📚 Year 12 Cambridge Further Mathematics: Vocabulary and Terminology Quick-Memorisation Guide | 剑桥Year 12进阶数学：词汇术语速记指南
Mastering the language of Further Mathematics is the first step to excelling in your exams. This guide pairs every essential term with a memorable explanation, helping you build confidence and fluency in both English and Chinese. Whether you are decoding complex numbers or navigating proof by induction, these bite-sized definitions and memory aids will keep you on track.
掌握进阶数学的语言是在考试中取得优异成绩的第一步。本指南为每个核心术语提供朗朗上口的解释和记忆方法，帮助你用中英双语建立信心和流利度。无论是解析复数还是探索归纳法证明，这些简短的定义和助记技巧都能助你一臂之力。
1. Complex Numbers | 复数
A complex number is written as z = x + iy, where x and y are real numbers and i² = −1. The real part is x, the imaginary part is y.
复数写作 z = x + iy，其中 x、y 为实数，且 i² = −1。x 称为实部，y 称为虚部。
The complex conjugate of z = x + iy is z* = x − iy. It reflects the number across the real axis on an Argand diagram.
共轭复数 z* = x − iy 由原复数关于实轴反射得到。记忆技巧：共轭就是虚部变号。
The modulus |z| = √(x² + y²) gives the distance from the origin. The argument arg(z) is the angle θ made with the positive real axis, usually in (−π, π].
模长 |z| = √(x² + y²) 是点到原点的距离，辐角 arg(z) 是从正实轴逆时针测量的角。联想记忆：模长像向量的长度，辐角像方向角。
Polar form: z = r(cosθ + i sinθ) = r eiθ. This is ideal for multiplication and division.
极坐标形式：z = r(cosθ + i sinθ) = r eiθ，乘法、除法时特别方便。记住欧拉公式 eiθ = cosθ + i sinθ。
2. Matrices | 矩阵
A matrix of order m × n has m rows and n columns. A square matrix has m = n.
m × n 矩阵有 m 行 n 列。方阵满足 m = n。记忆：行在前，列在后，“横行竖列”。
The identity matrix I is a square matrix with 1 on the leading diagonal and 0 elsewhere. It acts like the number 1 for multiplication.
单位矩阵 I 是主对角线为 1 其余为 0 的方阵，乘法中性元。好比普通乘法中的 1。
The inverse of matrix A, denoted A⁻¹, satisfies AA⁻¹ = A⁻¹A = I. Only non-singular matrices (det A ≠ 0) have an inverse.
逆矩阵 A⁻¹ 满足 AA⁻¹ = I，仅当行列式非零（非奇异）时存在。助记：非奇异 → 可逆。
The determinant of a 2×2 matrix [a b; c d] is ad − bc. The transpose AT swaps rows and columns.
2×2 矩阵的行列式为 ad − bc。转置 AT 将行变为列。
Transformation matrices can represent reflections, rotations, enlargements and shears in the plane.
变换矩阵可以表示平面内的反射、旋转、放大与剪切。记忆：反射行列式 −1，旋转行列式 1，面积缩放因子为 |det|。
3. Roots of Polynomials | 多项式的根
For a cubic equation ax³ + bx² + cx + d = 0 with roots α, β, γ, the sum of roots α+β+γ = −b/a, the sum of pairwise products αβ+βγ+γα = c/a, and the product αβγ = −d/a.
对于三次方程 ax³ + bx² + cx + d = 0，三根之和为 −b/a，两两积之和为 c/a，三根之积为 −d/a。口诀：和负 b，两两积正 c，积负 d（注意符号交替）。
These relationships extend to quartics with alternating signs. They are derived by comparing coefficients in the factorised form.
这些韦达定理关系可推广到四次方程，符号交替。通过展开因式乘积并对比系数得来。
Symmetric functions of roots, like α²+β²+γ², can be expressed using sum and product of roots without solving individually.
根的对称函数（如 α²+β²+γ²）可以用和与积表示，无需分别求出根。常用恒等式：α²+β²+γ² = (α+β+γ)² − 2(αβ+βγ+γα)。
4. Hyperbolic Functions | 双曲函数
The hyperbolic sine and cosine are defined by sinh x = (ex − e−x)/2 and cosh x = (ex + e−x)/2. They resemble ordinary trigonometric functions but lack periodicity.
双曲正弦 sinh x = (ex − e−x)/2，双曲余弦 cosh x = (ex + e−x)/2。它们类似三角函数但没有周期性。记忆：sinh 是奇函数，cosh 是偶函数。
The fundamental identity is cosh²x − sinh²x = 1. Compare with cos²x + sin²x = 1.
基本恒等式 cosh²x − sinh²x = 1，与三角函数符号相反。记忆：双曲函数平方差为 1。
Osborne’s rule helps convert trigonometric identities into hyperbolic identities by changing the sign of any term containing a product of two sines.
奥斯本规则：将三角恒等式中的 cos 换成 cosh，sin 换成 i sinh，但每出现两个 sinh 的乘积就要变号。简记：看到两个 sin 乘积加一个负号。
tanh x = sinh x / cosh x, and its range is (−1, 1), unlike tan x which ranges over all real numbers.
tanh x = sinh x / cosh x，值域为 (−1, 1)，而 tan x 的值域为全体实数。
5. Polar Coordinates | 极坐标
A point in polar coordinates is given by (r, θ), where r is the distance from the pole O and θ is the angle from the initial line.
极坐标下点表示为 (r, θ)，r 为极径，θ 为极角。极点为原点，极轴为正 x 轴。
The area enclosed by a polar curve r = f(θ) from θ=α to θ=β is ½ ∫ r² dθ.
极曲线 r = f(θ) 在角度 [α, β] 上围成的面积为 ½ ∫ r² dθ。记忆：扇形面积微元 ½ r² dθ 累加。
Common curves include the cardioid r = a(1+cosθ), the rose r = a cos(nθ), and the circle r = 2a cosθ.
常见极曲线：心形线 r = a(1+cosθ)，玫瑰线 r = a cos(nθ)（n 奇数时有 n 瓣），圆 r = 2a cosθ。心形线如其名。
Tangents parallel or perpendicular to the initial line are found by considering dy/dθ and dx/dθ where x = r cosθ, y = r sinθ.
切线平行或垂直于极轴时，通过参数方程 x = r cosθ, y = r sinθ 求导，令 dy/dθ = 0 或 dx/dθ = 0。
6. Differential Equations | 微分方程
A first-order separable differential equation can be written as dy/dx = g(x)h(y). The method: separate variables and integrate both sides.
一阶可分离变量微分方程形如 dy/dx = g(x)h(y)，解法：分离变量并积分。
For a linear first-order equation dy/dx + P(x)y = Q(x), the integrating factor is e∫P dx. Multiply through to make the left side an exact derivative.
线性一阶方程 dy/dx + P(x)y = Q(x) 的积分因子为 e∫P dx。乘上积分因子后左边变成乘积的导数，方便积分。
The general solution contains an arbitrary constant; applying boundary conditions gives a particular solution.
通解含任意常数，代入定解条件得到特解。记忆：条件个数应与方程阶数一致。
Second-order homogeneous equations with constant coefficients: ay” + by’ + cy = 0. Use the auxiliary equation am² + bm + c = 0 to find complementary function.
常系数二阶齐次方程 ay” + by’ + cy = 0，通过特征方程 am² + bm + c = 0 求余函数。实根对应指数解，复根对应振荡解。
7. Further Vectors | 进阶向量
The scalar (dot) product a · b = |a||b| cosθ is used to find angles and test perpendicularity. The vector (cross) product a × b yields a vector perpendicular to both a and b, with magnitude |a||b| sinθ.
数量积（点积）a · b = |a||b| cosθ，用于求夹角和判断垂直。向量积（叉积）a × b 得到垂直于 a 和 b 的向量，大小为 |a||b| sinθ。右手定则判方向。
The equation of a plane can be written in vector form: r · n = d, where n is a normal vector and d = a · n for a point a on the plane.
平面方程的向量形式：r · n = d，n 为法向量，d 为常数。知道一点与法向量即可写出。
The scalar triple product a · (b × c) gives the volume of the parallelepiped defined by the three vectors. If it is zero, the vectors are coplanar.
混合积 a · (b × c) 是三个向量张成平行六面体的体积。若值为零则三向量共面。
Finding the shortest distance from a point to a line or plane can be done using projection formulas or cross product magnitudes.
点到直线或平面的最短距离可利用投影公式或叉积大小求得。点到平面的距离 = |(a − p)·n|/|n|。
8. Proof | 证明
An implication “if P then Q” is written P ⇒ Q. The converse is Q ⇒ P, not automatically true. The contrapositive ¬Q ⇒ ¬P is logically equivalent.
蕴涵 “若 P 则 Q” 记作 P ⇒ Q。逆命题 Q ⇒ P 不一定成立。逆否命题 ¬Q ⇒ ¬P 与原命题等价，常用于证明。
A statement can be proved directly by logical steps from the hypothesis to the conclusion, or by contradiction by assuming the negation and deriving an impossibility.
直接证明从假设出发经由逻辑步骤得出结论。反证法假设结论不成立，导出矛盾，从而原命题成立。
Proof by induction: show a statement P(n) holds for n=1, then assume true for n=k and prove for n=k+1. Essential for sequences and divisibility.
数学归纳法：验证 n=1 成立；假设 n=k 成立，证明 n=k+1 也成立。特别适用于数列求和和整除性问题。
A counterexample disproves a universal claim by providing a single case where it fails.
反例用于推翻一个全称命题，只需一个不成立的特例。例如证明“所有质数都是奇数”是错的，可举出 2。
9. Inequalities | 不等式
Quadratic inequalities like ax²+bx+c > 0 are solved by sketching the graph and identifying intervals where it lies above the x-axis. Always consider the sign of ‘a’.
二次不等式如 ax²+bx+c > 0 的解集通过画抛物线草图确定。注意开口方向，找根并取两侧或中间区间。
Rational inequalities of the form f(x)/g(x) ≥ 0 require a sign table or critical values. Never multiply through by g(x) unless you are sure of its sign.
分式不等式如 f(x)/g(x) ≥ 0 要用符号表或临界值分析。切忌随意乘 g(x) 而不讨论符号，以免不等号方向错误。
The triangle inequality |a + b| ≤ |a| + |b| is fundamental in complex number and vector analysis.
三角不等式 |a + b| ≤ |a| + |b| 在复数和向量中都有应用，记忆：两边之和大于等于第三边。
Using modulus inequalities: |x − a| < d represents an interval (a−d, a+d). This idea extends to Argand diagrams for complex numbers.
模不等式 |x − a| < d 表示以 a 为中心、d 为半径的区间。在复平面上它表示圆形区域。
10. Summation and Series | 求和与级数
The method of differences is used to sum series where terms can be expressed as a difference, like Σ (1/(r(r+1))) = 1 − 1/(n+1). Most terms cancel telescopically.
裂项相消法适用于可将通项拆成差分的级数，例如 Σ (1/(r(r+1))) 变为 1 − 1/(n+1)。中间项互相抵消，只留首尾。
Standard series: Σ₁ⁿ r = n(n+1)/2, Σ₁ⁿ r² = n(n+1)(2n+1)/6, Σ₁ⁿ r³ = [n(n+1)/2]². Memorise these for quick summation manipulation.
标准结果：Σ r = n(n+1)/2, Σ r² = n(n+1)(2n+1)/6, Σ r³ = (Σ r)²。记住这些公式能快速化简求和式。
The Maclaurin series expansion of a function f(x) is f(x) = Σ (f⁽ⁿ⁾(0)/n!) xⁿ. It approximates functions as polynomials near x=0.
麦克劳林展开式 f(x) = Σ (f⁽ⁿ⁾(0)/n!) xⁿ 将函数在 x=0 附近展开成多项式。常用展开：eˣ, sin x, cos x, ln(1+x)。
Knowing the ranges of validity for Maclaurin series (e.g., ln(1+x) is valid for −1 < x ≤ 1) is essential for accurate approximations.
注意展开式的收敛区间，如 ln(1+x) 仅在 −1 < x ≤ 1 时有效。忽略收敛域会导致错误的近似值。
11. Further Algebra and Functions | 进阶代数与函数
The modulus function |x| gives the absolute value. Equations like |2x−1| = 3 split into two cases: 2x−1 = 3 or 2x−1 = −3.
绝对值函数 |x| 表示点到零的距离。绝对值方程需分情况讨论：|2x−1| = 3 等价于 2x−1 = ±3。
Functions can be combined by composition (f∘g)(x) = f(g(x)). The order matters; f∘g and g∘f are generally different.
函数复合 (f∘g)(x) = f(g(x))，注意顺序通常是不同的。例：f(x)=x², g(x)=x+1，则 f∘g = (x+1)²，g∘f = x²+1。
The inverse function f⁻¹ reverses the mapping: f⁻¹(f(x)) = x. It exists only if f is one-to-one.
反函数 f⁻¹ 将输出变回输入，要求原函数是单射。图像上 f⁻¹ 是 f 关于 y=x 的反射。
Partial fractions split a single rational expression into simpler fractions, useful for integration and series expansions.
部分分式将复杂分式拆成简单分式之和，便于积分和级数展开。分母的因式决定拆分形式。
12. Further Calculus | 进阶微积分
Differentiation from first principles: f'(x) = lim_{h→0} (f(x+h)−f(x))/h. This formal definition underpins all differentiation rules.
从第一性原理求导：f'(x) = lim_{h→0} (f(x+h)−f(x))/h。这是微分学的根基。
Leibniz’s theorem helps differentiate a product n times using binomial coefficients: (uv)⁽ⁿ⁾ = Σ (nCr) u⁽ⁿ⁻ʳ⁾ v⁽ʳ⁾.
莱布尼茨公式用于求乘积的高阶导数，结构与二项式展开相似。记住系数为组合数。
Volumes of revolution: rotation about the x-axis gives V = π ∫ y² dx. For rotation about the y-axis, use V = π ∫ x² dy.
旋转体体积：绕 x 轴旋转 V = π ∫ y² dx；绕 y 轴用 V = π ∫ x² dy。小心正确替换和限值。
Improper integrals where the limit of integration is infinite or the integrand is unbounded are evaluated using limits.
广义积分涉及无穷限或无界函数，通过极限来求值。判断收敛性至关重要。
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