📚 Vocabulary and Terminology Quick-Memorisation Guide for Year 13 WJEC Further Mathematics | Year 13 WJEC 进阶数学词汇术语速记指南
Mastering the precise vocabulary of A Level Further Mathematics is essential for interpreting questions correctly and communicating solutions effectively. This guide presents key terms across the Year 13 WJEC specification with concise definitions, memory aids, and matched Chinese-English explanations to accelerate your revision.
掌握 A Level 进阶数学的精确术语，对于正确理解题目和有效表达解答至关重要。本指南涵盖 WJEC 13 年级考纲中的关键术语，提供简明定义、记忆提示以及中英文对照解释，帮助你在复习中快速记忆。
1. Complex Numbers | 复数
Modulus-argument form: A complex number (z = x + iy) can be written as (z = r(costheta + isintheta)), where (r = |z|) is the modulus and (theta = arg z) is the argument. Memory tip: Think of polar coordinates — (r) is distance from origin, (theta) is counter‑clockwise angle from the positive real axis.
模-辐角形式：复数 (z = x + iy) 可表示为 (z = r(costheta + isintheta))，其中 (r = |z|) 是模，(theta = arg z) 是辐角。记忆提示：类比极坐标——(r) 是到原点的距离，(theta) 是从正实轴逆时针量起的角度。
Complex conjugate: The conjugate of (z = x + iy) is (z^* = bar{z} = x – iy). Reflects the point across the real axis on an Argand diagram. Tip: Changing the sign of the imaginary part mirrors the vector vertically.
共轭复数：(z = x + iy) 的共轭是 (z^* = bar{z} = x – iy)，在 Argand 图上关于实轴镜面对称。提示：改变虚部的符号相当于垂直镜像。
De Moivre’s theorem: ((costheta + isintheta)^n = cos(ntheta) + isin(ntheta)). Remember: Power multiplies the angle inside cosine and sine.
棣莫弗定理：((costheta + isintheta)^n = cos(ntheta) + isin(ntheta))。记忆：幂次与角度相乘。
Roots of unity: The solutions to (z^n = 1) lie on the unit circle and are equally spaced by angle (2pi/n). The sum of all (n)th roots of unity is zero. Visual cue: They form a regular (n)-gon on the Argand diagram.
单位根：方程 (z^n = 1) 的根位于单位圆上，且角度等间距为 (2pi/n)。所有 (n) 次单位根之和为零。视觉提示：它们在 Argand 图上构成正 (n) 边形。
Locus: A set of points satisfying a given condition, e.g., (|z – a| = r) represents a circle with centre (a) and radius (r). Tip: Interpret geometry on the Argand diagram immediately.
轨迹：满足给定条件的点集，例如 (|z – a| = r) 表示以 (a) 为圆心、半径为 (r) 的圆。提示：立即用 Argand 图进行几何解释。
2. Matrices and Linear Transformations | 矩阵与线性变换
Singular matrix: A square matrix whose determinant is zero, meaning it does not have an inverse. Memory: ‘Singular’ sounds like ‘single’ — it collapses dimensions, no inverse.
奇异矩阵：行列式为零的方阵，没有逆矩阵。记忆：“奇异”意味着退化——维度塌缩，不可逆。
Transpose AT: Swap rows and columns; (AT)ij = Aji. Tip: Reflect matrix across its main diagonal.
转置 AT：行列互换；(AT)ij = Aji。提示：沿主对角线进行镜像。
Determinant: For a 2×2 matrix [ begin{pmatrix} a & b \ c & d end{pmatrix} ] the determinant is (ad – bc). It represents the area scale factor of the transformation. Zero determinant means area collapses.
行列式：对于 2×2 矩阵[ begin{pmatrix} a & b \ c & d end{pmatrix} ]，行列式为 (ad – bc)。它表示变换的面积缩放因子。零行列式意味着面积退化为零。
Eigenvalue and eigenvector: For a matrix A, if Av = (lambda)v with v ≠ 0, then (lambda) is an eigenvalue and v is a corresponding eigenvector. Mnemonics: ‘Eigen’ (German for ‘own’) — the vector’s own direction does not change.
特征值与特征向量：对于矩阵 A，若 Av = (lambda)v 且 v ≠ 0，则 (lambda) 是特征值，v 是对应的特征向量。助记：德语 “eigen” 意为 “自己的”——向量自身方向不变。
Orthogonal matrix: A square matrix Q such that QTQ = I, i.e., its transpose equals its inverse. Its columns are orthonormal vectors. Think of it as a rotation or reflection matrix preserving lengths.
正交矩阵：满足 QTQ = I 的方阵，即转置等于逆。其列向量构成标准正交基。可视为保持长度的旋转或反射矩阵。
3. Hyperbolic Functions | 双曲函数
Definitions: (sinh x = frac{e^x – e^{-x}}{2}), (cosh x = frac{e^x + e^{-x}}{2}), (tanh x = frac{sinh x}{cosh x}). Remember: The ‘h’ comes from the hyperbola analogy, similar to circular trig.
定义：(sinh x = frac{e^x – e^{-x}}{2})，(cosh x = frac{e^x + e^{-x}}{2})，(tanh x = frac{sinh x}{cosh x})。记忆：’h’ 表明来自双曲线，类似于圆三角函数。
Osborn’s rule: Take any trigonometric identity and replace each trigonometric function with its hyperbolic counterpart, but change the sign of any product (or implied product) of two sines. Example: (cos^2theta + sin^2theta = 1) becomes (cosh^2 x – sinh^2 x = 1). The sign changes because sin × sin carries a negative sign.
奥斯本法则：取任何三角恒等式，将三角函数替换为对应的双曲函数，但需改变含有两个正弦乘积（或隐含乘积）的符号。例如：(cos^2theta + sin^2theta = 1) 变为 (cosh^2 x – sinh^2 x = 1)。符号改变是因为正弦×正弦带有负号。
Inverse hyperbolic functions: arsinh, arcosh, artanh. Can be expressed using natural logarithms, e.g., (mathrm{arsinh},x = ln(x + sqrt{x^2 + 1})). Tip: They appear in integration and differential equations.
反双曲函数：arsinh、arcosh、artanh，可用自然对数表示，如 (mathrm{arsinh},x = ln(x + sqrt{x^2 + 1}))。提示：它们常出现在积分和微分方程中。
4. Polar Coordinates | 极坐标
Polar form: Points are given by ((r, theta)), where (r geq 0) is the distance from the pole and (theta) is the angle from the initial line. Switch mental pictures from rectangular to radial grids.
极坐标形式：点用 ((r, theta)) 表示，(r geq 0) 是到极点的距离，(theta) 是从极轴量起的角度。思维图像从直角网格转为径向网格。
Cardioid: A polar curve of the form (r = a(1 + costheta)) or similar. It is heart‑shaped. Recall: ‘cardio-‘ means heart.
心形线：形如 (r = a(1 + costheta)) 的极曲线，形状像心脏。词根 ‘cardio-‘ 意为心脏。
Area enclosed: The area bounded by a polar curve (r = f(theta)) and half‑lines (theta = alpha, theta = beta) is (frac12 int_alpha^beta r^2 , dtheta). Think of tiny triangular slices of area (frac12 r cdot r dtheta).
围成面积：极曲线 (r = f(theta)) 与射线 (theta = alpha)、(theta = beta) 围成的面积为 (frac12 int_alpha^beta r^2 , dtheta)。设想微小三角形扇区面积为 (frac12 r cdot r dtheta)。
5. Series and Summation | 级数与求和
Maclaurin series: Expansion of a function about (x = 0): (f(x) = f(0) + f'(0)x + frac{f”(0)}{2!}x^2 + dots). Quick recall: It’s a Taylor series centred at zero.
麦克劳林级数：函数在 (x = 0) 处的展开：(f(x) = f(0) + f'(0)x + frac{f”(0)}{2!}x^2 + dots)。速记：以零为中心的泰勒级数。
Method of differences: Write a term (u_r) as (f(r) – f(r+1)) or similar so that the sum telescopes and most terms cancel. Look for the cancellation pattern vertically in the sum.
差分法：将项 (u_r) 写成 (f(r) – f(r+1)) 等形式，使得求和时各项可裂项相消。写出求和列，观察竖直方向的相消规律。
Standard results: Summation of (r), (r^2), (r^3) from (r=1) to (n): (sum r = frac12 n(n+1)), (sum r^2 = frac16 n(n+1)(2n+1)), (sum r^3 = frac14 n^2(n+1)^2). Memorise as formulas and verify by induction.
标准结果：从 (r=1) 到 (n) 的 (r)、(r^2)、(r^3) 求和：(sum r = frac12 n(n+1))，(sum r^2 = frac16 n(n+1)(2n+1))，(sum r^3 = frac14 n^2(n+1)^2)。记作公式并用归纳法验证。
6. Proof by Induction | 数学归纳法
Principle of mathematical induction: To prove a statement (P(n)) for all positive integers (n), show (1) base case (n = 1) is true, and (2) if (P(k)) is true then (P(k+1)) is true. Domino effect: one domino knocks the next.
数学归纳法原理：证明命题 (P(n)) 对所有正整数 (n) 成立，需证 (1) 基本情况 (n=1) 成立；(2) 若 (P(k)) 成立则 (P(k+1)) 成立。多米诺效应：一张骨牌碰倒下一张。
Inductive hypothesis: The assumption that (P(k)) holds for some arbitrary integer (k). You treat (P(k)) as temporarily given.
归纳假设：假设对某一任意整数 (k) 有 (P(k)) 成立。暂时将 (P(k)) 当作已知条件使用。
Inductive step: The logical argument that derives (P(k+1)) from (P(k)). Always link the (k+1) case back to the (k) case.
归纳递推：由 (P(k)) 推导出 (P(k+1)) 的逻辑论证。始终将 (k+1) 的情形与 (k) 的情形关联。
7. Differential Equations | 微分方程
First‑order linear: (frac{dy}{dx} + P(x)y = Q(x)). The integrating factor is (e^{int P(x) dx}). Multiply both sides by IF to get (frac{d}{dx}(y cdot IF) = Q cdot IF).
一阶线性微分方程：(frac{dy}{dx} + P(x)y = Q(x))，积分因子为 (e^{int P(x) dx})。两边乘以积分因子后可得 (frac{d}{dx}(y cdot mathrm{IF}) = Q cdot mathrm{IF})。
Auxiliary equation: For a linear second‑order ODE with constant coefficients, the equation (am^2 + bm + c = 0) obtained from (afrac{d^2y}{dx^2} + bfrac{dy}{dx} + cy = f(x)) by assuming (y = e^{mx}). Its roots determine the complementary function.
辅助方程：对常系数线性二阶常微分方程 (afrac{d^2y}{dx^2} + bfrac{dy}{dx} + cy = f(x))，设 (y = e^{mx}) 得到的代数方程 (am^2 + bm + c = 0)。其根决定补函数的形式。
Complementary function and particular integral: The general solution is (y = y_c + y_p), where (y_c) solves the homogeneous equation (f(x)=0) and (y_p) is any particular solution. Memory: ‘CF fits the left, PI fits the right.’
补函数与特解：通解为 (y = y_c + y_p)，其中 (y_c) 是对应齐次方程的解，(y_p) 是任一特解。记忆：“CF 匹配左端，PI 匹配右端。”
8. Further Vectors | 进阶向量
Vector (cross) product: a × b = |a||b| sin θ n̂, producing a vector perpendicular to both. Right‑hand rule gives direction; magnitude is area of parallelogram.
向量积（叉积）：a × b = |a||b| sin θ n̂，得到垂直于 a 和 b 的向量。右手定则确定方向；其大小为平行四边形面积。
Scalar triple product: a · (b × c) = the volume of the parallelepiped. It equals the determinant of the 3×3 matrix formed by the vectors. Cyclic permutations are equal; swapping two vectors changes sign.
标量三重积：a · (b × c) = 平行六面体的体积，等于由三向量组成的 3×3 行列式。循环置换值不变；交换两个向量变号。
Equation of a plane: r · n = d, where n is a normal vector and d is the perpendicular distance from the origin. Compare with point‑normal form: (r − a) · n = 0.
平面方程：r · n = d，其中 n 为法向量，d 为原点到平面的垂直距离。对比点法式：(r − a) · n = 0。
Distance from a point to a plane: For plane r · n = d and point with position vector p, distance = |p · n − d| / |n|. Projects the vector from a point on plane onto the unit normal.
点到平面的距离：对平面 r · n = d 和位置向量为 p 的点，距离 = |p · n − d| / |n|。将平面上任一点到 p 的向量投影到单位法向量上。
9. Functions and Inverse Functions | 函数与反函数
Domain and range: Domain is the set of allowed input values; range is the set of possible outputs. Write domain and range using interval notation or set notation.
定义域与值域：定义域是允许输入的集合；值域是所有可能输出的集合。用区间或集合符号书写。
One‑to‑one (injective): A function where different inputs give different outputs; distinct (x) values map to distinct (y) values. Necessary for an inverse to exist without restricting the domain.
单射（一对一的）：不同输入得到不同输出的函数；不同 (x) 值对应不同 (y) 值。这是其反函数存在而不需限制定义域的充要条件。
Onto (surjective): Every element of the codomain is an output of the function. Think ‘covers’ the entire target set.
满射：陪域中的每个元素都是函数的输出。想象“覆盖”整个目标集合。
Inverse function f−1: Swaps domain and range; (f^{-1}(y) = x) if and only if (f(x) = y). Graphically, reflect in the line (y = x).
反函数 f−1：交换定义域与值域；(f^{-1}(y) = x) 当且仅当 (f(x) = y)。图像上关于直线 (y = x) 对称。
10. Numerical Methods | 数值方法
Newton‑Raphson method: xn+1 = xn − f(xn)/f'(xn), used to solve f(x)=0. Memory: Starting from a guess, slide down the tangent to the x‑intercept.
牛顿-拉夫森法：xn+1 = xn − f(xn)/f'(xn)，用于求解 f(x)=0。记忆：从初值出发，沿切线滑向与 x 轴的交点。
Trapezium rule: Approximates (int_a^b f(x) dx) by dividing into (n) strips of width (h): (frac{h}{2}[f(x_0) + 2f(x_1) + dots + 2f(x_{n-1}) + f(x_n)]). Under‑ or over‑estimates depending on concavity.
梯形法则：将区间分成 (n) 个宽度为 (h)
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