📚 Pre-U CIE Further Maths: Vocabulary & Terminology Quick-Memorization Guide | Pre-U CIE 进阶数学：词汇术语速记指南
Mastering the distinct vocabulary of CIE Pre-U Further Mathematics is just as important as understanding the methods. The right terminology allows you to interpret questions quickly and express solutions with precision. This guide presents key terms you will meet across the syllabus, paired with memory-friendly explanations that help transfer them from short-term overload into long-term confidence.
掌握 CIE Pre-U 进阶数学的独特词汇与掌握方法同样重要。准确的术语能帮助你快速理解题意，精准表达解题过程。本指南汇集了考纲中反复出现的关键术语，并搭配易于记忆的解释，让它们从短期负担变成长期拿分的底气。
1. Complex Numbers | 复数
The imaginary unit i The complex conjugate is z* = a − bi, obtained by flipping the sign of the imaginary part.
虚数单位 i 满足 i² = −1。复数写作 z = a + bi，其中 a 为实部 Re(z)，b 为虚部 Im(z)。共轭复数 z* = a − bi，只需把虚部符号翻转即可得到。
The modulus |z| = √a² + b² gives the distance from the origin in the complex plane. The argument real axis, usually in (−π, π]. A quick memory link: modulus = magnitude, argument = angle.
模长 |z| = √a² + b² 表示复平面上点到原点的距离。辐角 arg(z) 是从正实轴测量的角度，通常范围是 (−π, π]。速记口诀：模长看长度，辐角看方向。
De Moivre's theorem states (cos θ + i sin θ)n in(nθ). It is the engine for raising complex numbers to powers and finding roots of unity.
棣莫弗定理：(cos θ + i sin θ)n = cos(nθ) + i sin(nθ)。它是复数乘方和求单位根的发动机。
2. Matrices | 矩阵
A matrix is a rectangular array of numbers. The determinant t = ad − bc. It tells you if the matrix is singular (det = 0) or invertible (det ≠ 0). The inverse is (1/det)(d −b; −c a).
矩阵是数字的矩形阵列。2×2 矩阵 (a b; c d) 的行列式为 det = ad − bc。用它判断矩阵是否奇异（det = 0 则不可逆）。逆矩阵为 (1/det)(d −b; −c a)。
The transpose AT swaps rows and columns. For an orthogonal matrix, AT = A−1. Eigenvalues λ satisfy det(A − λI) = 0, and eigenvectors x satisfy Ax = λx. Remember: “eigen” means “own”, so these vectors only get stretched.
转置 AT 交换行与列。正交矩阵满足 AT = A−1。特征值 λ 满足 det(A − λI) = 0，特征向量 x 满足 Ax = λx。记法：“eigen” 意为“自身的”，特征向量只被拉伸而不发生偏转。
To remember the order of multiplication in inverses: (AB)−1 = B−1A−1. Think of putting on socks then shoes; to reverse, remove shoes first.
逆矩阵乘积顺序：(AB)−1 = B−1A−1。想象先穿袜子再穿鞋，反向操作时就要先脱鞋。这个情景帮你记住顺序。
3. Vectors | 向量
The scalar (dot) product a · b = |a||b| cos θ gives a scalar. It is zero when vectors are perpendicular. The vector (cross) product a × b yields a vector perpendicular to both, with magnitude |a||b| sin θ.
标量积（点乘） a · b = |a||b| cos θ 结果为标量，当向量垂直时为零。向量积（叉乘） a × b 得到一个垂直于两者的向量，大小为 |a||b| sin θ。
The scalar triple product lanar. Vector equation of a line: r = a + t d; of a plane: r = a + λ u + μ v or r · n = p.
标量三重积 a · (b × c) 计算平行六面体的体积。若为零则三向量共面。直线的向量方程：r = a + t d；平面方程：r = a + λ u + μ v 或 r · n = p。
Think of 'dot' as 'direction overlap', and 'cross' as creating a new direction.
可以这么想：点乘看“方向重合度”，叉乘产生“新方向”。
4. Differential Equations | 微分方程
A first-order separable ODE can be written as g(y) dy = f(x) dx. Integrate both sides. The integrating factor method for y' + P(x)y = Q(x) uses μ(x) = e∫ P dx.
一阶可分离微分方程可写成 g(y) dy = f(x) dx，两侧积分即可。积分因子法用于 y' + P(x)y = Q(x)，因子为 μ(x) = e∫ P dx。
Second-order linear ODEs with constant coefficients: ay'' + by' + cy = f(x). Solve the complementary function (CF) by setting f(x)=0, then find a particular integral (PI) based on f(x). The general solution is y = CF + PI.
常系数二阶线性常微分方程：ay'' + by' + cy = f(x)。先解余函数（CF）即 f(x)=0 的情况，再根据 f(x) 形式求特解（PI），通解为 y = CF + PI。
The auxiliary equation is am² + bm + c = 0. Roots real/distinct → y = Aem₁x + Bem₂x; repeated → y = (A + Bx)emx; complex → y = epx(A cos qx + B sin qx).
辅助方程为 am² + bm + c = 0。实不等根 → y = Aem₁x + Bem₂x；重根 → y = (A + Bx)emx；复根 → y = epx(A cos qx + B sin qx)。
5. Hyperbolic Functions | 双曲函数
The hyperbolic sine and cosine are defined via exponentials: sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ lation to the hyperbola x² − y² = 1, just as trigonometric functions relate to the circle.
双曲正弦和余弦通过指数定义：sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2。它们名字中的“双曲”源于与双曲线 x² − y² = 1 的关系，恰如三角函数与单位圆的关系。
x = cosh x, d/dx cosh x = sinh x. Note the missing sign change compared to trig functions.
核心恒等式：cosh²x − sinh²x = 1。导数：d/dx sinh x = cosh x，d/dx cosh x = sinh x。留意这里没有三角函数的符号变化，这能帮你避免混淆。
Inverse hyperbolic functions can be expressed as logarithms: arsinh x = ln(x + √x²+1), arcosh x = ln(x + √x²−1), artanh x = ½ ln((1+x)/(1−x)).
反双曲函数可以用对数表示：arsinh x = ln(x + √x²+1)，arcosh x = ln(x + √x²−1)，artanh x = ½ ln((1+x)/(1−x))。
6. Polar Coordinates | 极坐标
θ is the angle from the initial line. Conversion: x = r cos θ, y = r sin θ, r² = x² + y².
点的极坐标表示为 (r, θ)，r 是到极点的距离，θ 是从极轴出发的角度。转换关系：x = r cos θ，y = r sin θ，r² = x² + y²。
The area bounded by a polar curve r = f(θ) and rays θ = α, β is ½ ∫αβ The arc length is ∫ √(r² + (dr/dθ)²) dθ. Think of the ½ r² dθ as a triangular area slice.
极坐标曲线 r = f(θ) 与射线 θ = α, β 围成的面积为 ½ ∫αβ r² dθ。弧长公式为 ∫ √(r² + (dr/dθ)²) dθ。面积微元 ½ r² dθ 可以形象地看成一个小三角形扇形。
to the initial line occur when d/dθ (r sin θ) = 0; perpendicular when d/dθ (r cos θ) = 0.
切线平行于极轴的条件是 d/dθ (r sin θ) = 0；垂直于极轴的条件是 d/dθ (r cos θ) = 0。
7. Proof by Induction | 归纳法证明
Mathematical induction consists of a basis step (prove true for n = 1), an inductive hypothesis (assume true for n = k), and an inductive step (prove true for n = k+1 using the assumption).
数学归纳法包含三个部分：奠基步骤（证明 n=1 时命题成立）；归纳假设（假设 n=k 时成立）；归纳递推（利用假设证明 n=k+1 时成立）。
This technique is heavily used for summation formulas, divisibility, matrix powers, and inequalities. Always state a conclusion: “By mathematical induction, true for all positive integers n.”
这一技术广泛用于求和公式、整除性、矩阵幂次和不等式证明。最后一定要写出结论：“由数学归纳法，对所有正整数 n 成立。”
A common slip: forgetting to use the inductive hypothesis in the inductive step. Check that your reasoning explicitly says “using the assumption P(k)”.
常见丢分点：在递推步骤中忘记用到归纳假设。务必让推理中明确出现“由假设 P(k) 可知”的字眼。
8. Series | 级数
The Maclaurin series expansion of a function about x = 0 is f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ... . The Taylor series about x = a follows the same pattern with (x−a).
函数在 x=0 处的麦克劳林展开为 f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ...。泰勒级数围绕 x=a 展开时把 x 替换为 (x−a) 即可。
Standard expansions to memorise: 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 − ... (|x| < 1).
必须熟记的标准展开式：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 − ... (|x| < 1)。
Summation notation Σ ar from r=1 to n is compact. The method of differences breaks a term as ur = vr+1 − vr so the series collapses.
连加符号 Σr=1n ar 让表达更简洁。差分法将项拆为 ur = vr+1 − vr，使级数前后抵消。
9. Numerical Methods | 数值方法
The Newton-Raphson method iterates xn+1 = xn − f(xn)/f'(xn) to approximate a root. Starting value must be sufficiently close to the root. The formula can be remembered as “current guess minus value over slope”.
牛顿-拉弗森法迭代公式为 xn+1 = xn − f(xn)/f'(xn)。初始值需要足够接近根。记忆技巧：“当前猜测值减去函数值除以斜率”。
The trapezium rule approximates ∫ab f(x) dx ≈ ½ h [y₀ + 2(y₁ + y₂ + ... + yn-1) + yn] with h = (b−a)/n. Simpson's rule (n even) gives more accuracy: ≈ ⅓ h [y₀ + 4y₁ + 2y₂ + 4y₃ + ... + yn].
梯形法则近似为 ∫ab f(x) dx ≈ ½ h [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：≈ ⅓ h [y₀ + 4y₁ + 2y₂ + 4y₃ + ... + yn]。
Remember trapezium: average endpoints with double interiors; Simpson: alternating 4 and 2.
梯形法则的系数规律：首尾系数 1，内部系数 2；辛普森法则：1, 4, 2, 4, ..., 1。记住这个交替节奏就能快速写出公式。
10. Group Theory | 群论
A group (G, *) is a set with a binary operation satisfying four axioms: closure, associativity, existence of an identity element, and existence of an inverse for every element. The mnemonic “CAIN” (Closure, Associativity, Identity, iNverse) helps retain the order.
群 (G, *) 是一个带有二元运算的集合，满足四条公理：封闭性、结合律、存在单位元、每个元素存在逆元。速记口诀 “CAIN” 或中文“封结位逆”可以帮你按顺序回忆。
An Abelian group additionally requires commutativity (a * b = b * a). A subgroup is a subset that is itself a group under the same operation. Use subgroup tests to verify.
阿贝尔群额外要求交换律 (a * b = b * a)。子群是原群的一个子集，在同种运算下本身也是一个群，验证时需用子群检验法。
The order nts; the order of an element is the smallest positive integer n such that aⁿ = e. Lagrange's theorem states that the order of a subgroup divides the order of the group.
群的阶指元素的个数；元素的阶是使得 aⁿ = e 的最小正整数 n。拉格朗日定理指出子群的阶一定整除群的阶。
11. Further Calculus | 进阶微积分
A reduction formula expresses an integral In in terms of In-1 or In-2 allowing repeated integration by parts to be condensed. Typical form: ∫ sinⁿ x dx, ∫ xⁿ eˣ dx.
递推公式将积分 In 用 In-1 或 In-2 表示，把多次分部积分压缩成一个公式。常见类型：∫ sinⁿ x dx，∫ xⁿ eˣ dx。
Arc length for a function y = f(x) from x=a to b is s = ∫ab dy/dt)²) dt. Surface area of revolution about the x-axis is S = 2π ∫ y √(1 + (dy/dx)²) dx.
函数 y = f(x) 从 x=a 到 b 的弧长为 s = ∫ab √(1 + (dy/dx)²) dx。参数曲线弧长：s = ∫ √((dx/dt)² + (dy/dt)²) dt。绕 x 轴旋转的曲面面积为 S = 2π ∫ y √(1 + (dy/dx)²) dx。
Mean value of a function over [a,b] is (1/(b−a)) ∫ab f(x) dx. It is the constant that gives the same area over that interval.
函数在 [a,b] 上的平均值为 (1/(b−a)) ∫ab f(x) dx，它是使得矩形面积等于原曲线下面积的那个常数值。
12. Roots of Polynomials | 多项式根
+ γ = −b/a, sum of pairwise products αβ + βγ + γα = c/a, product αβγ = −d/a. These are Viète's formulas.
对于三次方程 ax³ + bx² + cx + d = 0，根为 α, β, γ，有：α + β + γ = −b/a，两两积之和 αβ + βγ + γα = c/a，根的乘积 αβγ = −d/a。这些就是韦达定理。
d/a, sum of pairwise products = c/a, product = e/a. Pay attention to the alternating signs!
四次方程 ax⁴ + bx³ + cx² + dx + e = 0：根之和 = −b/a，三三乘积之和 = −d/a，两两积之和 = c/a，所有根的积 = e/a。注意符号正负交替的规律！
These relations are powerful when combined with symmetric sums. Convert expressions like α² + β² + γ² using (α+β+γ)² − 2(αβ+βγ+γα).
这些关系结合对称和式非常强大。例如 α² + β² + γ² 可转化为 (α+β+γ)² − 2(αβ+βγ+γα)。
(y−1)/2 into the original equation. Always verify the degree matches the number of roots.
求一个新多项式，使其根为原方程根经过某种变换得到（如 2α+1），方法是将 x = (y−1)/2 代入原方程。务必检查新方程的次数是否与根的个数一致。
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