Year 13 CCEA Further Mathematics: Fast-Track Glossary Guide | Year 13 CCEA 进阶数学：词汇术语速记指南

📚 Year 13 CCEA Further Mathematics: Fast-Track Glossary Guide | Year 13 CCEA 进阶数学：词汇术语速记指南

Mastering the precise terminology of CCEA Year 13 Further Mathematics is the bridge between confusion and clarity in exams. This fast‑track guide distils the essential vocabulary from pure maths, mechanics and statistics into concise definitions and memory hooks, so you can revise smarter and recall faster under pressure.

掌握 CCEA Year 13 进阶数学的精确术语，是在考试中化困惑为清晰的桥梁。本速记指南将纯数、力学和统计的核心词汇提炼成简明定义与记忆锚点，帮助你在压力下更聪明地复习、更快速地调用知识。

1. Complex Numbers | 复数

Imaginary unit (i) ows us to solve equations like x² + 1 = 0 and forms the backbone of complex numbers. Think “i am the root of −1”.

虚数单位 (i)：由 i² = −1 定义的数。它使 x² + 1 = 0 这类方程有解，构成复数的基础。联想“我 (i) 是 −1 的平方根”。

Complex number (z): A number of the form z = a + bi, where a, b ∈ ℝ e(z), b is the imaginary part Im(z). Visualise it as a point (a, b) on the Argand diagram.

复数 (z)：形如 z = a + bi 的数，a, b 为实数。a 是实部 Re(z)，b 是虚部 Im(z)。可将其可视化为阿干特图上的点 (a, b)。

Complex conjugate (z̅ or z*): The reflection of z across the real axis: z̅ = a − bi. Its magic: z × z̅ = a² + b² = |z|², a purely real number.

共轭复数 (z̅ 或 z*)：z 关于实轴的反射：z̅ = a − bi。其妙处：z × z̅ = a² + b² = |z|²，得到纯实数。

Modulus (|z|): The distance from the origin: |z| = √a² + b². Always non‑negative; it measures the size of the complex number.

模 (|z|)：到原点的距离：|z| = √a² + b²。始终非负，衡量复数的大小。

Argument (arg z) sured from the positive real axis, usually in (−π, π]. tan θ = b/a, but check the quadrant! Use CAST or sketch.

幅角 (arg z)：从正实轴量起的角度 θ，通常范围 (−π, π]。tan θ = b/a，但务必检查象限！用 CAST 图或画草图。

Polar form: z = r(cos θ + i sin θ) = r cis θ, with r = |z|. De Moivre: (r cis θ)ⁿ = rⁿ cis(nθ)

极坐标式：z = r(cos θ + i sin θ) = r cis θ，其中 r = |z|。棣莫弗定理：(r cis θ)ⁿ = rⁿ cis(nθ)

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2. Matrices | 矩阵

Matrix order: An m × n matrix has m rows and n columns. Pronounced “m by n”. A square matrix has m = n; its main diagonal runs top‑left to bottom‑right.

矩阵的阶：m × n 矩阵有 m 行、n 列，读作“m 乘 n”。方阵满足 m = n；其主对角线从左上角到右下角。

Determinant: A scalar value det(A) or |A| d only for square matrices. For 2×2, det = ad − bc. det = 0 → singular (no inverse). Think “determinant determines invertibility”.

行列式：一个标量值，记为 det(A) 或 |A|，仅对方阵定义。2×2 矩阵行列式为 ad − bc。det = 0 → 奇异矩阵（无可逆）。联想“行列式决定可逆性”。

Inverse matrix (A⁻¹): The matrix such that A A⁻¹ = A⁻¹ A = I. For 2×2, A⁻¹ = (1/det) × adjugate. Only non‑singular matrices have inverses.

逆矩阵 (A⁻¹)：满足 A A⁻¹ = A⁻¹ A = I 的矩阵。2×2 矩阵可用 (1/det) × 伴随矩阵求逆。仅非奇异矩阵有逆。

Transpose (Aᵀ): Flips rows and columns: (Aᵀ)ᵢⱼ = Aⱼᵢ. Useful identity: (AB)ᵀ = Bᵀ Aᵀ. A symmetric matrix satisfies Aᵀ = A.

转置 (Aᵀ)：行列互换：(Aᵀ)ᵢⱼ = Aⱼᵢ。有用恒等式：(AB)ᵀ = Bᵀ Aᵀ。对称矩阵满足 Aᵀ = A。

Eigenvalue (λ) and eigenvector: For a square matrix A, a non‑ λv. λ solves det(A − λI) = 0. Each eigenvalue comes with a direction that stays unchanged.

特征值 (λ) 与特征向量：对于方阵 A，存在非零向量 v 使 Av = λv。λ 由 det(A − λI) = 0 解得。每个特征值对应一个保持方向不变的向量。

Trace = sum of diagonal elements = sum of eigenvalues

迹 = 对角元素之和 = 特征值之和

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3. Vectors | 向量

Vector vs. scalar: A vector has magnitude and direction (e.g. velocity, force); a scalar has only magnitude (e.g. speed, mass). Notation: bold v or underlined.

向量与标量：向量有大小和方向（如速度、力）；标量仅有大小（如速率、质量）。记法：粗体 v 或下划线。

Dot product (scalar product): a · b = |a||b| a scalar. Perpendicular vectors give a · b = 0. In components: a · b = a₁b₁ + a₂b₂ + a₃b₃.

点积（标量积）：a · b = |a||b| cos θ，θ 为两向量夹角，结果是一个标量。垂直向量满足 a · b = 0。分量形式：a₁b₁ + a₂b₂ + a₃b₃。

Cross product (vector product): a × b yields a vector perpendicular to both a and b, magnitude |a||b| sin θ. Direction via right‑hand rule. Only defined in 3D. a × b = − b × a.

叉积（向量积）：a × b 得到同时垂直于 a 和 b 的向量，大小为 |a||b| sin θ，方向用右手定则。仅在三维空间定义。a × b = − b × a。

Unit vector: A vector of length 1. Often denoted ê. Any vector v gives a unit vector v̂ = v/|v|. Standard basis: i, j, k.

单位向量：长度为 1 的向量，常记为 ê。任何向量 v 可得到单位向量 v̂ = v/|v|。标准基：i, j, k。

Vector equation of a line line, d is direction vector, t scalar parameter. Planes use r · n = p or parametric form.

直线的向量方程：r = a + t d，a 为直线上一点，d 为方向向量，t 为标量参数。平面则用 r · n = p 或参数形式。

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4. Hyperbolic Functions | 双曲函数

Definitions: sinh x = (eˣ − e⁻ˣ)/2 (odd, stretches), cosh x = (eˣ + e⁻ˣ)/2 (even, hanging chain shape). tanh x = sinh x / cosh x. The ‘h’ signals a link to a hyperbola, just as trig links to circle.

定义：sinh x = (eˣ − e⁻ˣ)/2（奇函数，拉伸状），cosh x = (eˣ + e⁻ˣ)/2（偶函数，悬链线状）。tanh x = sinh x / cosh x。其中的 ‘h’ 提示与双曲线有关，正如三角函数与圆有关。

Key identity: cosh²x − sinh²x = 1. Compare with cos²x + sin²x = 1; the minus sign is the twist! Use it for integration and solving hyperbolic equations.

关键恒等式：cosh²x − sinh²x = 1。对比 cos²x + sin²x = 1；减号是关键区别！用于积分和求解双曲方程。

Osborn’s rule: To convert a trig identity into a hyperbolic identity, replace cos with cosh, sin with i sinh but then change the sign of any product (or implied product) of two sines. In practice: sin² becomes − sinh².

奥斯本法测：将三角恒等式转为双曲恒等式时，用 cosh 替换 cos，i sinh 替换 sin，但将两个正弦乘积的符号改变。实际操作中，sin² 变成 − sinh²。

Inverse hyperbolic functions: arsinh x = ln(x + √x²+1), arcosh x = ln(x + √x²−1) (x ≥ 1), artanh x = ½ ln((1+x)/(1−x)) (|x|<1). Logarithmic form is exam gold.

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)。对数形式是考试黄金考点。

Derivatives: d/dx sinh x = cosh x, d/dx cosh x = sinh x (no minus sign!)

导数：d/dx sinh x = cosh x，d/dx cosh x = sinh x（没有负号！）

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5. Polar Coordinates | 极坐标

Polar system (r, θ): r is distance from origin (pole), θ is angle from initial line (positive x‑ version: x = r cos θ, y = r sin θ. r² = x² + y², θ = arctan(y/x) with quadrant adjustment.

极坐标系 (r, θ)：r 为到原点（极点）的距离，θ 为从初始线（正 x 轴）量起的角度。转换公式：x = r cos θ, y = r sin θ。r² = x² + y², θ = arctan(y/x) 并结合象限调整。

Polar curves: Sketch by analysing r as a function of θ. Common shapes: cardioid r = a(1+cos θ), limaçon, rose curves r = a sin(nθ). Symmetry tests save time.

极坐标曲线：通过将 r 视作 θ 的函数来画图。常见形状：心形线 r = a(1+cos θ)、蚌线、玫瑰线 r = a sin(nθ)。利用对称性检验可节省时间。

Area in polar coordinates urve r = f(θ) from θ = α to β is A = ∫ ½ [f(θ)]² dθ over [α, β]. The ½ comes from the near‑triangular sector.

极坐标下的面积：曲线 r = f(θ) 在从 θ = α 到 β 之间扫过的面积 A = ∫ ½ [f(θ)]² dθ，积分区间 [α, β]。½ 源于近似的三角形扇形。

Tangents in polar form cos θ, y = r sin θ. Parallel to initial line when dy/dθ = 0; perpendicular when dx/dθ = 0.

极坐标下的切线：求斜率 dy/dx 用 dy/dx = (dy/dθ) / (dx/dθ)，其中 x = r cos θ, y = r sin θ。平行于初始线时 dy/dθ = 0；垂直时 dx/dθ = 0。

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6. Series and Summation | 级数与求和

Sigma notation Σ: Compact way to write sums. ∑_r=1ⁿ uᵣ means the sum of terms u₁ + u₂ + … + uₙ. Versatile for arithmetic, geometric and power series.

求和符号 Σ：简便记法。∑_r=1ⁿ uᵣ 表示项 u₁ + u₂ + … + uₙ 之和。适用于等差、等比和幂级数。

Arithmetic series m Sₙ = n/2 [2a + (n−1)d] or n/2 (first + last). The common difference d is constant. Check “same difference” as a quick test.

等差数列：和 Sₙ = n/2 [2a + (n−1)d] 或 n/2 (首项+末项)。公差 d 为常数。检验“差相同”即可快速判定。

Geometric series: Sum to n terms Sₙ = a(1−rⁿ)/(1−r) for r≠1. Infinite sum S∞ = a/(1−r) if |r|<1 (convergent). ‘r’ is the common ratio.

等比数列：前 n 项和 Sₙ = a(1−rⁿ)/(1−r), r≠1。无穷和 S∞ = a/(1−r) 当 |r|<1（收敛）。‘r’ 为公比。

Maclaurin series: A Taylor series about x = 0: f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + … + f⁽ⁿ⁾(0)xⁿ/n! + … Used for approximations and standard expansions like eˣ, sin x, ln(1+x).

麦克劳林级数：在 x=0 处的泰勒展开：f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + … + f⁽ⁿ⁾(0)xⁿ/n! + … 用于近似与标准展开，如 eˣ, sin x, ln(1+x)。

eˣ = 1 + x + x²/2! + x³/3! + …; sin x = x − x³/3! + x⁵/5! − …

eˣ = 1 + x + x²/2! + x³/3! + …; sin x = x − x³/3! + x⁵/5! − …

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7. Differential Equations | 微分方程

Order and degree: Order is the highest derivative; degree is the power of that derivative. First‑order ODEs often solved by separation of variables or integrating factor.

阶与次：阶指最高阶导数；次指该导数的幂。一阶常微分方程常通过分离变量或积分因子求解。

Separation of variables earrange as ∫ (1/g(y)) dy = ∫ f(x) dx. Integrate both sides and apply boundary conditions.

分离变量法：若 dy/dx = f(x)g(y)，重排为 ∫ (1/g(y)) dy = ∫ f(x) dx。两边积分并代入边界条件。

Integrating factor (I.F.): For dy/dx + P(x)y = Q(x), multiply by I.F. = e^{∫ P(x) dx}. Then LHS becomes d/dx (y × I.F.). Integrate directly.

积分因子 (I.F.)：对于 dy/dx + P(x)y = Q(x)，乘以 I.F. = e^{∫ P(x) dx}。此时左侧变为 d/dx (y × I.F.)，直接积分即可。

Second‑order linear ODEs with constant coefficients xiliary equation am² + bm + c = 0. Complementary function: if roots real distinct, y = A e^m₁x + B e^m₂x; if repeated, y = (A+Bx)e^mx; if complex, y = e^αx(A cos βx + B sin βx). Particular integral guessed from f(x).

常系数二阶线性常微分方程：a d²y/dx² + b dy/dx + c y = f(x)。解辅助方程 am² + bm + c = 0。补函数：若实根相异，y = A e^m₁x + B e^m₂x；重根时 y = (A+Bx)e^mx；共轭复根时 y = e^αx(A cos βx + B sin βx)。特解由 f(x) 的形式猜测。

Simple harmonic motion connection + ω²x = 0 gives x = C cos(ωt + φ). Model appears in mechanics with springs and pendulums.

简谐运动联结：d²x/dt² + ω²x = 0 的解为 x = C cos(ωt + φ)。该模型出现在力学中的弹簧和单摆。

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8. Further Calculus | 进阶微积分

Implicit differentiation: When y is not isolated, differentiate both sides w.r.t. x, treating y as a function of x. Multiply by dy/dx whenever you differentiate a y‑term. Gather dy/dx terms and factor.

隐函数微分：当 y 未显式表达时，对方程两边关于 x 求导，将 y 视作 x 的函数。每次对含 y 的项求导时乘上 dy/dx。然后合并含 dy/dx 的项并提取公因子。

Parametric differentiation , then dy/dx = (dy/dt) / (dx/dt). For second derivative, d²y/dx² = d/dt (dy/dx) / (dx/dt).

参数方程微分：已知 x = f(t), y = g(t)，则 dy/dx = (dy/dt) / (dx/dt)。二阶导数为 d²y/dx² = d/dt (dy/dx) / (dx/dt)。

Integration by substitution: Powerful with “reverse chain rule”. Choose u = g(x), compute du

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