Pre-U Edexcel Further Mathematics: Vocabulary and Terminology Quick Recall Guide | 进阶数学词汇术语速记指南

📚 Pre-U Edexcel Further Mathematics: Vocabulary and Terminology Quick Recall Guide | 进阶数学词汇术语速记指南

Success in Pre-U Edexcel Further Mathematics depends heavily on precise understanding and confident use of technical vocabulary. This guide distils essential terminology from the syllabus into bite-sized, bilingual explanations to accelerate your revision and support classroom learning. Each section presents key terms with concise definitions in English followed by their Chinese equivalents, helping you internalise both the concepts and the language of advanced mathematics.

在 Pre-U Edexcel 进阶数学中取得成功，离不开对专业术语的精准理解与自如运用。本指南将考纲中的核心词汇提炼为精炼的双语解释，助你加速复习、巩固课堂所学。每一节都以英文简明定义配中文对应表达，帮助你将高数概念与语言一同内化。

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1. Complex Numbers | 复数

Imaginary unit i: defined as i² = −1. All complex numbers can be written as z = x + i y, where x, y are real numbers.

虚数单位 i：满足 i² = −1。所有复数都可写成 z = x + i y，其中 x, y 为实数。

Real part and imaginary part: For z = x + i y, Re(z) = x, Im(z) = y (the real coefficient of i, not i y).

实部与虚部：对于 z = x + i y，Re(z) = x，Im(z) = y（即 i 的实系数，而非 i y）。

Modulus: |z| = √(x² + y²), the distance from the origin in the complex plane.

模：|z| = √(x² + y²)，表示复平面上点到原点的距离。

Argument: arg(z) = θ, the angle measured from the positive real axis, usually taken in (−π, π].

辐角：arg(z) = θ，从正实轴测量的角度，通常取值范围为 (−π, π]。

Complex conjugate: z̄ = x − i y, denoted by a bar over z. The product z z̄ = |z|².

共轭复数：z̄ = x − i y，符号为 z 上加一横。乘积 z z̄ = |z|²。

Polar form and Euler’s formula: z = r (cos θ + i sin θ) = r e^(iθ), where r = |z|, θ = arg(z).

极坐标形式与欧拉公式：z = r (cos θ + i sin θ) = r e^(iθ)，其中 r = |z|，θ = arg(z)。

De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ) for integer n.

棣莫弗定理：(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)，n 为整数。

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2. Matrices | 矩阵

Order of a matrix: An m × n matrix has m rows and n columns.

矩阵的阶：m × n 矩阵有 m 行和 n 列。

Transpose: A^T is obtained by swapping rows and columns; (A^T)ᵢⱼ = Aⱼᵢ.

转置：A^T 通过交换行与列得到；(A^T)ᵢⱼ = Aⱼᵢ。

Symmetric and skew-symmetric: A is symmetric if A^T = A; skew-symmetric if A^T = −A.

对称与反对称：若 A^T = A，则 A 为对称矩阵；若 A^T = −A，则为反对称矩阵。

Determinant: A scalar value computed from a square matrix, denoted det(A) or |A|; it is zero for singular matrices.

行列式：由方阵计算出的标量值，记作 det(A) 或 |A|；对于奇异矩阵，行列式为零。

Singular and non-singular: A square matrix is singular if det(A) = 0, otherwise non-singular. Only non-singular matrices have inverses.

奇异与非奇异：若 det(A) = 0，则方阵奇异；否则非奇异。只有非奇异矩阵才存在逆矩阵。

Inverse matrix: A⁻¹ satisfies A A⁻¹ = A⁻¹ A = I. For a 2×2 matrix, A⁻¹ = (1/det) × adjugate.

逆矩阵：A⁻¹ 满足 A A⁻¹ = A⁻¹ A = I。对于 2×2 矩阵，A⁻¹ = (1/det) × 伴随矩阵。

Orthogonal matrix: A square matrix such that A^T A = I; its inverse equals its transpose.

正交矩阵：满足 A^T A = I 的方阵；其逆等于其转置。

Eigenvalues and eigenvectors: For a square matrix A, if A v = λ v with v ≠ 0, then λ is an eigenvalue and v is the corresponding eigenvector.

特征值与特征向量：对于方阵 A，若 A v = λ v 且 v ≠ 0，则 λ 为特征值，v 为对应的特征向量。

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3. Vectors | 向量

Scalar (dot) product: a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃; result is a scalar.

标量积（点积）：a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃；结果是一个标量。

Vector (cross) product: a × b = |a||b| sin θ n̂; result is a vector perpendicular to both a and b.

向量积（叉积）：a × b = |a||b| sin θ n̂；结果是一个同时垂直于 a 和 b 的向量。

Scalar triple product: a · (b × c) gives the volume of the parallelepiped; it is cyclic: a·(b×c) = b·(c×a) = c·(a×b).

标量三重积：a · (b × c) 给出平行六面体的体积；具有循环性：a·(b×c) = b·(c×a) = c·(a×b)。

Vector triple product: a × (b × c) = (a·c)b − (a·b)c.

向量三重积：a × (b × c) = (a·c)b − (a·b)c。

Unit vector: A vector with magnitude 1; often written as â = a / |a|.

单位向量：长度为 1 的向量；常记作 â = a / |a|。

Equation of a line in 3D: r = a + λ d, where a is a point on the line and d is the direction vector.

三维直线方程：r = a + λ d，其中 a 是直线上一点，d 为方向向量。

Equation of a plane: r · n = a · n (scalar product form), or r = a + λ u + μ v (parametric form).

平面方程：r · n = a · n（数量积形式），或 r = a + λ u + μ v（参数形式）。

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4. Polar Coordinates | 极坐标

Polar coordinates (r, θ): r is the distance from the pole O, θ the angle from the initial line.

极坐标 (r, θ)：r 为到极点 O 的距离，θ 为从极轴测量的角度。

Cartesian conversion: x = r cos θ, y = r sin θ; r = √(x² + y²), θ = arctan(y/x) (adjusted for quadrant).

笛卡尔坐标转换：x = r cos θ, y = r sin θ；r = √(x² + y²), θ = arctan(y/x)（需根据象限调整）。

Polar curve sketching: Use symmetry, key values of θ, and the behaviour of r as θ varies. Common forms: cardioid, limacon, rose curves.

极坐标曲线草图绘制：利用对称性、θ 的关键值以及 r 随 θ 的变化趋势。常见形状：心形线、蜗线、玫瑰线。

Area enclosed by a polar curve: A = ½ ∫_α^β [r(θ)]² dθ.

极坐标曲线围成的面积：A = ½ ∫_α^β [r(θ)]² dθ。

Arc length in polar coordinates: s = ∫_α^β √(r² + (dr/dθ)²) dθ.

极坐标下的弧长：s = ∫_α^β √(r² + (dr/dθ)²) dθ。

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5. Hyperbolic Functions | 双曲函数

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: To convert a trigonometric identity into a hyperbolic identity, replace each trigonometric function with the corresponding hyperbolic function and change the sign of any term containing a product (or implied product) of two sines.

奥斯本法则：将三角恒等式转换为双曲恒等式时，将每个三角函数替换为对应的双曲函数，并改变含有两个正弦乘积（或隐含乘积）的项的符号。

Key identities: cosh² x − sinh² x = 1, sinh(2x) = 2 sinh x cosh x, cosh(2x) = cosh² x + sinh² x.

核心恒等式：cosh² x − sinh² x = 1，sinh(2x) = 2 sinh x cosh x，cosh(2x) = cosh² x + sinh² x。

Derivatives: d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech² x.

导数：d/dx (sinh x) = cosh x，d/dx (cosh x) = sinh x，d/dx (tanh x) = sech² x。

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）。

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6. Series and Sequences | 级数与序列

Arithmetic progression: a, a+d, a+2d, …; nth term uₙ = a + (n−1)d, sum Sₙ = n/2 [2a + (n−1)d].

等差数列：a, a+d, a+2d, …；第 n 项 uₙ = a + (n−1)d，前 n 项和 Sₙ = n/2 [2a + (n−1)d]。

Geometric progression: a, ar, ar², …; nth term uₙ = a rⁿ⁻¹, sum Sₙ = a(1−rⁿ)/(1−r) for r ≠ 1.

等比数列：a, ar, ar², …；第 n 项 uₙ = a rⁿ⁻¹，和 Sₙ = a(1−rⁿ)/(1−r)（r ≠ 1）。

Convergence of a series: An infinite series ∑ uₙ converges if the sequence of partial sums approaches a finite limit.

级数的收敛性：若部分和序列趋近于一个有限极限，则无穷级数 ∑ uₙ 收敛。

Maclaurin series: f(x) = f(0) + f'(0) x + f”(0)/2! x² + f”'(0)/3! x³ + …

麦克劳林级数：f(x) = f(0) + f'(0) x + f”(0)/2! x² + f”'(0)/3! x³ + …

Taylor series: f(x) = f(a) + f'(a)(x−a) + f”(a)/2! (x−a)² + …; Maclaurin is Taylor with a = 0.

泰勒级数：f(x) = f(a) + f'(a)(x−a) + f”(a)/2! (x−a)² + …；麦克劳林级数即 a=0 时的泰勒级数。

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7. Differential Equations | 微分方程

Order and degree: The order is the highest derivative; the degree is the power of the highest derivative when the equation is polynomial in derivatives.

阶与次：阶为最高阶导数的阶数；次为方程关于各阶导数为多项式时，最高阶导数的幂次。

Linear differential equation: An equation of the form aₙ(x) y^(n) + … + a₀(x) y = g(x), where y and its derivatives appear to the first power and are not multiplied together.

线性微分方程：形如 aₙ(x) y^(n) + … + a₀(x) y = g(x) 的方程，其中 y 及其各阶导数均为一次幂，且彼此不相乘。

Homogeneous vs non-homogeneous: A linear ODE is homogeneous if g(x) = 0; otherwise it is non-homogeneous (inhomogeneous).

齐次与非齐次：若 g(x) = 0，则线性常微分方程为齐次的；否则为非齐次的。

General solution and particular solution: The general solution contains arbitrary constants (number equal to the order). A particular solution satisfies given initial/boundary conditions.

通解与特解：通解包含任意常数（个数等于方程阶数）。特解满足给定的初始条件或边界条件。

Separation of variables: For dy/dx = f(x)g(y), rearrange to ∫ (1/g(y)) dy = ∫ f(x) dx.

分离变量法：对于 dy/dx = f(x)g(y)，变形为 ∫ (1/g(y)) dy = ∫ f(x) dx。

Integrating factor (first-order linear): For dy/dx + P(x) y = Q(x), the integrating factor is e^(∫ P dx). Multiply through to integrate.

积分因子（一阶线性）：对于 dy/dx + P(x) y = Q(x)，积分因子为 e^(∫ P dx)。两边同乘后再积分。

Auxiliary equation (second-order linear with constant coefficients): For a y” + b y’ + c y = 0, use the auxiliary equation a m² + b m + c = 0. The form of the general solution depends on the nature of the roots (real distinct, repeated, complex conjugate).

辅助方程（二阶常系数线性）：对于 a y” + b y’ + c y = 0，使用辅助方程 a m² + b m + c = 0。通解形式取决于根的性质（相异实根、重根、共轭复根）。

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8. Roots of Polynomials | 多项式根

Root: A value α such that P(α) = 0; (x − α) is a factor of the polynomial P(x).

根：使 P(α) = 0 的值 α；(x − α) 是多项式 P(x) 的一个因式。

Vieta’s formulas (symmetric sums of roots): For a cubic x³ + px² + qx + r = 0 with roots α, β, γ: Σα = −p, Σαβ = q, αβγ = −r.

韦达定理（根的对称和）：对于三次方程 x³ + px² + qx + r = 0，其根为 α, β, γ，有 Σα = −p，Σαβ = q，αβγ = −r。

Complex conjugate root theorem: If a polynomial has real coefficients, any non-real complex roots occur in conjugate pairs (α = a+ib and ᾱ = a−ib).

共轭复根定理：若多项式系数为实数，则所有非实复根均以共轭对形式出现（α = a+ib 与 ᾱ = a−ib）。

Transformations of roots: To obtain an equation whose roots are a function of the roots of a given equation, use substitution (e.g., y = x², y = 1/x, y = x + k).

根的变换：要得到一个方程，使其根为原方程根的某个函数，可利用代换（例如 y = x², y = 1/x, y = x + k）。

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9. Proof by Induction | 归纳法证明

Base case: Verify the statement holds for the smallest relevant integer, typically n = 1.

基础步骤：验证命题对最小的相关整数（通常为 n = 1）成立。

Inductive hypothesis: Assume the statement is true for some arbitrary integer n = k.

归纳假设：假设命题对某个整数 n = k 成立。

Inductive step: Using the hypothesis, prove the statement for n = k + 1. Then by mathematical induction, the statement holds for all n ≥ base value.

归纳步骤：利用假设，证明命题对 n = k + 1 也成立。从而根据数学归纳法，命题对所有 n ≥ 基础值成立。

Applications: Summation of series, divisibility statements, matrix powers, inequalities, and properties of sequences.

常见应用：级数求和、整除性命题、矩阵的幂、不等式以及序列性质的证明。

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10. Further Calculus | 进一步微积分

Improper integrals: Integrals with infinite limits or integrands with infinite discontinuities. Convergence requires the limit of the integral to exist.

反常积分：积分限为无穷或被积函数有无穷间断点的积分。若积分极限存在，则积分收敛。

Volume of revolution: About the x-axis: V = π ∫ [y(x)]² dx. About the y-axis: V = π ∫ [x(y)]² dy. Parametric and polar forms are also used.

旋转体体积：绕 x 轴：V = π ∫ [y(x)]² dx。绕 y 轴：V = π ∫ [x(y)]² dy。也可使用参数形式和极坐标形式。

Arc length (Cartesian): s = ∫ₐᵝ √(1 + (dy/dx)²) dx, or √((dx/dt)² + (dy/dt)²) dt for parametric curves.

弧长（笛卡尔坐标）：s = ∫ₐᵝ √(1 + (dy/dx)²) dx，参数曲线则用 √((dx/dt)² + (dy/dt)²) dt 积分。

Surface area of revolution: About the x-axis: S = 2π ∫ y √(1 + (dy/dx)²) dx (cartesian) or using ds in parametric/polar.

旋转体表面积：绕 x 轴：S = 2π ∫ y √(1 + (dy/dx)²) dx（笛卡尔），或利用参数/极坐标下的 ds。

Mean value of a function: f_mean = 1/(b−a) ∫ₐᵝ f(x) dx.

函数的平均值：f_mean = 1/(b−a) ∫ₐᵝ f(x) dx。

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