📚 Further Maths Vocabulary Quick-Memorization Guide | 进阶数学词汇术语速记指南
Mastering the terminology of CAIE A-Level Further Mathematics is crucial for understanding questions and communicating solutions precisely. This guide presents key terms from the Year 13 syllabus, complete with quick memory hooks to help you recall definitions, theorems, and methods efficiently. Each section pairs essential vocabulary with simple mental snapshots, ensuring you can translate exam wording into clear mathematical steps.
掌握 CAIE 进阶数学的术语对于理解题意和准确表达解答至关重要。本指南整理了 Year 13 课程的核心词汇，并附上快速记忆诀窍，帮助你高效回忆定义、定理和方法。每个小节都将关键术语与简洁的心理图景配对，让你能够把考题表述精准转化为数学步骤。
1. Complex Numbers | 复数核心术语
Complex numbers extend the real number system with the imaginary unit i, defined by i² = -1. A complex number z = a + bi has a real part Re(z) = a and an imaginary part Im(z) = b. The complex conjugate, denoted z* or z̅, flips the sign of the imaginary part: a – bi. Think of conjugation as a mirror reflection across the real axis — it merely changes the sign of the imaginary component.
复数通过虚数单位 i（定义 i² = -1）扩充了实数系统。复数 z = a + bi 的实部记作 Re(z)=a，虚部 Im(z)=b。共轭复数 z* 或 z̅ 改变虚部符号为 a – bi。把共轭想象成关于实轴的镜像反射 —— 只改变虚部的符号。
The modulus |z| = √(a² + b²) gives the distance from the origin in the complex plane. Picture it as the length of the vector from (0,0) to (a,b). The argument arg(z) = θ is the angle with the positive real axis, measured anticlockwise. The principal argument lies in the interval (-π, π]. Remember “principal” restricts the angle to one standard turn.
模长 |z| = √(a² + b²) 表示复平面中到原点的距离。可想象为从 (0,0) 到 (a,b) 的向量长度。辐角 arg(z)=θ 是与正实轴的夹角，通常逆时针测量。主辐角 限定在 (-π, π] 内。记忆技巧：”principal” 把角度限制在一个标准圈内。
z = r(cosθ + i sinθ) = reiθ
Polar form and exponential form are linked by Euler’s formula eiθ = cosθ + i sinθ. De Moivre’s theorem states (cosθ + i sinθ)n = cos(nθ) + i sin(nθ). To remember the difference: polar form is the trigonometric dress, exponential form is the compact Euler dress.
极坐标形式与指数形式由欧拉公式 eiθ=cosθ+i sinθ 连接。棣莫弗定理指出 (cosθ+i sinθ)n = cos(nθ)+i sin(nθ)。记忆窍门：极坐标形式是三角函数”外衣”，指数形式是简洁的欧拉”外衣”。
2. Matrices and Linear Transformations | 矩阵与线性变换
A matrix is a rectangular array of elements; its order is given as rows × columns. An identity matrix I acts like the number 1 — multiplying by I leaves a matrix unchanged. A square matrix has the same number of rows and columns. Its determinant det(A) is a scalar that indicates whether a matrix is singular (det=0, no inverse) or non-singular (det≠0, invertible). Think of the determinant as a “volume scaling factor” of the corresponding linear transformation.
矩阵是一个由元素组成的矩形阵列；其阶数表示为 行数 × 列数。单位矩阵 I 的作用类似于数字 1 —— 乘以 I 使矩阵不变。方阵 的行数与列数相等。其行列式 det(A) 是一个标量，用以判断矩阵是奇异的 (det=0，不可逆) 还是非奇异的 (det≠0，可逆)。把行列式联想为对应线性变换的”体积缩放因子”。
A matrix transformation maps points in a vector space. Standard transformations include reflections (mirror across a line), rotations (spinning about the origin) and shears (slanted push keeping one axis fixed). Pay attention to the key invariant: rotations preserve area and orientation, reflections preserve area but flip orientation, shears preserve area in some cases.
矩阵变换将向量空间中的点进行映射。常见变换包括反射（关于某直线镜像）、旋转（绕原点转动）和剪切（保持一轴不变地倾斜推挤）。牢记不变特征：旋转保面积和定向，反射保面积但反转定向，剪切在某些条件下保面积。
The inverse A⁻¹ satisfies AA⁻¹ = I. To find eigenvalues λ, solve det(A – λI) = 0. An eigenvalue’s corresponding eigenvector v satisfies Av = λv; geometrically, the vector stays on its own span after transformation. The mnemonic: “Eigen is own” — the vector remains its own direction.
逆矩阵 A⁻¹ 满足 AA⁻¹ = I。要求特征值 λ，解 det(A – λI)=0。特征向量 v 满足 Av = λv；几何意义是向量在变换后仍停留在自身张成的直线上。记忆法：”Eigen 就是自己的”，向量方向不变。
3. Hyperbolic Functions | 双曲函数速记
Hyperbolic functions are defined via exponentials: sinh x = (ex – e-x)/2 and cosh x = (ex + e-x)/2. Pronounce “sinh” as “shine” and “cosh” to rhyme with “posh”. The key identity cosh²x – sinh²x = 1 parallels the trigonometric identity but with a minus sign. Osborn’s rule says: replace a product of two sines in a trig identity by -sinh×sinh while changing cos to cosh, to obtain the hyperbolic analogue.
双曲函数由指数定义：sinh x = (ex – e-x)/2，cosh x = (ex + e-x)/2。把 “sinh” 读作 “shine”，”cosh” 与 “posh” 押韵。核心恒等式 cosh²x – sinh²x = 1 与三角恒等式类似，但符号为负。Osborn 规则指出：将三角恒等式中两个正弦的乘积替换为 -sinh×sinh，并将 cos 改为 cosh，即可得到双曲版本。
Inverse hyperbolic functions are written as arsinh, arcosh, artanh. Their logarithmic forms are handy: arsinh x = ln(x + √(x²+1)). To memorise, note that artanh x has domain |x| < 1, just like the range of tanh x. The derivatives d/dx(sinh x) = cosh x and d/dx(cosh x) = sinh x echo the trigonometric pattern but without extra signs.
反双曲函数记作 arsinh, arcosh, artanh。它们的对数形式很实用：arsinh x = ln(x + √(x²+1))。记忆时注意 artanh x 的定义域为 |x| < 1，与 tanh x 的值域一致。导数 d/dx(sinh x)=cosh x，d/dx(cosh x)=sinh x，与三角函数模式相同且无多余负号。
4. Differential Equations | 微分方程分类与术语
A differential equation involves an unknown function and its derivatives. Its order is the highest derivative present; its degree is the power of that highest derivative after the equation has been rationalised. For example, d²y/dx² + (dy/dx)³ = x is second order, first degree. The general solution contains arbitrary constants (equal to the order), while a particular solution satisfies given boundary or initial conditions.
微分方程包含未知函数及其导数。阶 (order) 是出现的最高阶导数；次 (degree) 是方程有理化后该最高阶导数的幂次。例如 d²y/dx² + (dy/dx)³ = x 是二阶一次。通解 包含与阶数相同数量的任意常数，特解 满足给定边界或初始条件。
The auxiliary equation for a homogeneous second-order linear ODE with constant coefficients is obtained by assuming y = emx. Its roots determine the form of the complementary function. For a non-homogeneous equation, the particular integral is found by guessing a form based on the RHS function (polynomial, exponential, trigonometric). Remember the structure: general solution = complementary function + particular integral.
常系数二阶线性齐次常微分方程的辅助方程通过假设 y = emx 得到。其根决定补函数的形式。对于非齐次方程，特积分 通过根据右端函数（多项式、指数、三角函数）猜测形式求得。牢记结构：通解 = 补函数 + 特积分。
For first-order linear equations of the form dy/dx + P(x)y = Q(x), the integrating factor is e∫P dx. Multiply through to make the left side an exact derivative. Think: “I-factor” turns the messy left into a single product rule outcome.
对于 dy/dx + P(x)y = Q(x) 形式的一阶线性方程，积分因子 为 e∫P dx。方程两边乘以积分因子，左侧便可写作一个乘积的导数。记忆口诀：”I 因子”把凌乱的左端变成直接积分的结果。
5. Polar Coordinates | 极坐标系统
In the polar system, a point is specified by (r, θ) where r is the distance from the pole (origin) and θ is the angle from the polar axis (positive x-axis). The conversion formulas are x = r cosθ, y = r sinθ and r² = x² + y². When sketching polar curves r = f(θ), pay close attention to symmetry and loops. A cardioid has equation r = a(1 + cosθ) or r = a(1 + sinθ), shaped like a heart; a limaçon can have an inner loop when r changes sign.
在极坐标系中，点由 (r, θ) 给出，r 是到极点（原点）的距离，θ 是从极轴（正 x 轴）量起的角度。转换公式为 x = r cosθ, y = r sinθ，以及 r² = x² + y²。绘制极坐标曲线 r = f(θ) 时，请关注对称性和环圈。心形线（cardioid）方程为 r = a(1+cosθ) 或 r = a(1+sinθ)，形如心脏；蜗线（limaçon）在 r 变号时会出现内环。
The area enclosed by a polar curve between θ = α and θ = β is given by ½ ∫ r² dθ. For arc length, use √(r² + (dr/dθ)²) dθ. Memorise the factor ½ for area — it comes from the circular sector formula Area = ½ r²θ.
极坐标曲线在 θ = α 到 θ = β 之间围成的面积由 ½ ∫ r² dθ 给出。弧长则使用 √(r² + (dr/dθ)²) dθ 积分。记住面积的 ½ 因子 —— 源自圆扇形公式 面积 = ½ r²θ。
6. Proof by Induction | 归纳法步骤术语
Mathematical induction is a domino-style proof technique for statements about natural numbers. It consists of four clearly named stages: the base case (proving the statement for n = 1 or the smallest relevant value), the inductive hypothesis (assuming the statement true for n = k), the inductive step (proving it for n = k + 1 using the hypothesis), and the conclusion (by induction the statement holds for all n). Remember the image of falling dominoes: knock the first one (base case) and ensure each domino knocks the next (inductive step).
数学归纳法是一种针对自然数命题的”多米诺”式证明技巧。它包含四个有明确名称的步骤：基础情形（证明 n = 1 或最小相关值成立）；归纳假设（假设 n = k 时命题真）；归纳递推（利用假设证明 n = k+1 时真）；结论（由归纳原理，命题对所有 n 成立）。记忆多米诺骨牌的画面：推倒第一块（基础情形），并确保每一块倒下都会推倒下一块（归纳递推）。
Careful use of terms: the inductive hypothesis is an assumption, not a proved fact within the proof. The inductive step often involves adding the (k+1)-th term to a sum or factorising an expression. Watch for the “true for n = k ⇒ true for n = k+1” logical structure.
术语使用注意：归纳假设是一种假定，在证明内部并非已证事实。归纳递推往往涉及向求和添加第 (k+1) 项或分解表达式。关注”若 n=k 真 ⇒ n=k+1 真”的逻辑结构。
7. Series and Summation | 级数与求和符号
Sigma notation Σr=1n ur compactly represents a sum. An arithmetic series has a constant difference d; its nth term is a + (n-1)d. A geometric series has constant ratio r; its nth term is arn-1. The sum to infinity of a geometric series converges when |r| < 1, giving S∞ = a/(1 – r). Remember “geometric ratio” as the key to infinite sum convergence.
求和符号 Σr=1n ur 简洁表示累加。等差数列 具有公差 d；第 n 项为 a + (n-1)d。等比数列 具有公比 r；第 n 项为 arn-1。当 |r| < 1 时，等比级数的无穷和收敛为 S∞ = a/(1-r)。记忆法：”geometric ratio” 是通向无穷和收敛的关键。
The method of differences simplifies a series by writing the general term as f(r) – f(r+1), so that intermediate terms cancel. This requires expressing the term in partial fractions or an equivalent telescoping form. Also, the Maclaurin series expands a function as f(x) = Σ f(n)(0)/n! xn, which is just a Taylor series about 0.
差分法 通过将一般项写为 f(r) – f(r+1) 的形式来化简级数，中间项相继抵消。这需要将一般项写成部分分式或等价的可裂项形式。此外，麦克劳林级数将函数展开为 f(x) = Σ f(n)(0)/n! xn，本质上是关于 0 点的泰勒级数。
8. Vectors in 3D | 空间向量
A vector in three dimensions is often written as a column with components i, j, k. The position vector of a point A is OA. The vector equation of a line is r = a + λb, where a is a point on the line and b is a direction vector. For a plane, the scalar product form r·n = p uses the normal vector n.
三维向量通常写作列向量，含有 i, j, k 分量。点 A 的位置向量是 OA。直线的向量方程为 r = a + λb，其中 a 是直线上一点，b 是方向向量。平面的点法式 r·n = p 使用法向量 n。
The dot (scalar) product a·b = |a||b| cosθ gives a scalar; it is zero when vectors are perpendicular. The cross (vector) product a×b produces a vector perpendicular to both, with magnitude |a×b| = |a||b| sinθ. Use the right-hand rule to determine direction. Mnemonic: “dot gives a number, cross gives a vector.”
点积（数量积）a·b = |a||b| cosθ 结果为一个标量；当向量垂直时点积为零。叉积（向量积）a×b 生成一个垂直于两者的向量，大小为 |a×b| = |a||b| sinθ。方向用右手定则判断。记忆口诀：”点积得数，叉积得向量”。
The scalar triple product a·(b×c) gives the volume of the parallelepiped. Coplanarity means three vectors lie in the same plane; their scalar triple product is zero. Shortest distance from a point to a line or between skew lines uses cross product and dot product.
标量三重积 a·(b×c) 给出平行六面体的体积。共面 意味着三个向量在同一平面内，其标量三重积为零。点到直线或两异面直线间的最短距离公式用到了叉积与点积。
9. Further Calculus Techniques | 进阶微积分技巧
Reduction formulae are recurrence relations that express an
Published by TutorHao | Year 13 进阶数学 Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导