📚 Year 13 OCR Further Mathematics: Vocabulary and Terminology Quick Guide | Year 13 OCR 进阶数学：词汇术语速记指南
Mastering the precise vocabulary of OCR Further Mathematics is essential for Year 13 students aiming for top grades. This quick-reference guide clarifies the most important terms, definitions and concepts from Core Pure 2 and common applied modules, helping you revise efficiently and avoid common misconceptions.
掌握 OCR 进阶数学的精准词汇是 Year 13 学生冲刺高分的必由之路。这份速记指南梳理了 Core Pure 2 及常见应用模块中最重要的术语、定义与概念，帮助你高效复习、避开易错点。
1. Complex Numbers: Polar Form & De Moivre’s Theorem | 复数：极坐标形式与棣莫弗定理
Modulus-argument (polar) form: A complex number z = a + bi can be written as z = r(cos θ + i sin θ), where r = |z| = √(a²+b²) is the modulus and θ = arg(z) is the argument, usually chosen with –π < θ ≤ π. This form makes multiplication and powers much simpler.
模辐角（极坐标）形式： 复数 z = a + bi 可以写成 z = r(cos θ + i sin θ)，其中 r = |z| = √(a²+b²) 为模，θ = arg(z) 为辐角，通常取 –π < θ ≤ π。该形式使乘法和幂运算变得非常简单。
De Moivre’s theorem: For any integer n, (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). It is the key to finding powers of complex numbers and to deriving trigonometric identities for cos nθ and sin nθ.
棣莫弗定理： 对任意整数 n，(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。此定理是求复数幂以及推导 cos nθ 和 sin nθ 三角恒等式的关键。
Geometric effect of multiplication: Multiplying two complex numbers in polar form multiplies their moduli and adds their arguments: r₁r₂[cos(θ₁+θ₂) + i sin(θ₁+θ₂)]. This corresponds to a rotation by θ₂ and an enlargement by factor r₂.
乘法的几何意义： 极坐标形式下两复数相乘，模长相乘，辐角相加：r₁r₂[cos(θ₁+θ₂) + i sin(θ₁+θ₂)]。几何上对应旋转 θ₂ 角度并按因子 r₂ 缩放。
2. Complex Loci and Roots of Unity | 复数轨迹与单位根
nth roots of unity: The solutions of zⁿ = 1 are zₖ = e^(i·2πk/n) = cos(2πk/n) + i sin(2πk/n) for k = 0,1,…,n-1. They lie symmetrically on the unit circle and form a regular n-gon with one vertex at 1.
n 次单位根： 方程 zⁿ = 1 的解为 zₖ = e^(i·2πk/n) = cos(2πk/n) + i sin(2πk/n)，k = 0,1,…,n-1。它们对称地分布在单位圆上，构成一个以 1 为一个顶点的正 n 边形。
Loci in the complex plane: Common loci include |z – a| = r (a circle centre a, radius r), arg(z – a) = α (a half‑line from a at angle α), and |z – a| = |z – b| (the perpendicular bisector of the segment joining a and b).
复平面上的轨迹： 常见轨迹包括 |z – a| = r（以 a 为圆心、r 为半径的圆），arg(z – a) = α（从 a 出发且倾角为 α 的射线），以及 |z – a| = |z – b|（连接 a 和 b 的线段的中垂线）。
Maximum and minimum arguments and moduli: When a locus is drawn, you may be asked for the greatest possible value of |z| or arg(z). These can often be found by drawing a tangent from the origin to the locus or using geometry.
模与辐角的最值： 给定轨迹后，常需求出 |z| 或 arg(z) 的最大可能值。通常可通过从原点向轨迹作切线或利用几何关系求得。
3. Matrices: Invariants, Eigenvalues and Diagonalisation | 矩阵：不变点、特征值与对角化
Invariant points and lines: For a linear transformation with matrix M, a point p is invariant if M p = p. An invariant line has all points mapped to another point on the same line (the direction vector v satisfies M v = λ v).
不变点与不变线： 对于矩阵 M 表示的线性变换，若 M p = p 则 p 为不变点。不变线上的每一点都被映射到同一线上的另一点（方向向量 v 满足 M v = λ v）。
Eigenvalues and eigenvectors: An eigenvalue λ and corresponding eigenvector v satisfy M v = λ v, v ≠ 0. They are found by solving det(M – λ I) = 0 and then (M – λ I)v = 0.
特征值与特征向量： 特征值 λ 与对应的特征向量 v 满足 M v = λ v，v ≠ 0。通过解 det(M – λ I) = 0 得到特征值，再代入 (M – λ I)v = 0 求出特征向量。
Diagonalisation and matrix powers: If M can be written as M = P D P⁻¹ with diagonal D, then Mⁿ = P Dⁿ P⁻¹, useful for powers of transformations. D contains the eigenvalues on the diagonal, and P holds corresponding eigenvectors as columns.
对角化与矩阵的高次幂： 若 M 可表示为 M = P D P⁻¹，其中 D 为对角阵，则 Mⁿ = P Dⁿ P⁻¹，便于求变换的高次幂。对角阵 D 的对角元为特征值，P 的列为对应的特征向量。
4. Hyperbolic Functions: Definitions and Identities | 双曲函数：定义与恒等式
Definitions: sinh x = (eˣ – e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x. These functions are not periodic and appear frequently when integrating using trigonometric substitution analogues.
定义： sinh x = (eˣ – e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x。这些函数不是周期函数，在使用类似三角代换的积分时频繁出现。
Key identities and Osborn’s rule: cosh² x – sinh² x = 1, and similar identities mirror trigonometric ones. To convert a trig identity into a hyperbolic one, replace cos with cosh, sin with i sinh, and check signs – any product of two ‘sinh’ terms changes sign (Osborn’s rule).
核心恒等式与 Osborn 法则： cosh² x – sinh² x = 1，许多恒等式与三角恒等式类似。将三角恒等式转换为双曲恒等式时，将 cos 换为 cosh，sin 换为 i sinh，并检查符号——任何两个“sinh”项的乘积需要变号（Osborn 法则）。
Inverse hyperbolic functions: arsinh x = ln(x + √(x²+1)), arcosh x = ln(x + √(x²-1)) for x ≥ 1, artanh x = ½ln((1+x)/(1-x)) for |x| < 1. These logarithmic forms are essential for integration and differentiation.
反双曲函数： arsinh x = ln(x + √(x²+1))，arcosh x = ln(x + √(x²-1))（x ≥ 1），artanh x = ½ln((1+x)/(1-x))（|x| < 1）。这些对数形式对积分与求导至关重要。
5. Polar Coordinates: Curves and Areas | 极坐标：曲线与面积
Polar equation and curve types: A curve is defined as r = f(θ). Common shapes include cardioids (r = a(1 ± cos θ) or r = a(1 ± sin θ)), limaçons, roses, and circles. You must be able to sketch them and find tangents at the pole.
极坐标方程与曲线类型： 曲线由 r = f(θ) 定义。常见形状有心形线（r = a(1 ± cos θ) 或 r = a(1 ± sin θ)）、蜗线、玫瑰线和圆。必须能绘制草图并求曲线在极点处的切线。
Area of a polar sector: The area enclosed by a polar curve r = f(θ) from θ = α to β is A = ½ ∫[α, β] r² dθ. This formula is derived by considering infinitesimal triangular sectors and is used for finding total area or area between two curves.
极坐标扇形面积： 极坐标曲线 r = f(θ) 从 θ = α 到 β 所围面积为 A = ½ ∫[α, β] r² dθ。该公式由微小三角扇形推导而来，用于求总面积或两曲线之间的面积。
Tangents parallel and perpendicular to the initial line: By using the formula dy/dx in terms of r and θ, you can determine points where the tangent is horizontal (dy/dθ = 0) or vertical (dx/dθ = 0). Often combined with the curve’s symmetry to locate key points.
平行或垂直于极轴的切线： 借助用 r 和 θ 表达的 dy/dx 公式，可以确定切线水平（dy/dθ = 0）或竖直（dx/dθ = 0）的点。常结合曲线的对称性来定位关键点。
6. Differential Equations: Integrating Factors and Second Order | 微分方程：积分因子与二阶方程
First-order linear equation and integrating factor: For dy/dx + P(x)y = Q(x), the integrating factor is μ(x) = e^(∫ P dx). Multiplying through by μ turns the left side into d(μ y)/dx, allowing integration.
一阶线性方程与积分因子： 对于 dy/dx + P(x)y = Q(x)，积分因子为 μ(x) = e^(∫ P dx)。方程两边同乘 μ 后，左边变为 d(μ y)/dx，即可积分求解。
Second-order homogeneous equation: a d²y/dx² + b dy/dx + c y = 0. Its auxiliary equation is a m² + b m + c = 0. The complementary function (CF) depends on the nature of the roots: real distinct, repeated real, or complex conjugates.
二阶齐次微分方程： a d²y/dx² + b dy/dx + c y = 0。其辅助方程为 a m² + b m + c = 0。余函数（CF）的形式取决于辅助方程根的类型：相异实根、重实根或共轭复根。
Non-homogeneous equation and particular integral: For g(x) = polynomial, e^(kx), cos kx, sin kx or their combinations, the particular integral (PI) is found by assuming a similar form with unknown coefficients and substituting into the ODE. General solution = CF + PI.
非齐次方程与特解： 当 g(x) 为多项式、e^(kx)、cos kx、sin kx 或其组合时，特解（PI）通过假设含待定系数的相似形式并代入方程求得。通解 = CF + PI。
7. Further Vectors: Dot Product, Cross Product & Planes | 进阶向量：标积、矢积与平面
Scalar (dot) product: For vectors a, b, a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ. Used for finding angles, projections, and determining perpendicularity (a·b = 0).
标积（点积）： 对向量 a, b，a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ。用于求夹角、投影以及判断垂直（a·b = 0）。
Vector (cross) product: a × b = |a||b| sin θ n̂, where n̂ is a unit vector perpendicular to both a and b following the right-hand rule. The magnitude |a × b| gives the area of the parallelogram spanned by a and b.
矢积（叉积）： a × b = |a||b| sin θ n̂，其中 n̂ 为按右手法则同时垂直于 a 和 b 的单位向量。大小 |a × b| 等于 a 和 b 张成的平行四边形面积。
Planes and distance from a point: A plane has vector equation r·n = p (n is a normal vector). The cartesian form is ax+by+cz=d. The perpendicular distance from point (x₀,y₀,z₀) to this plane is |ax₀+by₀+cz₀−d| / √(a²+b²+c²).
平面及点到平面距离： 平面的向量方程为 r·n = p（n 为法向量），笛卡尔形式为 ax+by+cz=d。点 (x₀,y₀,z₀) 到该平面的垂直距离为 |ax₀+by₀+cz₀−d| / √(a²+b²+c²)。
8. Maclaurin Series and Evaluating Limits | 麦克劳林级数与求极限
Maclaurin series formula: f(x) = f(0) + f ‘(0)x + f ”(0)x²/2! + f ”'(0)x³/3! + … . You must be able to derive series for standard functions and compound functions using substitution.
麦克劳林级数公式： f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + …。必须能够导出标准函数及通过代换得到的复合函数的级数。
Standard expansions: eˣ = 1 + x + x²/2! + …, sin x = x – x³/3! + x⁵/5! – …, cos x = 1 – x²/2! + x⁴/4! – …, ln(1+x) = x – x²/2 + x³/3 – … (valid |x|<1). These are often used up to the x³ or x⁴ term.
标准展开式： eˣ = 1 + x + x²/2! + …，sin x = x – x³/3! + x⁵/5! – …，cos x = 1 – x²/2! + x⁴/4! – …，ln(1+x) = x – x²/2 + x³/3 – …（|x|<1 时有效）。通常用到 x³ 或 x⁴ 项。
Using series to find limits: When evaluating a limit such as (sin x – x)/x³ as x→0, replace sin x with its Maclaurin series, simplify, and cancel powers. This avoids repeated use of L’Hôpital’s rule and isolates the leading term.
利用级数求极限： 求极限例如 lim(x→0) (sin x – x)/x³ 时，用 sin x 的麦克劳林级数代换，化简并约去幂次，可避免反复使用洛必达法则，直接提取首项。
9. Further Mechanics: Impulse, Centres of Mass and Energy | 进阶力学：冲量、质心与能量
Impulse–momentum principle: Impulse I = FΔt = mv – mu (change in momentum). For a particle, the impulse vector equals the change in its momentum vector. In collisions, the total momentum of a system is conserved when no external impulse acts.
冲量–动量原理： 冲量 I = FΔt = mv – mu（动量改变量）。对质点而言，冲量矢量等于其动量矢量的改变量。碰撞中，若无外冲量作用，系统总动量守恒。
Centres of mass: For a system of particles, the centre of mass position vector r̄ satisfies M r̄ = Σ m
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