📚 Year 12 Cambridge Further Mathematics: Formula & Theorem Quick Reference Guide | 剑桥12年级进阶数学：公式定理速查手册
This quick reference handbook brings together the essential formulas and theorems for the Cambridge Year 12 Further Mathematics course. Each section provides concise statements with bilingual explanations, helping you review core concepts quickly before exams.
本速查手册汇集了剑桥12年级进阶数学课程的核心公式与定理。每个部分以简洁的双语解释呈现，帮助你在考前快速回顾核心概念。
1. Complex Numbers | 复数
A complex number is written as z = a + bi, where a, b ∈ ℝ and i² = −1. The conjugate is z* = a − bi, and the modulus is |z| = √(a² + b²). The argument θ = arg(z) satisfies tanθ = b/a, with the quadrant of z taken into account.
复数写作 z = a + bi，其中 a, b 为实数且 i² = −1。共轭复数 z* = a − bi，模长为 |z| = √(a² + b²)。辐角 θ = arg(z) 满足 tanθ = b/a，并需根据 z 所在象限确定。
eiθ = cosθ + i sinθ
De Moivre’s Theorem: (cosθ + i sinθ)n = cos(nθ) + i sin(nθ) for integer n. This is used to find powers and roots of complex numbers.
德莫弗定理：(cosθ + i sinθ)n = cos(nθ) + i sin(nθ)（n 为整数）。用于求复数的幂与根。
The n nth roots of unity are given by zk = e2πi k/n for k = 0, 1, …, n−1. Their sum is zero.
n 次单位根为 zk = e2πi k/n，k = 0, 1, …, n−1。它们的和为零。
2. Matrices and Linear Transformations | 矩阵与线性变换
Matrix multiplication is defined only when the number of columns of the first equals the number of rows of the second. The identity matrix I has ones on the main diagonal and zeros elsewhere: AI = IA = A.
矩阵乘法仅当第一个矩阵的列数等于第二个矩阵的行数时才有定义。单位矩阵 I 的主对角线元素为 1，其余为 0：AI = IA = A。
The determinant of a 2×2 matrix A = [a b; c d] is det(A) = ad − bc. A is invertible if and only if det(A) ≠ 0, and A⁻¹ = (1/det(A)) [d −b; −c a].
2×2 矩阵 A = [a b; c d] 的行列式为 det(A) = ad − bc。A 可逆当且仅当 det(A) ≠ 0，逆矩阵为 A⁻¹ = (1/det(A)) [d −b; −c a]。
| Transformation | Matrix |
| Rotation by θ (anticlockwise) | [cosθ −sinθ; sinθ cosθ] |
| Reflection in the line y = (tanθ)x | [cos2θ sin2θ; sin2θ −cos2θ] |
| Enlargement scale factor k | [k 0; 0 k] |
线性变换矩阵如上表。复合变换对应于矩阵从右向左相乘。
3. Vectors and 3D Geometry | 向量与空间几何
The scalar (dot) product: a · b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃. The vector (cross) product: a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k. |a × b| = |a||b| sinθ.
标量积（点积）：a · b = |a||b| cosθ = a₁b₁ + a₂b₂ + a₃b₃。向量积（叉积）：a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k，且 |a × b| = |a||b| sinθ。
The vector equation of a line is r = a + λb, where a is a point on the line and b is a direction vector. The plane r · n = p has normal n and distance p/|n| from the origin.
直线向量方程：r = a + λb，其中 a 为直线上一点，b 为方向向量。平面方程 r · n = p 以 n 为法向量，原点到平面的距离为 p/|n|。
The shortest distance from a point p to the line r = a + λb is |(p − a) × b̂|, where b̂ = b/|b|.
点 p 到直线 r = a + λb 的最短距离为 |(p − a) × b̂|，其中 b̂ = b/|b|。
4. 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.
定义：
sinh x = (ex − e−x)/2,
cosh x = (ex + e−x)/2,
tanh x = sinh x / cosh x。
cosh²x − sinh²x = 1
Osborn’s rule: To convert a trigonometric identity into a hyperbolic one, replace each cos by cosh and sin by i sinh, then change the sign of any product (or implied product) of two sinh terms.
奥斯本法则：将三角恒等式变为双曲恒等式时，将 cos 替换为 cosh、sin 替换为 i sinh，然后改变任两个 sinh 乘积的符号。
Derivatives: d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech²x. Inverse hyperbolic derivatives: d/dx (arsinh x) = 1/√(x²+1), d/dx (arcosh x) = 1/√(x²−1) (x >1).
导数：d/dx (sinh x) = cosh x，d/dx (cosh x) = sinh x，d/dx (tanh x) = sech²x。反函数导数：d/dx (arsinh x) = 1/√(x²+1)，d/dx (arcosh x) = 1/√(x²−1)（x >1）。
5. Differentiation Techniques | 微分技巧
Product rule: d/dx (uv) = u’v + uv’.
Quotient rule: d/dx (u/v) = (u’v − uv’)/v².
乘积法则：d/dx (uv) = u’v + uv’。
商法则：d/dx (u/v) = (u’v − uv’)/v²。
Chain rule: If y = f(u) and u = g(x), then dy/dx = dy/du × du/dx. For parametric equations x = f(t), y = g(t), dy/dx = (dy/dt) / (dx/dt) and d²y/dx² = (d/dt (dy/dx)) / (dx/dt).
链式法则：若 y = f(u) 且 u = g(x)，则 dy/dx = dy/du × du/dx。对于参数方程 x = f(t), y = g(t)，dy/dx = (dy/dt) / (dx/dt) 且 d²y/dx² = (d/dt (dy/dx)) / (dx/dt)。
Implicit differentiation: Differentiate every term with respect to x, treating y as a function of x, then solve for dy/dx. Remember d/dx (y²) = 2y dy/dx.
隐函数微分：将每一项对 x 求导，视 y 为 x 的函数，然后解出 dy/dx。注意 d/dx (y²) = 2y dy/dx。
6. Integration Techniques | 积分技巧
Integration by parts: ∫ u dv = uv − ∫ v du. A useful order for choosing u is LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential).
分部积分：∫ u dv = uv − ∫ v du。选择 u 的常用顺序为 LIATE（对数、反三角、代数、三角、指数）。
Standard integrals involving hyperbolic functions:
∫ sinh x dx = cosh x + C,
∫ cosh x dx = sinh x + C,
∫ 1/√(x²+1) dx = arsinh x + C,
∫ 1/√(x²−1) dx = arcosh x + C (x>1).
涉及双曲函数的标准积分：
∫ sinh x dx = cosh x + C,
∫ cosh x dx = sinh x + C,
∫ 1/√(x²+1) dx = arsinh x + C,
∫ 1/√(x²−1) dx = arcosh x + C (x>1)。
Partial fractions are used to split a rational expression into simpler fractions before integrating, e.g., 1/(x²−a²) = 1/(2a)[1/(x−a) − 1/(x+a)].
部分分式用于在积分前将有理式拆分为更简单分式，如 1/(x²−a²) = 1/(2a)[1/(x−a) − 1/(x+a)]。
7. Taylor and Maclaurin Series | 泰勒与麦克劳林级数
The Maclaurin series of a function f(x) is f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … , valid for |x| < R.
函数 f(x) 的麦克劳林级数为 f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + …，在 |x| < R 内成立。
The Taylor series about x = a: f(x) = f(a) + f'(a)(x−a) + f”(a)(x−a)²/2! + …
在 x = a 处的泰勒级数：f(x) = f(a) + f'(a)(x−a) + f”(a)(x−a)²/2! + …
Key expansions:
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 − … ( −1 < x ≤ 1 )
重要展开式：
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 − … ( −1 < x ≤ 1 )
8. Polar Coordinates | 极坐标
Conversion: x = r cosθ, y = r sinθ, r = √(x² + y²), θ = arctan(y/x). A polar curve has equation r = f(θ).
转换关系：x = r cosθ, y = r sinθ，r = √(x² + y²)，θ = arctan(y/x)。极坐标曲线方程为 r = f(θ)。
The area bounded by a polar curve and the half-lines θ = α and θ = β is A = ½ ∫αβ r² dθ.
极坐标曲线与射线 θ = α、θ = β 所围面积为 A = ½ ∫αβ r² dθ。
The arc length of a polar curve from θ = α to θ = β is s = ∫αβ √(r² + (dr/dθ)²) dθ.
从 θ = α 到 θ = β 的极坐标曲线弧长为 s = ∫αβ √(r² + (dr/dθ)²) dθ。
9. Differential Equations | 微分方程
First-order separable: dy/dx = f(x)g(y) ⇒ ∫ (1/g(y)) dy = ∫ f(x) dx.
一阶可分离：dy/dx = f(x)g(y) ⇒ ∫ (1/g(y)) dy = ∫ f(x) dx。
Linear first-order ODE: dy/dx + P(x)y = Q(x). The integrating factor is μ(x) = e∫P dx. Multiply through to get d/dx (μ y) = μ Q, then integrate.
线性一阶常微分方程：dy/dx + P(x)y = Q(x)。积分因子为 μ(x) = e∫P dx。两边乘以 μ 得 d/dx (μ y) = μ Q，再积分求解。
Second-order linear homogeneous ODEs with constant coefficients: ay” + by’ + cy = 0. Solve the auxiliary equation am² + bm + c = 0. If roots are real and distinct (m₁ ≠ m₂): y = Aem₁x + Bem₂x; repeated root m: y = (A + Bx)emx; complex roots α ± iβ: y = eαx(A cos βx + B sin βx).
二阶常系数线性齐次常微分方程：ay” + by’ + cy = 0。求解辅助方程 am² + bm + c = 0。若根为不等实根 m₁ ≠ m₂：y = Aem₁x + Bem₂x；重根 m：y = (A + Bx)emx；复根 α ± iβ：y = eαx(A cos βx + B sin βx)。
10. Proof by Induction | 数学归纳法
Steps:
1. Base case: verify the statement for the smallest integer (usually n = 1).
2. Inductive hypothesis: assume the statement is true for n = k.
3. Inductive step: prove that the statement for n = k+1 follows from the hypothesis.
4. Conclusion: by mathematical induction, the statement holds for all positive integers n.
步骤：
1. 基础情形：验证最小整数（通常 n = 1）时命题成立。
2. 归纳假设：假设 n = k 时命题成立。
3. 归纳递推：证明 n = k+1 时命题由假设推出。
4. 结论：由数学归纳法，命题对所有正整数 n 成立。
A typical summation example: Prove Σr=1n r = n(n+1)/2. The inductive step requires adding the (k+1) term to both sides of the assumed equality.
典型求和例子：证明 Σr=1n r = n(n+1)/2。归纳递推需在假设等式两边同时加上第 (k+1) 项。
11. Summation and Binomial Theorem | 求和与二项式定理
Standard summations:
Σr=1n r = n(n+1)/2,
Σr=1n r² = n(n+1)(2n+1)/6,
Σr=1n r³ = [n(n+1)/2]².
标准求和公式：
Σr=1n r = n(n+1)/2,
Σr=1n r² = n(n+1)(2n+1)/6,
Σr=1n r³ = [n(n+1)/2]²。
Binomial expansion: For any rational n, (1+x)n = 1 + n x + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + … , valid for |x| < 1. The general term is (n choose r) xr where (n choose r) = n(n−1)…(n−r+1)/r!.
二项式展开：对任意有理数 n，(1+x)n = 1 + n x + [n(n−1)/2!] x² + [n(n−1)(n−2)/3!] x³ + … ，在 |x| < 1 时收敛。通项为 C(n, r) xr，其中 C(n, r) = n(n−1)…(n−r+1)/r!。
The formula can be adapted for (a + b)n by writing (a + b)n = an(1 + b/a)n and expanding, provided |b/a| < 1.
可通过写成 (a + b)n = an(1 + b/a)n 再展开（要求 |b/a| < 1）。
12. Further Applications | 拓展应用
Volumes of revolution: About the x-axis, V = π ∫ab y² dx. About the y-axis, V = π ∫cd x² dy. For parametric curves, substitute x(t) and y(t) and change the limits accordingly.
旋转体体积：绕 x 轴旋转时，V = π ∫ab y² dx。绕 y 轴旋转时，V = π ∫cd x² dy。对参数曲线，代入 x(t), y(t) 并相应改变
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