📚 AS Further Maths Vocabulary Memory Hacks | AS进阶数学词汇术语速记指南
Mastering the precise vocabulary is half the battle in AS Further Mathematics. This guide pairs each key term with a memory hack so you can recall definitions instantly under exam pressure.
在AS进阶数学中，精准掌握词汇是成功的一半。本指南将每个关键术语搭配一个记忆技巧，帮助你在考试压力下立即回想起定义。
1. Roots and Coefficients of Polynomials | 根与系数的关系
Root (α, β, γ): A value that makes the polynomial equal to zero. Think of roots as the 'invisible numbers' hidden inside the equation.
根 (α, β, γ)： 使多项式等于零的值。把根想象成隐藏在方程里的“隐形数字”。
Sum of roots: For quadratic ax²+bx+c=0, sum α+β = –b/a. Memory aid: 'SUM of roots is Opposite sign of b over a' (SUM = –b/a).
根之和： 对于二次式 ax²+bx+c=0，和 α+β = –b/a。记忆：“和是 b 的相反数除以 a”。
Product of roots: For a quadratic, αβ = c/a. For a cubic ax³+bx²+cx+d=0, αβγ = –d/a. Link: product of roots keeps the sign pattern (alternating).
根之积： 二次式 αβ = c/a；三次式 αβγ = –d/a。规律：乘积的符号交替变化。
Sum of pairwise products: For cubic, αβ+βγ+γα = c/a. Visualise choosing pairs of roots and multiplying them.
根的两两乘积之和： 三次式为 αβ+βγ+γα = c/a。想象从三个根中选取两两组合相乘再求和。
2. Transformations of Roots | 根的变换
Root transformation: s are related to the original roots, e.g. each root increased by k. Use substitution x = y – k to reverse-engineer.
根的变换： 求一个新多项式，使其根与原多项式的根有特定关系，例如每个根加 k。用替换 x = y – k 来逆向推导。
Reciprocal roots: New roots are 1/α, 1/β. Strategy: replace x by 1/x and clear denominators. Memory: 'flip it' – x becomes 1/x.
倒数根： 新根为 1/α, 1/β。方法：用 1/x 替换 x 并去分母。记忆：“翻转”——x 变成 1/x。
Squared roots: Roots become α², β². Use x = √y, but careful with sign. Often handled by symmetric functions of α²+β² etc.
平方根： 根变成 α², β²。使用 x = √y，但要注意符号。通常借助对称式 α²+β² 等处理。
3. Rational Functions: Asymptotes and Sketching | 有理函数：渐近线与图像
Vertical asymptote: Occurs where denominator = 0 but numerator ≠ 0. The graph shoots off to infinity. Memory: 'VA where the function blows up'.
垂直渐近线： 出现在分母为零但分子不为零之处，图像在此趋向无穷。记忆：“函数爆炸的地方”。
Horizontal asymptote: are degrees: if deg(num) < deg(den), HA is y=0; if equal, HA is leading coefficient ratio.
水平渐近线： 当 x → ±∞ 时的趋势。比较次数：分子次数小于分母，则 y=0；相等则为最高次系数之比。
Oblique asymptote: : When degree of numerator is exactly one more than denominator. Divide numerator by denominator to find the slant line y = mx + c.
斜渐近线： 分子次数比分母刚好高一次时出现。用长除法求斜线 y = mx + c。
Intercepts: x-intercept(s): set numerator = 0 (and not denominator=0). y-intercept: set x = 0. 'Crossing points mark the story of the graph.'
截距： x 截距：令分子=0（且分母不为0）。y 截距：令 x=0。“截距是图像故事的关键节点”。
4. Partial Fractions | 部分分式
Proper rational function: Degree of numerator < degree of denominator. If improper, first perform polynomial division.
真有理函数： 分子次数小于分母次数。如果是假分式，先进行多项式除法。
Linear factors: For denominator (ax+b)(cx+d), write as A/(ax+b) + B/(cx+d). Memory: each linear factor gets a constant numerator.
线性因式： 对于分母 (ax+b)(cx+d)，拆成 A/(ax+b) + B/(cx+d)。记忆：每个线性因式上方放一个常数。
Repeated linear factors: (ax+b)² → A/(ax+b) + B/(ax+b)². The power increases stepwise. Think: 'repeat with raised powers'.
重复线性因式： (ax+b)² 拆成 A/(ax+b) + B/(ax+b)²。方幂逐级递增。记忆：“重复并提升方次”。
Quadratic factor: ax²+bx+c (cannot be factorised) gets numerator Cx+D. 'Quadratic denominator, linear numerator.'
二次因式： 不可分解的 ax²+bx+c 上方放分子 Cx+D。“二次分母对应一次分子”。
5. Summation of Series | 级数求和
Sigma notation ∑: Compact way to write sum. ∑r=1n ur. Memory: 'Sigma squeezes many terms into one symbol.'
求和符号 ∑： 紧凑表示求和。∑r=1n ur。记忆：“Sigma 把许多项压缩成一个符号”。
Standard results: ∑r = n(n+1)/2, ∑r² = n(n+1)(2n+1)/6, ∑r³ = (n(n+1)/2)². Learn these by heart: 'Sum of cubes is the square of the sum of integers.'
标准结果： ∑r = n(n+1)/2，∑r² = n(n+1)(2n+1)/6，∑r³ = (n(n+1)/2)²。记忆：“立方和是整数和的平方”。
Method of differences: Express term as f(r) – f(r+1) or similar to make intermediate terms cancel. Look for telescoping patterns.
差分法： 将通项写成 f(r) – f(r+1) 等形式，让中间项相互抵消。寻找裂项相消的规律。
Summation of polynomial series: Expand and use linearity: ∑(a·ur + b·vr) = a∑ur + b∑vr.
多项式级数求和： 展开并利用线性性质：∑(a·ur + b·vr) = a∑ur + b∑vr。
6. Matrices: Operations and Determinant | 矩阵：运算与行列式
Order of a matrix: Rows × columns. A 2×3 matrix has 2 rows, 3 columns. 'Row first, column second.'
矩阵的阶： 行 × 列。2×3 矩阵有 2 行 3 列。“先行后列”。
Matrix multiplication: Non-commutative: AB ≠ BA in general. The entry in row i, column j comes from dot product of row i of A with column j of B.
矩阵乘法： 不满足交换律：通常 AB ≠ BA。第 i 行第 j 列元素来自 A 的 i 行与 B 的 j 列的点积。
Identity matrix I: x with 1s on the main diagonal, 0s elsewhere. AI = IA = A. Like the number 1 for matrices.
单位矩阵 I： 方阵，主对角线为 1，其余为 0。AI = IA = A。就像矩阵世界的“1”。
Determinant of 2×2: For M = [a b; c d], det(M) = ad – bc. Zero determinant means the matrix is singular (no inverse).
2×2 行列式： M = [a b; c d] 的行列式为 ad – bc。行列式为零说明矩阵是奇异的（无逆矩阵）。
Singular vs non-singular: Singular = determinant = 0, which means the rows/columns are linearly dependent. 'Singular means single-failure – no inverse.'
奇异与非奇异： 奇异 ↔ 行列式为 0，意味着行或列线性相关。“奇异是独败——没有逆”。
7. Inverse of a Matrix and Solving Equations | 逆矩阵与方程求解
Inverse of 2×2 matrix: For [a b; c d], inverse = 1/(ad-bc) [ d -b; -c a ]. 'Swap a and d, change signs of b and c, divide by determinant.'
2×2 逆矩阵： [a b; c d] 的逆 = 1/(ad-bc) [ d -b; -c a ]。“交换 a 和 d，b 和 c 变号，除以行列式”。
Solving AX = B: If A is non-singular, X = A⁻¹B. This gives a unique solution. 'Invert and multiply.'
解 AX = B： 若 A 非奇异，则 X = A⁻¹B，给出唯一解。“求逆再乘”。
Consistency of system: m may have no solution or infinitely many solutions. Check if equations are contradictory.
方程组相容性： 若 det(A)=0，可能无解或无穷多解。检查等式是否存在矛盾。
Adjugate (adjoint): The transpose of the cofactor matrix. Useful for inverses of larger matrices (though not always required in AS).
伴随矩阵： 余子式矩阵的转置，用于求逆（AS 可能不深入）。
8. Polar Coordinates: Basics and Graphing | 极坐标：基础与图像
Polar coordinates (r, θ): r = distance from pole (origin), θ = angle from initial line (positive x-axis). 'Polar pairs: distance then direction.'
极坐标 (r, θ)： r 是到极点的距离，θ 是从极轴（正 x 轴）起算的角度。“极坐标对：先距离后方向”。
Simple polar curves: e radius a; θ = α → ray at angle α; r = aθ → Archimedean spiral; r = a(1+cosθ) → cardioid.
简单极坐标曲线： r = a → 半径为 a 的圆；θ = α → 角度 α 的射线；r = aθ → 阿基米德螺线；r = a(1+cosθ) → 心形线。
Symmetry: If equation unchanged when θ → –θ, curve is symmetric about initial line. If unchanged when r → –r, symmetry about pole.
对称性： 若 θ→–θ 时方程不变，则曲线关于极轴对称；若 r→–r 不变，则关于极点对称。
Area in polar coordinates: Area = ½ ∫ r² dθ. 'Half the integral of r-squared d-theta.' Limits from smallest to largest θ of the region.
极坐标中的面积： 面积 = ½ ∫ r² dθ。“r 平方对 θ 积分的一半”。积分限取区域的最小到最大 θ。
9. Vectors: Lines and Planes | 向量：线与面
Vector equation of a line: b, where a is position vector of a point, b is direction vector, t is a scalar parameter. 'Anchor a, direction b.'
直线的向量方程： r = a + t b，a 是定点位置向量，b 是方向向量，t 是标量参数。“定点 a，方向 b”。
Dot product (scalar product): a·b = |a||b|cosθ. Used to find angles and perpendicularity (a·b = 0). 'Dot gives cos.'
点积（标量积）： a·b = |a||b|cosθ。用于求角度和判断垂直（a·b=0）。“点积出余弦”。
Equation of a plane: r·n = d, where n is a normal vector and d is the perpendicular distance from origin times |n|. 'Plane needs a normal.'
平面方程： r·n = d，n 是法向量，d 是原点到平面距离乘以 |n|。“平面需要一个法向量”。
Distance from point to plane: |(a·n – d)| / |n|, where a is the point's position vector. 'Projection onto normal, minus d, divided by |n|.'
点到平面的距离： |(a·n – d)| / |n|，a 为点的位置向量。“向法向投影，减去 d，除以 |n|”。
Angle between planes: Use the acute angle between their normals: cosθ = |n₁·n₂|/(|n₁||n₂|).
平面夹角： 用两个法向之间的锐角：cosθ = |n₁·n₂|/(|n₁||n₂|)。
10. Proof by Induction | 归纳证明
Mathematical induction: A method to prove statements for all positive integers. Think of a row of dominoes: if the first falls, and each knocks the next, all fall.
数学归纳法： 用于证明对所有正整数成立的命题。想象一排多米诺骨牌：第一张倒下，且每张推倒下一张，则全部倒下。
Base case: erify the statement for the smallest value, usually n = 1. Foundation of the domino chain.
基础步骤： 验证命题对最小取值（通常 n=1）成立。是多米诺链条的起点。
Inductive hypothesis: Assume the statement is true for some arbitrary positive integer k. 'Suppose it's true for k.'
归纳假设： 假设命题对某个正整数 k 成立。“假设 k 时成立”。
Inductive step: Prove that if true for n = k, then it must be true for n = k+1. This is the 'knock-on' logic.
归纳步骤： 证明若 n=k 时成立，则 n=k+1 时也成立。这是“传递”逻辑。
Conclusion: : By induction, the statement holds for all n ≥ starting value. Always write the formal closing sentence.
结论： 由归纳法，命题对所有 n ≥ 初始值成立。务必写出规范总结句。
Common applications: Summation formulas, divisibility (e.g. 3ⁿ – 1 divisible by 2), inequalities, matrix powers, recurrence relations.
常见应用： 求和公式、整除性（如 3ⁿ – 1 被 2 整除）、不等式、矩阵幂、递推关系。
Published by TutorHao | Further Mathematics Revision Series | aleveler.com
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