📚 Year 12 Cambridge Further Maths: Terminology Quick-Memorisation Guide | Year 12 剑桥进阶数学：词汇术语速记指南
Mastering Cambridge Further Mathematics at Year 12 starts with building a solid mental dictionary. This guide pairs every essential term with a short, memorable explanation in English and Chinese, followed by quick memory hooks to help you lock definitions into long-term recall. Use it alongside your past paper practice and revision notes.
攻克 Year 12 剑桥进阶数学的第一步，是建立起扎实的术语词典。本指南为每一个核心术语配上了简短好记的英文和中文解释，并附上速记提示，帮你把定义牢牢锁进长期记忆。配合历年真题和复习笔记使用效果更佳。
1. Complex Numbers Essentials | 复数基础术语
Imaginary unit i mplex number can be written in the form z = x + iy, with real part x and imaginary part y.
虚数单位 i：定义为 i² = −1。任何复数都可写成 z = x + iy 的形式，其中 x 为实部，y 为虚部。
Complex conjugate * = x − iy. The product z z* = x² + y² is always real and gives the square of the modulus.
共轭复数：对于 z = x + iy，其共轭为 z* = x − iy。乘积 z z* = x² + y² 恒为实数，等于模的平方。
Modulus and argument: The modulus |z| = √x² + y² argument arg(z) = θ is the angle measured from the positive real axis, usually in (−π, π].
模与辐角：模 |z| = √x² + y² 表示到原点的距离。辐角 arg(z) = θ 是从正实轴量起的有向角，通常取主值区间 (−π, π]。
2. Argand Diagrams & Polar Form | 阿甘特图与极形式
Argand diagram: A plot where the x-axis represents the real part and the y-axis represents the imaginary part of a complex number.
阿甘特图：一种平面图示，x 轴代表复数的实部，y 轴代表虚部。
Modulus-argument form: z = r(cosθ + i sinθ) where r = |z| and θ = arg(z). Multiplication then follows: multiply moduli, add arguments.
模-辐角形式：z = r(cosθ + i sinθ)，其中 r = |z|，θ = arg(z)。乘法规则变为“模相乘，辐角相加”。
Exponential form: z = reⁱθ, derived from Euler's formula. It dramatically simplifies multiplication, division and powers (de Moivre's theorem).
指数形式：z = reⁱθ，由欧拉公式导出。它能极大化简乘、除及乘方运算（棣莫弗定理）。
3. Matrices & Determinants Crash | 矩阵与行列式速记
Order of a matrix: An m × n matrix has m rows and n columns. A square matrix has m = n.
矩阵的阶：m × n 矩阵具有 m 行 n 列。方阵满足 m = n。
Singular vs non-singular: A square matrix is singular if its determinant is zero, meaning it has no inverse. Non-singular matrices have a non-zero determinant and are invertible.
奇异矩阵 与 非奇异矩阵：方阵若行列式为零，则为奇异矩阵，不可逆。非奇异矩阵的行列式不为零，可逆。
Determinant of a 2×2 = [a b; c d], det(M) = ad − bc. If det(M) = 0 the matrix collapses the plane onto a line.
2×2 行列式：对于 M = [a b; c d]，det(M) = ad − bc。若 det(M) = 0，该矩阵将平面压缩到一条直线上。
4. Linear Transformations Language | 线性变换术语
Transformation matrix: A matrix M maps a point with position vector r to r' = Mr. The origin is always fixed.
变换矩阵：矩阵 M 将位置向量为 r 的点映射到 r' = Mr。原点始终是定点。
Invariant points and lines: A point is invariant if Mr = r. An invariant line is mapped onto itself (points may move along the line).
不变点与不变线：若满足 Mr = r，该点为不变点。不变线被映射到自身直线上（点可沿线滑动）。
Shear and stretch: A shear leaves area unchanged and keeps one line fixed. A stretch (with factor k) scales distances perpendicular to a fixed line by k.
剪切与拉伸：剪切保持面积不变且固定一条直线。拉伸（因子为 k）将垂直于不变线的距离缩放 k 倍。
Rotation about origin: The matrix [[cosθ, −sinθ], [sinθ, cosθ]] rotates points anti-clockwise by angle θ.
绕原点旋转：矩阵 [[cosθ, −sinθ], [sinθ, cosθ]] 将点绕原点逆时针旋转角度 θ。
5. Roots of Polynomials | 多项式根的关系
Sum of roots: For a cubic az³ + bz² + cz + d = 0, the sum of roots α + β + γ = −b/a.
根之和：对于三次方程 az³ + bz² + cz + d = 0，根之和 α + β + γ = −b/a。
Sum of pairwise products ubic, αβ + βγ + γα = c/a. These symmetric sums are fundamental for building new equations.
根的两两乘积之和：对于该三次方程，αβ + βγ + γα = c/a。这些对称和在构造新方程时至关重要。
Product of roots: αβγ = −d/a (note the sign alternates with degree). Useful for checking answers quickly.
根之积：αβγ = −d/a（注意符号随次数交替变化）。常用于快速检验结果。
6. Polar Coordinates | 极坐标核心词
Polar coordinates (r, θ): r is the radial distance from the pole O, θ is the angle measured from the initial line (positive x-axis).
极坐标 (r, θ)：r 是到极点 O 的径向距离，θ 是从极轴（正 x 轴）量起的有向角。
Cardioid ation r = a(1 + cosθ) or r = a(1 − cosθ), shaped like a heart. The loop size depends on a.
心脏线：方程为 r = a(1 + cosθ) 或 r = a(1 − cosθ) 的曲线，形状像一颗心脏。环路大小由 a 决定。
Area enclosed by a polar curve: Area = ½ ∫ r² dθ from α to β. Remember the ½ factor – a common slip.
极曲线围成面积：面积 = ½ ∫ r² dθ，积分限从 α 到 β。务记 ½ 因子，这是常犯的小错。
7. Hyperbolic Functions | 双曲函数术语
Definitions: sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x.
定义：sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x。
Osborn's rule: Convert a trigonometric identity to a hyperbolic one by replacing circular functions with their hyperbolic equivalents, but flip the sign whenever two sinh terms are multiplied.
奥斯本法则：将三角恒等式转化为双曲恒等式时，把圆函数换成对应的双曲函数，但凡出现两个 sinh 相乘的项，符号反转。
Inverse hyperbolic functions: arsinh x = ln(x + √x²+1), arcosh x = ln(x + √x²−1) for x ≥ 1, artanh x = ½ ln((1+x)/(1−x)) for |x|<1.
反双曲函数：arsinh x = ln(x + √x²+1)，arcosh x = ln(x + √x²−1) （x ≥ 1），artanh x = ½ ln((1+x)/(1−x)) （|x|<1）。
8. Series & Summation Techniques | 级数与求和方法
Standard sums: Σr = n(n+1)/2, Σr² = n(n+1)(2n+1)/6, Σr³ = n²(n+1)²/4. Know these by heart.
标准求和公式：Σr = n(n+1)/2，Σr² = n(n+1)(2n+1)/6，Σr³ = n²(n+1)²/4。务必熟记。
Method of differences: Express a term as f(r) − f(r+1) so that intermediate terms cancel when summed, leaving only the start and end. Perfect for fractions like 1/(r(r+1)).
差分法：将通项写成 f(r) − f(r+1) 的形式，使求和时中间项互相抵消，只留下首尾两项。非常适合处理形如 1/(r(r+1)) 的分式。
Summation notation Σ: Always check the lower and upper limits; the number of terms is (upper − lower + 1).
求和符号 Σ：始终核对上下限；项的个数为 (上限 − 下限 + 1)。
9. Further Differentiation | 进阶微分速查
Derivative of inverse trig: d/dx (arcsin x) = 1/√1 − x², d/dx (arctan x) = 1/(1 + x²). Remember the positive sign for arcsin.
反三角函数的导数：d/dx (arcsin x) = 1/√1 − x²，d/dx (arctan x) = 1/(1 + x²)。注意 arcsin 的导数为正。
Derivative of hyperbolic functions /dx (sinh x) = cosh x, d/dx (cosh x) = sinh x (no sign change, unlike circular functions).
双曲函数的导数：d/dx (sinh x) = cosh x，d/dx (cosh x) = sinh x（与三角函数不同，符号不变）。
Implicit differentiation: Differentiate every term with respect to x, treating y as a function and attaching dy/dx whenever you differentiate a y-term.
隐函数微分：每项同时对 x 求导，将 y 视为 x 的函数，每当对含 y 的项求导时补上 dy/dx。
10. Further Integration | 进阶积分必备
Standard integrals to recognise: ∫ 1/√a² − x² dx = arcsin(x/a) + C, ∫ 1/(a² + x²) dx = (1/a) arctan(x/a) + C.
必须识别的标准积分：∫ 1/√a² − x² dx = arcsin(x/a) + C，∫ 1/(a² + x²) dx = (1/a) arctan(x/a) + C。
Integration using partial fractions: Split a rational expression into simpler fractions, each integrating to a logarithm or inverse trig function. Always cover repeated linear factors carefully.
部分分式积分：将有理式拆分成更简单的分式，每一项积分为对数或反三角函数。遇到重根线性因子要格外细致。
Integration by substitution For definite integrals, remember to change the limits according to the substitution. Write 'let u = ...' clearly.
换元积分法：对于定积分，记得按代换式同步变换积分限。步骤中清晰写出“令 u = ...”。
11. Differential Equations | 微分方程术语
General solution: Contains an arbitrary constant (or constants). For a first-order equation, the general solution is a family of curves.
通解：包含任意常数（或多个常数）。对于一阶方程，通解是一族曲线。
Particular solution: Found by substituting given initial conditions into the general solution to determine the constant(s).
特解：将给定的初始条件代入通解，确定常数后得到的解。
Integrating factor = Q(x), the factor is e^∫ P dx. Multiplying by it makes the left side an exact derivative.
积分因子：对于 dy/dx + P(x)y = Q(x)，积分因子为 e^∫ P dx。方程两边同乘该因子后，左边化为全导数。
Separable equation ge to (1/g(y)) dy = f(x) dx and integrate both sides. Always add the constant immediately.
可分离变量方程：若 dy/dx = f(x)g(y)，重排为 (1/g(y)) dy = f(x) dx 后两边积分。积分时立刻加上常数。
12. Proof by Induction & Vectors | 归纳法与向量点睛
Induction basis ement is true for the smallest value (usually n = 1). If this step fails, the proof fails.
归纳奠基：证明命题对最小取值（通常 n = 1）成立。若此步失败，整个证明无效。
Inductive step ove for n = k + 1 using the assumption. Clearly link the algebra to the target expression.
归纳递推：假设 n = k 时成立，利用该假设证明 n = k + 1 时成立。代数推导要清晰指向目标表达式。
Vector equation of a line he line, d is a direction vector, and λ is a real parameter. This works in both 2D and 3D.
直线的向量方程：r = a + λd，其中 a 是线上一个点的位置向量，d 为方向向量，λ 为实数参数。该形式在二维和三维中都适用。
Scalar product for angle: cosθ = (a·b) / (|a||b|). Use this to test perpendicularity (a·b = 0) or to find the angle between two lines.
数量积求夹角：cosθ = (a·b) / (|a||b|)。可用来检验垂直（a·b = 0）或求两直线间的夹角。
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