📚 Formula & Theorem Quick Reference for OCR Further Mathematics | A-Level OCR 进阶数学：公式定理速查手册
This comprehensive quick‑reference guide distils the essential formulae and theorems required for OCR A‑Level Further Mathematics. Designed for rapid revision, it covers pure core topics from complex numbers and matrices through to calculus, polar coordinates, differential equations and beyond. Each section presents key results with concise explanations, paired bilingually to support conceptual clarity. Whether you are reviewing for mocks or the final examination, keep this handbook at your side to reinforce fundamentals and to sharpen problem‑solving skills.
本综合速查手册浓缩了 OCR A‑Level 进阶数学考试所需的核心公式与定理。内容旨在快速复习，涵盖从复数与矩阵到微积分、极坐标、微分方程等纯数核心主题。每一节都用简明解释配合双语对照，以加深概念理解。无论是准备模拟考还是最终大考，都可以把本手册放在手边，扎实基础、提升解题能力。
1. Complex Numbers | 复数
A complex number is written as z = a + bi, where a, b ∈ ℝ and i² = −1. The real part is Re(z) = a and the imaginary part is Im(z) = b.
复数写作 z = a + bi，其中 a, b 为实数且 i² = −1。实部为 Re(z) = a，虚部为 Im(z) = b。
The complex conjugate of z is z* = a − bi. It satisfies zz* = a² + b² = |z|², where the modulus is |z| = √(a² + b²).
z 的共轭复数是 z* = a − bi。满足 zz* = a² + b² = |z|²，其中模为 |z| = √(a² + b²)。
The argument (Arg) is the angle θ = arctan(b/a) measured from the positive real axis, taking the quadrant of z into account. Thus z = r(cos θ + i sin θ) with r = |z|.
辐角 (Arg) 是以正实轴为基准的角度 θ = arctan(b/a)，需考虑 z 所在象限。因此 z = r(cos θ + i sin θ)，其中 r = |z|。
De Moivre’s theorem: [r(cos θ + i sin θ)]ⁿ = rⁿ (cos(nθ) + i sin(nθ)) for integer n. It is used to find powers and roots of complex numbers. The n–th roots of unity are given by z = cos(2kπ/n) + i sin(2kπ/n), k = 0, 1, …, n−1.
棣莫弗定理：[r(cos θ + i sin θ)]ⁿ = rⁿ (cos(nθ) + i sin(nθ))，n 为整数。用于求复数的乘幂与方根。n 次单位根为 z = cos(2kπ/n) + i sin(2kπ/n)，k = 0, 1, …, n−1。
Euler’s formula links exponential and trigonometric forms: e^(iθ) = cos θ + i sin θ, so z = re^(iθ). Multiplication then becomes re^(iθ) × se^(iφ) = rse^(i(θ+φ)).
欧拉公式将指数式与三角式联系起来：e^(iθ) = cos θ + i sin θ，因此 z = re^(iθ)。乘法变为 re^(iθ) × se^(iφ) = rse^(i(θ+φ))。
2. Matrices and Linear Transformations | 矩阵与线性变换
A 2×2 matrix M = [[a, b], [c, d]] represents a linear transformation. Its determinant is det(M) = ad − bc. An inverse exists if det(M) ≠ 0, and is given by M⁻¹ = (1/det(M))[[d, −b], [−c, a]].
2×2 矩阵 M = [[a, b], [c, d]] 表示一个线性变换。其行列式为 det(M) = ad − bc。若 det(M) ≠ 0，则存在逆矩阵 M⁻¹ = (1/det(M))[[d, −b], [−c, a]]。
For 3×3 matrices, the determinant can be expanded along any row or column using the checkerboard sign pattern (+ − +). The inverse is given by (1/det(A)) adj(A), where adj(A) is the transpose of the cofactor matrix.
对于 3×3 矩阵，行列式可按任一行（列）展开，符号遵循正负交替的棋盘模式（+ − +）。逆矩阵为 (1/det(A)) adj(A)，其中 adj(A) 是余子式矩阵的转置。
Matrices can represent rotations, reflections, enlargements and shears. For rotation by angle θ anticlockwise: [[cos θ, −sin θ], [sin θ, cos θ]]. Reflection in the line y = (tan θ)x has matrix [[cos 2θ, sin 2θ], [sin 2θ, −cos 2θ]].
矩阵可表示旋转、反射、拉伸和剪切。逆时针旋转 θ 角的矩阵为 [[cos θ, −sin θ], [sin θ, cos θ]]。关于直线 y = (tan θ)x 的反射矩阵为 [[cos 2θ, sin 2θ], [sin 2θ, −cos 2θ]]。
The product of two matrices corresponds to the composition of transformations, applied from right to left. The identity matrix I = [[1,0],[0,1]] leaves every vector unchanged.
两个矩阵的乘积对应于变换的复合，从右向左应用。单位矩阵 I = [[1,0],[0,1]] 不改变任何向量。
3. Vectors: Products and Geometry | 向量：乘积与几何
The scalar (dot) product of vectors a and b is a·b = |a||b| cos θ, where θ is the angle between them. In component form, if a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), then a·b = a₁b₁ + a₂b₂ + a₃b₃. It is zero when vectors are perpendicular.
向量 a 与 b 的数量积（点乘）为 a·b = |a||b| cos θ，其中 θ 为夹角。在分量形式下，a·b = a₁b₁ + a₂b₂ + a₃b₃。当两向量垂直时点乘为零。
The vector (cross) product a × b yields a vector perpendicular to both a and b, with magnitude |a||b| sin θ. In i, j, k notation: a × b = |i j k; a₁ a₂ a₃; b₁ b₂ b₃| = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k.
向量积（叉乘）a × b 产生一个既垂直于 a 又垂直于 b 的向量，大小为 |a||b| sin θ。用 i, j, k 表示：a × b = |i j k; a₁ a₂ a₃; b₁ b₂ b₃| = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k。
The equation of a line in vector form is r = a + λb, where a is a point on the line and b is a direction vector. A plane is defined by r·n = d (normal form) or r = a + λb + μc (parametric). The acute angle θ between two planes with normals n₁, n₂ satisfies cos θ = |n₁·n₂| / (|n₁||n₂|).
直线的向量方程为 r = a + λb，其中 a 为线上一点，b 为方向向量。平面可由法线式 r·n = d 或参数式 r = a + λb + μc 定义。两平面（法向量为 n₁, n₂）的锐角 θ 满足 cos θ = |n₁·n₂| / (|n₁||n₂|)。
The shortest distance from a point P with position vector p to the line r = a + λb is |(a − p) × b| / |b|.
点 P（位置向量 p）到直线 r = a + λb 的最短距离为 |(a − p) × b| / |b|。
4. Hyperbolic Functions | 双曲函数
The hyperbolic sine and cosine are defined by sinh x = (eˣ − e⁻ˣ)/2 and cosh x = (eˣ + e⁻ˣ)/2. Their domains and derivatives mirror trigonometric functions with important sign differences: d/dx(sinh x) = cosh x, d/dx(cosh x) = sinh x.
双曲正弦和双曲余弦定义为 sinh x = (eˣ − e⁻ˣ)/2 与 cosh x = (eˣ + e⁻ˣ)/2。它们的定义域和导数与三角函数相似，但有重要的符号区别：d/dx(sinh x) = cosh x，d/dx(cosh x) = sinh x。
Fundamental identity: cosh² x − sinh² x = 1. Other hyperbolic functions: tanh x = sinh x / cosh x, coth x = 1/tanh x, sech x = 1/cosh x, cosech x = 1/sinh x.
基本恒等式：cosh² x − sinh² x = 1。其他双曲函数：tanh x = sinh x / cosh x，coth x = 1/tanh x，sech x = 1/cosh x，cosech x = 1/sinh x。
Inverse hyperbolic functions are often expressed via 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.
反双曲函数常用对数表示：arsinh x = ln(x + √(x²+1))，arcosh x = ln(x + √(x²−1))（x ≥ 1），artanh x = ½ ln((1+x)/(1−x))（|x|<1）。
Derivatives of inverse hyperbolics: d/dx(arsinh x) = 1/√(x²+1), d/dx(arcosh x) = 1/√(x²−1), d/dx(artanh x) = 1/(1−x²).
反双曲函数的导数：d/dx(arsinh x) = 1/√(x²+1)，d/dx(arcosh x) = 1/√(x²−1)，d/dx(artanh x) = 1/(1−x²)。
5. Further Calculus: Differentiation and Integration | 进阶微积分：微分与积分
Standard derivatives beyond basic ones include: d/dx(e^(kx)) = ke^(kx), d/dx(aˣ) = aˣ ln a, d/dx(ln x) = 1/x, d/dx(sin⁻¹ x) = 1/√(1−x²), d/dx(cos⁻¹ x) = −1/√(1−x²), d/dx(tan⁻¹ x) = 1/(1+x²).
超越基本函数的常见导数包括：d/dx(e^(kx)) = ke^(kx)，d/dx(aˣ) = aˣ ln a，d/dx(ln x) = 1/x，d/dx(sin⁻¹ x) = 1/√(1−x²)，d/dx(cos⁻¹ x) = −1/√(1−x²)，d/dx(tan⁻¹ x) = 1/(1+x²)。
The chain rule, product rule and quotient rule are essential: if y = f(g(x)) then dy/dx = f'(g(x))·g'(x); d/dx(uv) = u’v + uv’; d/dx(u/v) = (u’v − uv’)/v².
链式法则、乘积法则与商法是必须掌握的：若 y = f(g(x))，则 dy/dx = f'(g(x))·g'(x)；d/dx(uv) = u’v + uv’；d/dx(u/v) = (u’v − uv’)/v²。
Integration techniques include integration by parts: ∫ u dv/dx dx = uv − ∫ v du/dx dx, often used when the integrand is a product of a polynomial and an exponential/trigonometric/logarithmic function. Substitution method: if x = g(u) then ∫ f(x) dx = ∫ f(g(u)) g'(u) du. Standard forms: ∫ f'(x)/f(x) dx = ln|f(x)| + C, ∫ f'(x) e^f(x) dx = e^f(x) + C.
积分技巧包括分部积分法：∫ u dv/dx dx = uv − ∫ v du/dx dx，常用于多项式与指数/三角/对数函数的乘积。换元法：若 x = g(u)，则 ∫ f(x) dx = ∫ f(g(u)) g'(u) du。标准形式：∫ f'(x)/f(x) dx = ln|f(x)| + C，∫ f'(x) e^f(x) dx = e^f(x) + C。
Improper integrals have infinite limits or integrands with vertical asymptotes; evaluate by replacing the singularity with a bound and taking the limit, e.g. ∫₁^∞ 1/x² dx = lim_(b→∞) [−1/x]₁^b = 1.
广义积分有无穷积分限或被积函数有垂直渐近线；通过用有限变量替换奇点并取极限来计算，例如 ∫₁^∞ 1/x² dx = lim_(b→∞) [−1/x]₁^b = 1。
6. Series and Sequences | 级数与序列
The Maclaurin series expansion for a function f(x) is f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … . Standard 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 − … (valid for |x|<1).
函数 f(x) 的麦克劳林展开式为 f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + …。标准展开式：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 − …（当 |x|<1 时有效）。
The binomial expansion for (1+x)ⁿ is (1+x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + …, valid for |x|<1 when n is not a positive integer.
(1+x)ⁿ 的二项展开式为 (1+x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + …，当 n 不是正整数时，收敛区间为 |x|<1。
Method of differences: for sequences where term can be expressed as uᵣ = f(r) − f(r+1), the sum Σ_(r=1)^n uᵣ telescopes to f(1) − f(n+1). This is powerful for summing rational functions or trigonometric series.
差分法：当项可写为 uᵣ = f(r) − f(r+1) 的形式时，求和 Σ_(r=1)^n uᵣ 可化简为 f(1) − f(n+1)。这在求有理函数或三角级数的和时非常有效。
7. Polar Coordinates | 极坐标
A point in polar coordinates is given by (r, θ), where r is the distance from the origin and θ the angle from the initial line (positive x‑axis). Conversion: x = r cos θ, y = r sin θ, r = √(x²+y²), θ = arctan(y/x).
极坐标中的一个点由 (r, θ) 表示，其中 r 是到原点的距离，θ 是从初始线（正 x 轴）测量的角度。转换关系：x = r cos θ，y = r sin θ，r = √(x²+y²)，θ = arctan(y/x)。
The area enclosed by a polar curve r = f(θ) between angles α and β is A = ½ ∫_α^β r² dθ. For a curve defined parametrically or in Cartesian form, use the appropriate Jacobian.
极坐标曲线 r = f(θ) 在角度 α 到 β 之间所围成的面积为 A = ½ ∫_α^β r² dθ。对于参数式或直角坐标形式，使用相应的雅可比变换。
The arc length of a polar curve is s = ∫_α^β √(r² + (dr/dθ)²) dθ. Common curves: cardioid r = a(1+cos θ), circle r = 2a cos θ, rose curves r = a sin(nθ).
极坐标曲线的弧长为 s = ∫_α^β √(r² + (dr/dθ)²) dθ。常见曲线：心形线 r = a(1+cos θ)，圆 r = 2a cos θ，玫瑰线 r = a sin(nθ)。
8. Differential Equations | 微分方程
A first‑order linear differential equation has the form dy/dx + P(x)y = Q(x). The integrating factor is μ = e^(∫ P(x) dx). Multiply through by μ to get d/dx(μ y) = μ Q(x), then integrate.
一阶线性微分方程具有形式 dy/dx + P(x)y = Q(x)。积分因子为 μ = e^(∫ P(x) dx)。将方程乘以 μ 得到 d/dx(μ y) = μ Q(x)，再积分即可。
Separable equations can be written as g(y) dy = f(x) dx; integrate both sides. Homogeneous equations dy/dx = F(y/x) use the substitution v = y/x, giving a separable equation in v and x.
可分离变量的方程可写为 g(y) dy = f(x) dx，两边积分。齐次方程 dy/dx = F(y/x) 使用代换 v = y/x，化为关于 v 与 x 的可分离变量方程。
Second‑order linear homogeneous equations with constant coefficients take the form 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); repeated real root m: y = (A + Bx)e^(mx); complex roots α ± iβ: y = e^(αx)(A cos βx + B sin βx).
二阶常系数齐次线性方程形式为 a d²y/dx² + b dy/dx + c y = 0。辅助方程为 am² + bm + c = 0。若根 m₁, m₂ 为不等实根：通解 y = Ae^(m₁x) + Be^(m₂x)；重实根 m：y = (A + Bx)e^(mx)；复根 α ± iβ：y = e^(αx)(A cos βx + B sin βx)。
For the non‑homogeneous case a d²y/dx² + b dy/dx + c y = f(x), the general solution is complementary function + particular integral. Select a trial function based on f(x): polynomial, exponential, trigonometric form, and determine coefficients by substitution.
对于非齐次方程 a d²y/dx² + b dy/dx + c y = f(x)，通解为余函数（齐次通解）加特解。根据 f(x) 类型选择试验函数：多项式、指数函数或三角函数形式，通过代入确定系数。
9. Conic Sections | 圆锥曲线
Conic sections (parabola, ellipse, hyperbola) can be described in Cartesian and parametric forms. The standard parabola y² = 4ax has focus (a,0) and directrix x = −a. In parametric form: x = at², y = 2at.
圆锥曲线（抛物线、椭圆、双曲线）可用直角坐标和参数形式描述。标准抛物线 y² = 4ax 的焦点为 (a,0)，准线为 x = −a。参数形式：x = at²，y = 2at。
The ellipse x²/a² + y²/b² = 1 has foci at (±ae, 0) where e² = 1 − b²/a² (eccentricity). Parametric equations: x = a cos θ, y = b sin θ. The sum of distances from any point on ellipse to the foci is constant 2a.
椭圆 x²/a² + y²/b² = 1 的焦点在 (±ae, 0)，其中离心率 e² = 1 − b²/a²。参数方程为 x = a cos θ，y = b sin θ。椭圆上任意一点到两焦点的距离之和为常数 2a。
The hyperbola x²/a² − y²/b² = 1 has foci (±ae, 0) with e² = 1 + b²/a², asymptotes y = ±(b/a)x. Parametric form using hyperbolic functions: x = a cosh t, y = b sinh t (right branch). The rectangular hyperbola xy = c² has asymptotes along the axes.
双曲线 x²/a² − y²/b² = 1 的焦点为 (±ae, 0)，其中 e² = 1 + b²/a²，渐近线为 y = ±(b/a)x。用双曲函数的参数形式：x = a cosh t，y = b sinh t（右支）。等轴双曲线 xy = c² 的渐近线为坐标轴。
Tangents and normals to these curves are found using differentiation, often exploiting parametric differentiation: dy/dx = (dy/dt)/(dx/dt). The gradient of the tangent then gives the equation via y − y₁ = m(x − x₁).
这些曲线的切线与法线通过微分（通常利用参数求导：dy/dx = (dy/dt)/(dx/dt)）得出。切线的斜率用于点斜式方程 y − y₁ = m(x − x₁)。
10. Proof by Induction and Numerical Methods | 归纳证明与数值方法
Proof by mathematical induction is a standard technique for verifying statements involving natural numbers. The structure: (i) Base case: verify the statement for n = 1 (or initial value). (ii) Inductive hypothesis: assume true for n = k. (iii) Inductive step: prove true for n = k+1 using the assumption. (iv) Conclude by induction that the statement holds for all n ≥ 1.
数学归纳法是验证涉及自然数命题的标准方法。结构为：(i) 基础情形：验证 n = 1（或初始值）时命题成立；(ii) 归纳假设：假设 n = k 时成立；(iii) 归纳步骤：利用假设证明 n = k+1 时成立；(iv) 由归纳原理得出命题对所有 n ≥ 1 成立。
Common applications: summation of series, divisibility proofs, matrix powers, and inequalities.
常见应用：级数求和、整除性证明、矩阵的乘幂以及不等式。
Numerical methods approximate solutions to equations and integrals when exact methods fail. The Newton‑Raphson formula for solving f(x)=0: xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ). It converges quadratically near a root if f'(xₙ) ≠ 0. Iterative formulas fixed‑point iteration xₙ₊₁ = g(xₙ) require |g'(x)| < 1 near the root for convergence.
当解析方法失效时，数值方法可以逼近方程的解和积分的值。求解 f(x)=0 的牛顿‑拉夫森公式：xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ)。若 f'(xₙ) ≠ 0，在根附近具有二次收敛性。迭代公式 xₙ₊₁ = g(xₙ)（不动点迭代）需要在根附近满足 |g'(x)| < 1 才能收敛。
Trapezium rule for approximate integration: ∫_a^b f(x) dx ≈ (h/2)[y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ], where h = (b−a)/n. Simpson’s rule (n even) offers higher accuracy: ≈ (h/3)[y₀ + 4(y₁+y₃+…)+2(y₂+y₄+…)+yₙ].
梯形法则用于近似积分：∫_a^b 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）精度更高：≈ (h/3)[y₀ + 4(y₁+y₃+…)+2(y₂+y₄+…)+yₙ]。
Step‑by‑step iteration and error bounds: for the trapezium rule the error is bounded by (b−a)³/(12n²) max|f”(x)|. Always check convergence criteria when applying iterative methods.
逐步迭代与误差界：梯形法则的误差界为 (b−a)³/(12n²) max|f”(x)|。应用迭代法时务必检查收敛条件。
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