📚 Year 13 Cambridge Further Mathematics: Formula & Theorem Quick Reference Handbook | Year 13 剑桥进阶数学：公式定理速查手册
This quick reference handbook provides a concise summary of key formulas and theorems from the Cambridge Further Mathematics syllabus. It is designed to help Year 13 students consolidate essential pure mathematics concepts for revision and examination preparation.
本速查手册简要总结了剑桥进阶数学课程中的核心公式与定理，旨在帮助 Year 13 学生巩固纯数学关键概念，便于复习与备考。
1. Summation of Series | 级数求和
Standard summation formulas for the first n natural numbers, their squares, and cubes are essential. For ∑r=1n r = n(n+1)/2, ∑r=1n r2 = n(n+1)(2n+1)/6, and ∑r=1n r3 = n2(n+1)2/4. The method of differences expresses a sum as a telescoping series, e.g. ∑ (f(r) − f(r+1)) = f(1) − f(n+1).
前 n 个自然数之和、平方和与立方和的标准公式如下：∑r=1n r = n(n+1)/2，∑r=1n r2 = n(n+1)(2n+1)/6，∑r=1n r3 = n2(n+1)2/4。差分法可将和式化为望远镜级数，例如 ∑ (f(r) − f(r+1)) = f(1) − f(n+1)。
2. Proof by Induction | 数学归纳法
The principle of mathematical induction has three main steps: prove the statement is true for an initial value (base case, commonly n = 1), assume it holds for n = k (inductive hypothesis), and then show it must be true for n = k+1. The conclusion follows for all positive integers n. For example, to prove ∑r=1n r = n(n+1)/2, verify for n = 1, assume true for k, and show that adding (k+1) gives the formula for k+1.
数学归纳法包含三个主要步骤：证明初始值（通常 n = 1）时命题成立（基础情况）；假设 n = k 时成立（归纳假设）；进而证明 n = k+1 时也成立。由此推出对所有正整数 n 成立。例如，为了证明 ∑r=1n r = n(n+1)/2，先验证 n = 1，假设对 k 成立，再通过加上 (k+1) 证明对 k+1 也成立。
3. Complex Numbers | 复数
A complex number z = x + iy can be written in polar form z = r(cos θ + i sin θ), where r = |z| = √(x2+y2) and θ = arg z. Multiplication and division become simple: z1z2 = r1r2(cos(θ1+θ2) + i sin(θ1+θ2)). De Moivre’s theorem states (cos θ + i sin θ)n = cos(nθ) + i sin(nθ), which is powerful for finding powers and roots of complex numbers.
复数 z = x + iy 可写为极坐标形式 z = r(cos θ + i sin θ)，其中 r = |z| = √(x2+y2)，θ = arg z。乘法和除法变得简单：z1z2 = r1r2(cos(θ1+θ2) + i sin(θ1+θ2))。棣莫弗定理指出 (cos θ + i sin θ)n = cos(nθ) + i sin(nθ)，该定理在计算复数的乘幂和根时非常有用。
The nth roots of a complex number are given by zk = r1/n (cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)) for k = 0,1,…,n−1. The nth roots of unity satisfy zn = 1 and lie symmetrically on the unit circle. Loci of points in the complex plane, such as |z − z0| = r (circle) or arg(z − z0) = α (ray), are frequently examined.
复数的 n 次根由 zk = r1/n (cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)) 给出，k = 0,1,…,n−1。单位根满足 zn = 1，并对称地分布在单位圆上。复平面上的轨迹，如 |z − z0| = r（圆）或 arg(z − z0) = α（射线），经常出现在考试中。
4. Matrices | 矩阵
The determinant of a 2×2 matrix A = [a b; c d] is det A = ad − bc. For a 3×3 matrix, the determinant can be expanded using minors. The inverse of a non-singular matrix A is A−1 = (1/det A) adj A, where adj A is the transpose of the cofactor matrix. For a 2×2 case, A−1 = (1/(ad−bc)) [d −b; −c a].
2×2 矩阵 A = [a b; c d] 的行列式为 det A = ad − bc。对于 3×3 矩阵，可以使用余子式展开计算行列式。非奇异矩阵 A 的逆为 A−1 = (1/det A) adj A，其中 adj A 是余子式矩阵的转置。在 2×2 情形下，A−1 = (1/(ad−bc)) [d −b; −c a]。
Eigenvalues and eigenvectors are fundamental for diagonalisation. For matrix A, solve det(A − λI) = 0 to find eigenvalues λ. If a matrix has n distinct eigenvectors, it can be diagonalised as A = PDP−1, where D is a diagonal matrix of eigenvalues and P has the corresponding eigenvectors as columns. This simplifies computing powers of A: An = PDnP−1.
特征值与特征向量是对角化的基础。对于矩阵 A，求解 det(A − λI) = 0 可得特征值 λ。如果一个矩阵有 n 个线性无关的特征向量，则可对角化为 A = PDP−1，其中 D 是对角矩阵，对角线元素为特征值；P 的列为对应的特征向量。这大大简化了矩阵幂的计算：An = PDnP−1。
5. Vectors | 向量
The scalar (dot) product of two vectors a · b = |a||b| cos θ is used to find angles and test perpendicularity. The vector (cross) product a × b = |a||b| sin θ n̂ gives a vector perpendicular to both a and b. A line can be written as r = a + t d, and a plane as r · n = d or r = a + λu + μv. The distance from a point to a line or plane can be found using vector projections and cross products.
向量的数量（点）积 a · b = |a||b| cos θ 常用于求角度和判断垂直关系。向量（叉）积 a × b = |a||b| sin θ n̂ 产生一个垂直于 a 和 b 的向量。直线可表示为 r = a + t d，平面可表示为 r · n = d 或 r = a + λu + μv。点到直线的距离以及点到平面的距离可利用向量投影和叉积求得。
6. Hyperbolic Functions | 双曲函数
The basic hyperbolic functions are defined by sinh x = (ex − e−x)/2, cosh x = (ex + e−x)/2, and tanh x = sinh x / cosh x. Important identities mirror trigonometric ones but with sign differences: cosh2x − sinh2x = 1, sinh 2x = 2 sinh x cosh x, and cosh 2x = cosh2x + sinh2x.
基本双曲函数定义为 sinh x = (ex − e−x)/2，cosh x = (ex + e−x)/2，tanh x = sinh x / cosh x。重要的恒等式与三角恒等式类似，但符号有所不同：cosh2x − sinh2x = 1，sinh 2x = 2 sinh x cosh x，cosh 2x = cosh2x + sinh2x。
Derivatives are straightforward: d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, and d/dx (tanh x) = sech2 x. The inverse hyperbolic functions can be expressed as logarithms, e.g. arsinh x = ln(x + √(x2+1)), and their derivatives are d/dx (arsinh x) = 1/√(1+x2), d/dx (arcosh x) = 1/√(x2−1) for x > 1.
导数计算直接明了：d/dx (sinh x) = cosh x，d/dx (cosh x) = sinh x，d/dx (tanh x) = sech2 x。反双曲函数可表示为对数形式，例如 arsinh x = ln(x + √(x2+1))。其导数分别为 d/dx (arsinh x) = 1/√(1+x2)，d/dx (arcosh x) = 1/√(x2−1) (x > 1)。
7. Differentiation | 微分
Leibniz’s theorem gives the nth derivative of a product: dn/dxn(uv) = ∑k=0n C(n, k) u(n−k) v(k), where C(n, k) = n!/(k!(n−k)!). Standard derivatives include d/dx (arctan x) = 1/(1+x2), d/dx (arcsin x) = 1/√(1−x2), and corresponding hyperbolic forms. Repeated differentiation can be applied to find Maclaurin series.
莱布尼兹定理给出乘积的 n 阶导数：dn/dxn(uv) = ∑k=0n C(n, k) u(n−k) v(k)，其中 C(n, k) = n!/(k!(n−k)!)。常用的标准导数包括 d/dx (arctan x) = 1/(1+x2)，d/dx (arcsin x) = 1/√(1−x2)，以及相应的双曲形式。反复求导可用于求麦克劳林展开式。
8. Integration | 积分
Reduction formulas are used to integrate powers of trigonometric functions recursively. For example, if In = ∫ sinn x dx, then In = −(1/n) sinn−1 x cos x + ((n−1)/n) In−2. Similarly for cosn x. These are derived using integration by parts and allow evaluation of definite integrals like ∫0π/2 sinn x dx.
约化公式用于递推求解三角函数的幂次积分。例如，若 In = ∫ sinn x dx，则 In = −(1/n) sinn−1 x cos x + ((n−1)/n) In−2。cosn x 的情形类似。这些公式通过分部积分法得出，并可用来计算形如 ∫0π/2 sinn x dx 的定积分。
Further integration techniques include arc length s = ∫ √(1 + (dy/dx)2) dx and surface area of revolution S = ∫ 2πy √(1 + (dy/dx)2) dx for curves rotated about the x-axis. For parametric curves, use s = ∫ √((dx/dt)2 + (dy/dt)2) dt and similarly for surface area.
进一步的积分应用包括弧长公式 s = ∫ √(1 + (dy/dx)2) dx 和绕 x 轴旋转的曲面面积公式 S = ∫ 2πy √(1 + (dy/dx)2) dx。对于参数曲线，使用 s = ∫ √((dx/dt)2 + (dy/dt)2) dt，曲面面积也做相应的替换。
9. Differential Equations | 微分方程
A first-order linear differential equation of the form dy/dx + P(x) y = Q(x) can be solved using the integrating factor μ(x) = e∫ P dx. Multiplying through by μ gives d/dx (μy) = μQ, which can be integrated directly. This method is a core technique throughout the syllabus.
形如 dy/dx + P(x) y = Q(x) 的一
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