
7. Series Expansions – Maclaurin and Taylor Series | 级数展开 – 麦克劳林与泰勒级数
The Maclaurin series expansion of a function f(x) about x = 0 is f(x) = f(0) + f ‘(0)x + f ”(0)x²/2! + f ”'(0)x³/3! + … . The Taylor series about x = a is f(x) = f(a) + f ‘(a)(x−a) + f ”(a)(x−a)²/2! + … .
函数 f(x) 在 x = 0 处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! + … 。在 x = a 处的泰勒级数为 f(x) = f(a) + f ‘(a)(x−a) + f ”(a)(x−a)²/2! + … 。
Standard Maclaurin series include: eˣ = 1 + x + x²/2! + x³/3! + … ; sin x = x − x³/3! + x⁵/5! − … ; cos x = 1 − x²/2! + x⁴/4! − … ; ln(1+x) = x − x²/2 + x³/3 − … (valid −1 < x ≤ 1). These can be used to approximate functions and to evaluate limits.
标准麦克劳林级数包括：eˣ = 1 + x + x²/2! + x³/3! + … ；sin x = x − x³/3! + x⁵/5! − … ；cos x = 1 − x²/2! + x⁴/4! − … ；ln(1+x) = x − x²/2 + x³/3 − … （有效区间 −1 < x ≤ 1）。这些展开可用于近似计算和求极限。
8. Vector Geometry – Lines and Planes in 3D | 向量几何 – 空间直线与平面
A line in 3D can be expressed in vector form as r = a + λb, where a is a position vector of a point on the line and b is a direction vector. The Cartesian form is (x−x₁)/l = (y−y₁)/m = (z−z₁)/n. A plane can be written as r·n = p or in scalar form ax + by + cz = d, where n = (a, b, c) is a normal vector.
三维空间中的直线可用向量方程表示为 r = a + λb，其中 a 为直线上一点的位矢，b 为方向向量。笛卡尔形式为 (x−x₁)/l = (y−y₁)/m = (z−z₁)/n。平面可表示为 r·n = p 或标量形式 ax + by + cz = d，其中 n = (a, b, c) 为法向量。
The angle between two lines is given by cos θ = |b₁·b₂|/(|b₁||b₂|). The angle between a line and a plane is sin θ = |b·n|/(|b||n|). The distance from a point with position vector a to the plane r·n = p is |a·n − p|/|n|. The intersection of two planes is a line, and the intersection of three planes can be a point, a line, or no solution.
两直线的夹角满足 cos θ = |b₁·b₂|/(|b₁||b₂|)。直线与平面的夹角满足 sin θ = |b·n|/(|b||n|)。点（位矢为 a）到平面 r·n = p 的距离为 |a·n − p|/|n|。两平面相交于一条直线，三平面相交于一点、一条线或无解。
9. Numerical Methods – Root Finding and Integration | 数值方法 – 求根与积分
The Newton-Raphson method for solving f(x)=0 uses the iterative formula xₙ₊₁ = xₙ − f(xₙ)/f ‘(xₙ). It converges quickly if the initial guess is sufficiently close to the root. The formula is derived from the tangent approximation.
求解方程 f(x)=0 的牛顿-拉夫森法使用迭代公式 xₙ₊₁ = xₙ − f(xₙ)/f ‘(xₙ)。若初始值足够接近根，该方法收敛速度很快。该公式由切线近似导出。
For numerical integration, the trapezium rule estimates ∫[a,b] y dx ≈ h[(y₀+yₙ)/2 + y₁ + y₂ + … + yₙ₋₁] with step size h = (b−a)/n. Simpson’s rule (n even) is more accurate: ∫[a,b] y dx ≈ (h/3)[y₀ + yₙ + 4(y₁+y₃+…) + 2(y₂+y₄+…)]. Another iterative method is fixed-point iteration xₙ₊₁ = g(xₙ), which converges if |g'(x)| < 1 near the root.
数值积分方面，梯形法则用步长 h = (b−a)/n 估计 ∫[a,b] y dx ≈ h[(y₀+yₙ)/2 + y₁ + y₂ + … + yₙ₋₁]。辛普森法则（n 为偶数）精度更高：∫[a,b] y dx ≈ (h/3)[y₀ + yₙ + 4(y₁+y₃+…) + 2(y₂+y₄+…)]。另一种迭代法是固定点迭代 xₙ₊₁ = g(xₙ)，当在根附近 |g'(x)| < 1 时收敛。
10. Further Calculus – Reduction Formulae and Arc Length | 进阶微积分 – 归约公式与弧长
Reduction formulas are used to evaluate integrals of powers of functions. For example, define Iₙ = ∫ sinⁿ x dx. Integration by parts gives Iₙ = −(1/n) sinⁿ⁻¹ x cos x + ((n−1)/n) Iₙ₋₂. Similarly, for Iₙ = ∫ cosⁿ x dx, a similar pattern arises. These formulas recursively reduce the power until a base integral is reached.
归约公式用于计算函数幂次的积分。例如，定义 Iₙ = ∫ sinⁿ x dx，利用分部积分可得 Iₙ = −(1/n) sinⁿ⁻¹ x cos x + ((n−1)/n) Iₙ₋₂。对于 Iₙ = ∫ cosⁿ x dx，也有类似的递推关系。这些公式递归地降低幂次，直至达到基础积分。
Arc length of a curve y = f(x) from x=a to x=b is L = ∫[a,b] √(1 + (dy/dx)²) dx. For a parametric curve (x(t), y(t)), L = ∫[t₁,t₂] √((dx/dt)² + (dy/dt)²) dt. The surface area of revolution about the x-axis for y = f(x) is S = 2π ∫[a,b] y √(1 + (dy/dx)²) dx.
曲线 y = f(x) 从 x=a 到 x=b 的弧长为 L = ∫[a,b] √(1 + (dy/dx)²) dx。对于参数曲线 (x(t), y(t))，弧长 L = ∫[t₁,t₂] √((dx/dt)² + (dy/dt)²) dt。曲线绕 x 轴旋转的旋转面面积为 S = 2π ∫[a,b] y √(1 + (dy/dx)²) dx。
11. Summation of Series – Standard Results and Method of Differences | 级数求和 – 标准结果与差分法
Standard summation formulas: Σ[r=1 to n] 1 = n; Σ[r=1 to n] r = n(n+1)/2; Σ[r=1 to n] r² = n(n+1)(2n+1)/6; Σ[r=1 to n] r³ = [n(n+1)/2]². These can be used to find sums of polynomial series.
标准求和公式：Σ[r=1 to n] 1 = n；Σ[r=1 to n] r = n(n+1)/2；Σ[r=1 to n] r² = n(n+1)(2n+1)/6；Σ[r=1 to n] r³ = [n(n+1)/2]²。这些可用于求多项式级数的和。
The method of differences applies when a term u_r can be expressed as f(r) − f(r+1) or f(r+1) − f(r). Then Σ[r=1 to n] u_r telescopes to f(1) − f(n+1) or similar. This is particularly useful for rational functions where partial fractions reveal cancellations.
差分法适用于某项 u_r 可表示为 f(r) − f(r+1) 或 f(r+1) − f(r) 的情形。此时 Σ[r=1 to n] u_r 可错位相消为 f(1) − f(n+1) 等形式。对于有理函数，通过部分分式常能发现抵消除项，此法尤为有效。
12. Proof by Induction – Framework and Examples | 归纳证明法 – 框架与示例
The principle of mathematical induction involves proving a statement P(n) for all positive integers n. Steps: 1) Base case: verify P(1) is true. 2) Inductive step: assume P(k) true for some arbitrary k ≥ 1, then prove P(k+1) is true using this assumption. 3) Conclusion: by induction, P(n) holds for all n ∈ ℕ.
数学归纳法原理用于证明对所有正整数 n 的命题 P(n)。步骤：1）归纳基础：验证 P(1) 为真。2）归纳递推：假设对某个任意 k ≥ 1，P(k) 为真，据此证明 P(k+1) 也为真。3）结论：由归纳法可知，对所有 n ∈ ℕ，P(n) 成立。
Common induction proofs include divisibility (e.g., 3²ⁿ − 1 is divisible by 8), summation formulas (e.g., Σr = n(n+1)/2), recurrence relations (e.g., proving closed form of uₙ from uₙ₊₁ = Auₙ + B), and matrix powers (e.g., Mⁿ has a specific form). Induction is also used to prove inequalities such as 2ⁿ > n² for n ≥ 5.
常见的归纳证明包括整除性（如 3²ⁿ − 1 被 8 整除）、求和公式（如 Σr = n(n+1)/2）、递推关系（如由 uₙ₊₁ = Auₙ + B 证明 uₙ 的封闭形式）以及矩阵幂次（如 Mⁿ 具有特定形式）。归纳法也用来证明不等式，例如当 n ≥ 5 时，2ⁿ > n²。
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