📚 Year 13 WJEC Further Maths: Quick Glossary Guide | Year 13 WJEC 进阶数学：词汇术语速记指南
Mastering the technical vocabulary of A‑level Further Mathematics is as critical as solving equations. This quick‑reference guide walks through the essential terms you will meet in the Year 13 WJEC course — from complex numbers and hyperbolic functions to vectors, mechanics and statistics — with bilingual definitions designed to boost understanding and exam confidence.
掌握 A‑level 进阶数学的专业术语与解方程同样重要。这份速记指南梳理了 Year 13 WJEC 课程中你将遇到的关键术语——从复数、双曲函数到向量、力学与统计——通过双语定义，帮助你深化理解，增强考试信心。
1. Complex Numbers | 复数
A complex number has the form z = x + iy, where x is the real part Re(z) and y is the imaginary part Im(z), and i² = −1.
复数的形式为 z = x + iy，其中 x 是实部 Re(z)，y 是虚部 Im(z)，且 i² = −1。
The complex conjugate of z is z* = x − iy. It reflects the number in the real axis.
复数的共轭为 z* = x − iy，它是原数关于实轴的反射。
The modulus is |z| = √(x² + y²), representing the distance from the origin in the Argand diagram.
模长 |z| = √(x² + y²)，表示在 Argand 图中点到原点的距离。
The argument arg(z) = θ, where tan θ = y/x, measured from the positive real axis, usually in (−π, π].
辐角 arg(z) = θ，满足 tan θ = y/x，从正实轴起量，主值通常在 (−π, π]。
Euler’s relation eⁱ⁰ = cos θ + i sin θ links exponential and trigonometric forms.
欧拉公式 eⁱ⁰ = cos θ + i sin θ 将指数形式与三角形式联系起来。
De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ for integer n.
棣莫弗定理：(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ，适用于整数 n。
An nth root of unity is any complex solution of zⁿ = 1; there are n distinct roots equally spaced around the unit circle.
单位根是指满足 zⁿ = 1 的复数解；共有 n 个不同的根，均匀分布在单位圆上。
2. Matrices and Linear Transformations | 矩阵与线性变换
A matrix is a rectangular array of numbers; an m×n matrix has m rows and n columns.
矩阵是数字的矩形阵列；m×n 矩阵有 m 行和 n 列。
The identity matrix I has ones on the main diagonal and zeros elsewhere; for any conformable matrix A, AI = IA = A.
单位矩阵 I 的主对角线元素为 1，其余为 0；对任意可乘矩阵 A，有 AI = IA = A。
The determinant of a 2×2 matrix A = [[a, b],[c, d]] is det(A) = ad − bc.
二阶矩阵 A = [[a, b],[c, d]] 的行列式为 det(A) = ad − bc。
A singular matrix has zero determinant and no inverse; a non‑singular matrix has a non‑zero determinant and possesses an inverse A⁻¹.
奇异矩阵行列式为零，不可逆；非奇异矩阵的行列式不为零，存在逆矩阵 A⁻¹。
A linear transformation maps vectors by matrix multiplication; standard transformations include rotations, reflections, stretches and shears.
线性变换通过矩阵乘法将向量映射起来；标准的变换包括旋转、反射、伸缩和剪切。
The inverse transformation reverses the effect of a transformation and is represented by the inverse matrix.
逆变换逆转原变换的效果，由逆矩阵表示。
Invariant points satisfy Mx = x; invariant lines are lines that map to themselves under the transformation.
不动点满足 Mx = x；不变线是在变换下映射到自身的直线。
3. Hyperbolic Functions | 双曲函数
Hyperbolic functions are defined using exponentials: sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2.
双曲函数用指数定义：sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2。
tanh x = sinh x / cosh x; coth x = 1/tanh x; sech x = 1/cosh x; cosech x = 1/sinh x.
tanh x = sinh x / cosh x；coth x = 1/tanh x；sech x = 1/cosh x；cosech x = 1/sinh x。
The fundamental identity is cosh² x − sinh² x = 1 (compare with cos² x + sin² x = 1).
基本恒等式为 cosh² x − sinh² x = 1（对比 cos² x + sin² x = 1）。
Osborn’s rule converts trigonometric identities into hyperbolic identities by replacing cos² with cosh² and changing the sign of any product of two sines.
奥斯本法则是将三角恒等式转换为双曲恒等式的方法：把 cos² 换成 cosh²，并将任意两个正弦的乘积变号。
Inverse hyperbolic functions are denoted arsinh, arcosh, artanh; they can be expressed in logarithmic form.
反双曲函数记作 arsinh, arcosh, artanh；它们均可用对数形式表示。
Differentiation: d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech² x.
求导：d/dx (sinh x) = cosh x，d/dx (cosh x) = sinh x，d/dx (tanh x) = sech² x。
4. Polar Coordinates | 极坐标
In the polar coordinate system, a point is given by (r, θ), where r is the radial distance from the pole O and θ is the angle from the initial line.
在极坐标系中，点用 (r, θ) 表示，其中 r 是从极点 O 出发的径向距离，θ 是从极轴起的角度。
Cartesian conversion: x = r cos θ, y = r sin θ; also r = √(x² + y²), θ = arctan(y/x) considering the quadrant.
与直角坐标的转换：x = r cos θ，y = r sin θ；同时 r = √(x² + y²)，θ = arctan(y/x) 需根据象限确定。
A polar curve is defined by r = f(θ); common examples include cardioids (r = a(1 + cos θ)) and limaçons.
极坐标曲线由 r = f(θ) 定义；常见例子有心形线 (r = a(1 + cos θ)) 和蚌线。
The area enclosed by a polar curve is A = ½ ∫[α,β] r² dθ.
极坐标曲线所围成的面积为 A = ½ ∫[α,β] r² dθ.
The arc length of a polar curve from θ = α to θ = β is s = ∫[α,β] √(r² + (dr/dθ)²) dθ.
极坐标曲线在 θ = α 到 θ = β 之间的弧长为 s = ∫[α,β] √(r² + (dr/dθ)²) dθ.
5. Differential Equations | 微分方程
A differential equation (DE) involves an unknown function and its derivatives; the order is the highest derivative present.
微分方程涉及未知函数及其导数；阶数是指方程中出现的最高阶导数的阶数。
A first‑order separable DE can be written as g(y) dy = f(x) dx, solved by direct integration.
一阶可分离变量的微分方程可写成 g(y) dy = f(x) dx，通过直接积分求解。
An integrating factor μ(x) = e^(∫P(x)dx) is used to solve a first‑order linear DE of the form dy/dx + P(x)y = Q(x).
积分因子 μ(x) = e^(∫P(x)dx) 用于求解一阶线性微分方程 dy/dx + P(x)y = Q(x)。
A second‑order homogeneous linear DE with constant coefficients has the auxiliary equation am² + bm + c = 0; the form of the general solution depends on whether roots are real & distinct, repeated, or complex.
常系数二阶齐次线性微分方程的辅助方程为 am² + bm + c = 0；通解的形式取决于根是相异实根、重根还是共轭复根。
The particular integral (PI) is a trial function used to find a particular solution when the equation is non‑homogeneous; it must be linearly independent of the complementary function (CF).
特解 (PI) 是在非齐次条件下求特解的试探函数，且必须与补函数 (CF) 线性无关。
General solution: y = CF + PI, combining the complementary function and a particular integral.
通解为 y = CF + PI，即补函数与特解之和。
6. Proof by Induction | 归纳法证明
Mathematical induction proves a statement P(n) for all positive integers n: base case (n = 1), inductive hypothesis (assume true for n = k), inductive step (prove for n = k+1).
数学归纳法用于证明对所有正整数 n 都成立的命题 P(n)：基础步骤 (n=1)、归纳假设 (假设 n=k 成立)、归纳步骤 (证明 n=k+1 成立)。
The inductive hypothesis P(k) must be used explicitly in the proof of P(k+1).
在证明 P(k+1) 时必须明确使用归纳假设 P(k)。
Strong induction assumes the statement holds for all integers less than or equal to k, which is useful for recurrence relations of order greater than 1.
强归纳法假设命题对所有小于等于 k 的整数都成立，适用于高于一阶的递推关系。
Induction is frequently used to prove divisibility, matrix powers, summation formulae, and inequalities.
归纳法常用于证明整除性、矩阵的幂、求和公式以及不等式。
7. Vectors and 3D Geometry | 向量与三维几何
The vector equation of a line is r = a + λb, where a is a position vector on the line and b is the direction vector.
直线的向量方程为 r = a + λb，其中 a 是直线上一点的位置向量，b 是方向向量。
The scalar (dot) product a·b = |a||b| cos θ is used to find angles between vectors and to test orthogonality (a·b = 0).
数量的点积 a·b = |a||b| cos θ 用于求向量间的角度以及检验正交性 (a·b = 0)。
The vector (cross) product a×b = |a||b| sin θ n̂ is a vector perpendicular to both a and b; its magnitude gives the area of the parallelogram they span.
向量的叉积 a×b = |a||b| sin θ n̂ 是一个垂直于 a 和 b 的向量；其模长等于以 a、b 为邻边的平行四边形的面积。
The vector equation of a plane can be written as r·n = d, where n is a normal vector to the plane.
平面的向量方程可写作 r·n = d，其中 n 是平面的法向量。
The angle between two planes is the acute angle between their normals.
两平面的夹角是指其法向量间的锐角。
The shortest distance from a point to a line, or from a point to a plane, can be found using perpendicular vectors or projection formulae.
点到直线或点到平面的最短距离可利用垂向量或投影公式求得。
8. Further Mechanics: Impulse and Work-Energy | 进阶力学：冲量与功-能
Impulse J = F Δt = Δ(mv) is the change in momentum caused by a force acting over a time interval.
冲量 J = F Δt = Δ(mv)，是力在一段时间内作用所引起的动量变化。
The principle of conservation of momentum states that for a system with no external forces, total momentum before collision equals total momentum after collision.
动量守恒定律指出，在无外力作用的系统中，碰撞前的总动量等于碰撞后的总动量。
Work done by a constant force is W = Fs cos θ, where s is displacement and θ is the angle between force and displacement.
恒力所做的功为 W = Fs cos θ，其中 s 是位移，θ 是力与位移之间的夹角。
Kinetic energy K = ½mv²; gravitational potential energy U = mgh near the Earth’s surface.
动能 K = ½mv²；地表附近的重力势能 U = mgh。
The work–energy principle states that the net work done is equal to the change in kinetic energy (W = ΔK).
功能原理指出，合力所做的功等于动能的变化量 (W = ΔK)。
Power is the rate of doing work: P = Fv for a constant force acting in the direction of motion.
功率是做功的速率：当恒力沿运动方向作用时，P = Fv。
9. Further Statistics: Discrete and Continuous Distributions | 进阶统计：离散与连续分布
The Poisson distribution models the number of events occurring in a fixed interval, with parameter λ (mean = variance = λ). Probability: P(X = r) = e⁻λ λʳ/r!.
泊松分布用于模拟固定区间内事件发生的次数，参数为 λ (均值 = 方差 = λ)。概率公式：P(X = r) = e⁻λ λʳ/r!。
The binomial distribution B(n, p) can be approximated by Poisson when n is large and p is small, with λ = np.
当 n 很大且 p 很小时，二项分布 B(n, p) 可用泊松分布近似，此时 λ = np。
A continuous random variable has a probability density function (p.d.f.) f(x) ≥ 0, and the area under the curve over its domain equals 1.
连续随机变量具有概率密度函数 f(x) ≥ 0，且其定义域内曲线下的总面积为 1。
The cumulative distribution function (c.d.f.) F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt.
累积分布函数 F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt。
Mean (expectation) for continuous distributions: E(X) = ∫ x f(x) dx; variance Var(X) = E(X²) − [E(X)]².
连续分布的均值（期望）：E(X) = ∫ x f(x) dx；方差 Var(X) = E(X²) − [E(X)]²。
The normal distribution N(μ, σ²) is a bell‑shaped continuous distribution with parameters μ (mean) and σ² (variance); standardisation Z = (X − μ)/σ ∼ N(0, 1).
正态分布 N(μ, σ²) 是一种钟形连续分布，参数为 μ (均值) 和 σ² (方差)；标准化后 Z = (X − μ)/σ ∼ N(0, 1)。
10. Hypothesis Testing and Chi‑squared Tests | 假设检验与卡方检验
A hypothesis test assesses evidence against a null hypothesis H₀. The significance level α is the probability of rejecting H₀ when it is true (Type I error).
假设检验用于评估反对原假设 H₀ 的证据。显著性水平 α 是当 H₀ 成立时拒绝 H₀ 的概率（第一类错误）。
The p‑value is the probability of obtaining a test statistic at least as extreme as the observed one, assuming H₀ is true.
p 值是指在原假设 H₀ 成立的情况下，获得至少与观察值同样极端的检验统计量的概率。
For a normal distribution mean test with known variance, the test statistic is Z = (x̄ − μ)/(σ/√n).
对于已知方差的正态分布均值检验，检验统计量为 Z = (x̄ − μ)/(σ/√n)。
The chi‑squared (χ²) test for goodness of fit compares observed frequencies Oᵢ with expected frequencies Eᵢ: χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ.
拟合优度的卡方 (χ²) 检验比较观测频数 Oᵢ 和期望频数 Eᵢ：χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ。
The degrees of freedom ν for a goodness‑of‑fit test is (number of categories − 1 − number of estimated parameters).
拟合优度检验的自由度 ν = 类别数 − 1 − 需估计的参数个数。
A contingency table test for independence between two categorical variables also uses the χ² statistic with ν = (r − 1)(c − 1).
两个分类变量独立性的列联表检验也使用 χ² 统计量，自由度 ν = (行数−1)(列数−1)。
11. Complex Transformations and Loci | 复数变换与轨迹
A complex function w = f(z) maps points from the z‑plane to the w‑plane; common mappings include w = kz, w = z², and Möbius transformations w = (az + b)/(cz + d).
复变函数 w = f(z) 将 z 平面上的点映射到 w 平面；常见映射包括 w = kz、w = z² 和莫比乌斯变换 w = (az + b)/(cz + d)。
The locus of |z − a| = r is a circle centred at a with radius r.
满足 |z − a| = r 的轨迹是以 a 为圆心、半径为 r 的圆。
The locus of |z − a| = |z − b| is the perpendicular bisector of the segment joining a and b.
满足 |z − a| = |z − b| 的轨迹是连接 a、b 两点的线段的垂直平分线。
arg(z − a) = θ defines a half‑line starting from a (excluding a) making an angle θ with the positive real axis.
arg(z − a) = θ 定义了一条从 a 出发（不含 a）且与正实轴夹角为 θ 的射线。
Combining modulus and argument inequalities yields regions such as annular sectors and half‑planes.
将模与辐角的不等式组合可以得到圆环扇形、半平面等区域。
12. Further Calculus: Maclaurin Series and Integration Techniques | 进阶微积分：麦克劳林级数与积分技巧
The Maclaurin series of a function f(x) is f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + ….
函数 f(x) 的麦克劳林级数为 f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + …。
Standard series include eˣ = Σ xⁿ/n!, sin x = Σ (−1)ⁿx²ⁿ⁺¹/(2n+1)!, cos x = Σ (−1)ⁿx²ⁿ/(2n)!, ln(1+x) = Σ (−1)ⁿ⁻¹xⁿ/n.
标准级数包括：eˣ = Σ xⁿ/n!，sin x = Σ (−1)ⁿx²ⁿ⁺¹/(2n+1)!，cos x = Σ (−1)ⁿx²ⁿ/(2n)!，ln(1+x) = Σ (−1)ⁿ⁻¹xⁿ/n。
Integration by substitution uses the chain rule in reverse: let u = g(x) so that ∫ f(g(x))g'(x)dx = ∫ f(u)du.
换元积分法逆向使用链式法则：令 u = g(x)，则 ∫ f(g(x))g'(x)dx = ∫ f(u)du。
Integration by parts: ∫ u dv = uv − ∫ v du, essential for products of functions.
分部积分法：∫ u dv = uv − ∫ v du，对于函数乘积的积分至关重要。
Partial fractions decompose a rational function into simpler fractions, enabling direct integration when the denominator factorises.
部分分式法将有理函数分解为更简单的分式之和，当分母可因式分解时可直接积分。
Improper integrals have either an infinite limit of integration or an integrand with an infinite discontinuity; they are evaluated as limits.
反常积分的积分限为无穷大或被积函数在积分区间内有无穷间断点；其计算通过取极限得到。
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