Year 13 CIE Further Mathematics: Formula & Theorem Quick Reference Handbook | Year 13 CIE 进阶数学：公式定理速查手册

📚 Year 13 CIE Further Mathematics: Formula & Theorem Quick Reference Handbook | Year 13 CIE 进阶数学：公式定理速查手册

This concise handbook collates the essential formulas, definitions, and theorems required for the Year 13 CIE Further Mathematics syllabus. It is designed for rapid revision and quick lookup before exams, covering Further Pure topics as well as key results from mechanics and statistics components.

本简明手册汇集了 Year 13 CIE 进阶数学所需的基本公式、定义与定理，旨在便于考前快速复习与查阅，涵盖进阶纯数以及力学和统计部分的主要结论。

1. Hyperbolic Functions | 双曲函数

Hyperbolic sine and cosine are defined in terms of exponential functions.

双曲正弦与双曲余弦由指数函数定义。

sinh x = (eˣ − e⁻ˣ) / 2, cosh x = (eˣ + e⁻ˣ) / 2

The fundamental hyperbolic identity is cosh²x − sinh²x = 1, analogous to the trigonometric identity.

基本双曲恒等式为 cosh²x − sinh²x = 1，与三角恒等式类似。

cosh²x − sinh²x = 1, 1 − tanh²x = sech²x, coth²x − 1 = csch²x

Addition formulas mirror those for sine and cosine but with sign differences.

加法公式与正弦、余弦相似，但符号存在差异。

sinh(A ± B) = sinh A cosh B ± cosh A sinh B
cosh(A ± B) = cosh A cosh B ± sinh A sinh B

Derivatives of hyperbolic functions are cyclic, with no negative sign appearing on sinh.

双曲函数的导数呈现循环关系，且 sinh 的导数不含负号。

d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech² x

2. Inverse Hyperbolic Functions | 反双曲函数

The inverse hyperbolic functions can be expressed using natural logarithms.

反双曲函数可用自然对数表达。

arsinh x = ln(x + √(x² + 1)) for all x
arcosh x = ln(x + √(x² − 1)) for x ≥ 1
artanh x = ½ ln((1 + x)/(1 − x)) for |x| < 1

Their derivatives are rational or algebraic, providing standard integrals.

其导数是有理函数或代数函数，可作为标准积分使用。

d/dx (arsinh x) = 1/√(x² + 1), d/dx (arcosh x) = 1/√(x² − 1) (x > 1), d/dx (artanh x) = 1/(1 − x²) (|x| < 1)

3. Matrices and Determinants | 矩阵与行列式

For a 2×2 matrix A = [a b; c d], the determinant is det(A) = ad − bc.

对于 2×2 矩阵 A = [a b; c d]，行列式为 det(A) = ad − bc。

det(A) = ad − bc

For a 3×3 matrix, the determinant expands along the first row using cofactors.

对于 3×3 矩阵，行列式按第一行展开并用余子式计算。

det(A) = a(ei − fh) − b(di − fg) + c(dh − eg) for rows a,b,c; d,e,f; g,h,i

The inverse of a matrix is given by A⁻¹ = (1/det A) adj(A), where adj(A) is the adjugate (transpose of cofactor matrix). A 2×2 rotation matrix anticlockwise by θ is [[cos θ, -sin θ], [sin θ, cos θ]].

逆矩阵为 A⁻¹ = (1/det A) adj(A)，其中 adj(A) 为伴随矩阵（余子式矩阵的转置）。绕原点逆时针旋转 θ 的 2×2 旋转矩阵为 [[cos θ, -sin θ], [sin θ, cos θ]]。

4. Eigenvalues and Eigenvectors | 特征值与特征向量

For a square matrix A, an eigenvalue λ satisfies det(A − λI) = 0. The corresponding eigenvector v fulfills Av = λv.

对于方阵 A，特征值 λ 满足 det(A − λI) = 0。对应的特征向量 v 满足 Av = λv。

det(A − λI) = 0

Diagonalisation uses the matrix P whose columns are eigenvectors, giving P⁻¹AP = D, where D is diagonal with eigenvalues.

对角化利用特征向量构成的矩阵 P，使得 P⁻¹AP = D，其中 D 为以特征值为对角元的对角矩阵。

P⁻¹AP = D

5. Complex Numbers (Polar Form & De Moivre) | 复数（极坐标形式与棣莫弗定理）

A complex number z = x + iy has modulus r = |z| = √(x² + y²) and argument θ = arg(z). The polar form is z = r(cos θ + i sin θ) = r e^(iθ).

复数 z = x + iy 的模为 r = |z| = √(x² + y²)，辐角为 θ = arg(z)。极坐标形式为 z = r(cos θ + i sin θ) = r e^(iθ)。

Multiplication and division in polar form are: z₁z₂ = r₁r₂ e^(i(θ₁+θ₂)), z₁/z₂ = (r₁/r₂) e^(i(θ₁−θ₂)).

极坐标形式的乘除法为：z₁z₂ = r₁r₂ e^(i(θ₁+θ₂))，z₁/z₂ = (r₁/r₂) e^(i(θ₁−θ₂))。

(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ (De Moivre’s theorem / 棣莫弗定理)

The nth roots of unity are given by z = r^(1/n) e^(i(θ+2kπ)/n) for k = 0, 1, …, n−1.

n 次方根由 z = r^(1/n) e^(i(θ+2kπ)/n) 给出，k = 0,1,…, n−1。

6. Further Calculus Integration Techniques | 进阶微积分（积分技巧）

Standard integrals involving inverse trigonometric and hyperbolic functions are vital.

涉及反三角函数和反双曲函数的标准积分至关重要。

∫ 1/√(a² − x²) dx = sin⁻¹(x/a) + C, ∫ 1/(a² + x²) dx = (1/a) tan⁻¹(x/a) + C
∫ 1/√(x² + a²) dx = arsinh(x/a) + C, ∫ 1/√(x² − a²) dx = arcosh(x/a) + C (x > a)

Integration by parts is given by ∫ u dv = uv − ∫ v du. Partial fractions and substitution (including hyperbolic substitution) are common methods.

分部积分公式为 ∫ u dv = uv − ∫ v du。部分分式与换元法（包括双曲换元）是常用方法。

7. Arc Length and Surface Area of Revolution | 弧长与旋转体表面积

The arc length of a curve y = f(x) from x = a to x = b is s = ∫ₐᵝ √(1 + (dy/dx)²) dx. For parametric equations x(t), y(t), use s = ∫ √((dx/dt)² + (dy/dt)²) dt.

曲线 y = f(x) 在 x = a 到 x = b 的弧长为 s = ∫ₐᵝ √(1 + (dy/dx)²) dx。对于参数方程 x(t), y(t)，使用 s = ∫ √((dx/dt)² + (dy/dt)²) dt。

s = ∫ₐᵝ √(1 + (dy/dx)²) dx or s = ∫ √((dx/dt)² + (dy/dt)²) dt

The surface area generated by rotating the curve y = f(x) about the x-axis is S = 2π ∫ y √(1 + (dy/dx)²) dx. For rotation about the y-axis, replace y with x and dy/dx with dx/dy.

曲线 y = f(x) 绕 x 轴旋转所得表面积为 S = 2π ∫ y √(1 + (dy/dx)²) dx。绕 y 轴旋转时，将 y 替换为 x，dy/dx 替换为 dx/dy。

S = 2π ∫ y ds

8. Differential Equations (First & Second Order) | 微分方程（一阶与二阶）

A first‑order linear ODE dy/dx + P(x)y = Q(x) is solved with the integrating factor μ = e^(∫ P dx). Then (y μ)’ = μ Q, so y μ = ∫ μ Q dx + C.

一阶线性常微分方程 dy/dx + P(x)y = Q(x) 使用积分因子 μ = e^(∫ P dx) 求解，得 y μ = ∫ μ Q dx + C。

μ = e^(∫ P dx), y μ = ∫ μ Q dx + C

For a second‑order homogeneous ODE a d²y/dx² + b dy/dx + c y = 0, solve the characteristic equation a m² + b m + c = 0. The general solution depends on the roots: distinct real m₁, m₂ → y = A e^(m₁x) + B e^(m₂x); repeated real m → y = (A + Bx) e^(mx); complex α ± iβ → y = e^(αx)(A cos βx + B sin βx).

对于二阶齐次常微分方程 a d²y/dx² + b dy/dx + c y = 0，求解特征方程 a m² + b m + c = 0。通解取决于根的类型：相异实根 m₁, m₂ → y = A e^(m₁x) + B e^(m₂x)；重根 m → y = (A + Bx) e^(mx)；复根 α ± iβ → y = e^(αx)(A cos βx + B sin βx)。

a m² + b m + c = 0

For non‑homogeneous equations, a particular integral yₚ is found using undetermined coefficients (e.g., polynomial, exponential, trigonometric forms) and added to the complementary function.

对于非齐次方程，用待定系数法（多项式、指数、三角函数等形式）求出特解 yₚ，并与余函数相加。

9. Polar Coordinates | 极坐标

The area enclosed by a polar curve r = f(θ) between θ = α and θ = β is A = ½ ∫ r² dθ.

极坐标曲线 r = f(θ) 在 θ = α 到 θ = β 之间所围面积为 A = ½ ∫ r² dθ。

A = ½ ∫ r² dθ

The gradient of the tangent to a polar curve is obtained via dy/dx = (dy/dθ)/(dx/dθ), using x = r cos θ, y = r sin θ. This yields dy/dx = (r′ sin θ + r cos θ) / (r′ cos θ − r sin θ).

极坐标曲线的切线斜率通过 dy/dx = (dy/dθ)/(dx/dθ) 求得，利用 x = r cos θ, y = r sin θ，得 dy/dx = (r′ sin θ + r cos θ) / (r′ cos θ − r sin θ)。

dy/dx = (r′ sin θ + r cos θ) / (r′ cos θ − r sin θ)

10. Series and Summations | 级数与求和

The Maclaurin series expansion of a function f(x) is f(x) = Σ_{n=0}^{∞} f^(n)(0) xⁿ / n!, valid within the radius of convergence.

函数 f(x) 的麦克劳林级数展开为 f(x) = Σ_{n=0}^{∞} f^(n)(0) xⁿ / n!，在收敛半径内成立。

eˣ = Σ xⁿ/n!, sin x

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Year 13 CIE Further Mathematics: Formula & Theorem Quick Reference Handbook | Year 13 CIE 进阶数学：公式定理速查手册