📚 Year 13 AQA Further Mathematics: Vocabulary and Terminology Quick Memorization Guide | AQA 进阶数学 Year 13 词汇术语速记指南
Mastering AQA Further Mathematics at Year 13 demands not only problem-solving prowess but also a precise understanding of crucial terminology. This guide compiles essential vocabulary from Pure, Mechanics, and Statistics strands, offering clear definitions and memory aids to boost your exam confidence.
掌握 Year 13 AQA 进阶数学不仅需要解题能力，更需要精确理解关键术语。本指南汇编了纯数、力学和统计模块的核心词汇，提供清晰定义与记忆诀窍，助力提升考试信心。
1. Complex Numbers Terminology | 复数术语
Complex Number (z) A number of the form z = a + bi, where a, b ∈ ℝ and i² = −1. Here a is the real part Re(z) and b is the imaginary part Im(z).
复数 (z) 形如 z = a + bi 的数，其中 a, b ∈ ℝ 且 i² = −1。a 称为实部 Re(z)，b 称为虚部 Im(z)。
Complex Conjugate The conjugate of z = a + bi is z* = a − bi. Geometrically, it is the reflection of z across the real axis. Product z z* = a² + b².
共轭复数 z = a + bi 的共轭为 z* = a − bi，几何上关于实轴对称。满足 z z* = a² + b²。
Modulus The distance from the origin to the point representing z, given by |z| = √(a² + b²). It is always non-negative.
模 从原点到表示 z 的点的距离，由 |z| = √(a² + b²) 给出，始终非负。
Argument The angle θ that the line joining z to the origin makes with the positive real axis, usually chosen in (−π, π] as the principal argument, denoted arg(z).
辐角 连接 z 与原点的直线与正实轴的夹角 θ，通常取主值区间 (−π, π]，记为 arg(z)。
Polar (Modulus-Argument) Form z = r(cosθ + i sinθ) where r = |z| and θ = arg(z). This representation simplifies multiplication and division.
极坐标形式（模-辐角形式） z = r(cosθ + i sinθ)，其中 r = |z|，θ = arg(z)。此形式简化乘除法。
Euler’s Relation e^(iθ) = cosθ + i sinθ. It connects exponential and trigonometric functions, giving the compact form z = r e^(iθ).
欧拉公式 e^(iθ) = cosθ + i sinθ，将指数函数与三角函数联系起来，得到紧凑形式 z = r e^(iθ)。
De Moivre’s Theorem For any integer n, (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ). Used to find powers and roots of complex numbers.
棣莫弗定理 对任意整数 n，(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ)，用于求复数的幂与根。
Roots of Unity The n solutions to zⁿ = 1, given by e^(2πik/n) for k = 0, 1, …, n−1. They are equally spaced on the unit circle.
单位根 方程 zⁿ = 1 的 n 个解，表示为 e^(2πik/n)（k = 0, 1, …, n−1），在单位圆上均匀分布。
2. Matrices and Linear Transformations | 矩阵与线性变换
Matrix A rectangular array of entries with rows and columns. The order is rows × columns. A column vector is an n×1 matrix.
矩阵 由行列排列的矩形数表，阶数为 行数 × 列数。列向量是 n×1 矩阵。
Identity Matrix I A square matrix with 1s on the leading diagonal and 0s elsewhere; AI = IA = A for compatible matrices.
单位矩阵 I 主对角线为 1、其余为 0 的方阵，满足 AI = IA = A。
Determinant det(A) A scalar associated with a square matrix. For 2×2, det(A) = ad − bc. If det(A)=0, the matrix is singular (non-invertible); otherwise non-singular.
行列式 det(A) 方阵对应的标量。对 2×2 矩阵，det(A) = ad − bc。若 det(A)=0，矩阵奇异（不可逆）；否则非奇异。
Inverse A⁻¹ The matrix such that A⁻¹A = AA⁻¹ = I. Exists only when det(A) ≠ 0. For 2×2, A⁻¹ = (1/det) × [[d, -b], [-c, a]].
逆矩阵 A⁻¹ 满足 A⁻¹A = AA⁻¹ = I 的矩阵，仅当 det(A) ≠ 0 时存在。2×2 矩阵逆公式为 (1/det) × [[d, -b], [-c, a]]。
Transpose Aᵀ Formed by swapping rows and columns; (Aᵀ)ᵢⱼ = Aⱼᵢ. Useful for orthogonal matrices where Aᵀ = A⁻¹.
转置矩阵 Aᵀ 行列互换得到，(Aᵀ)ᵢⱼ = Aⱼᵢ。正交矩阵满足 Aᵀ = A⁻¹。
Eigenvalues and Eigenvectors For a square matrix A, a non-zero vector v satisfying A v = λ v is an eigenvector with eigenvalue λ. Solve det(A − λI) = 0, the characteristic equation, to find λ.
特征值与特征向量 对方阵 A，非零向量 v 满足 A v = λ v，则 λ 为特征值，v 为特征向量。通过解特征方程 det(A − λI) = 0 求 λ。
Invariant Lines A line mapped to itself under a linear transformation; points on it may move along the line. Invariant lines often correspond to eigenvectors.
不变线 在线性变换下映射到自身的直线，线上点可沿该线移动。不变线常对应特征向量方向。
Linear Transformation A mapping from vectors to vectors that preserves addition and scalar multiplication. Represented by a matrix M: x ↦ Mx. Examples: rotations, reflections, shears, enlargements.
线性变换 保持向量加法和标量乘法的映射，可用矩阵 M 表示：x ↦ Mx。例如旋转、反射、剪切、放大。
3. Hyperbolic Functions | 双曲函数
Hyperbolic Sine sinh x = (eˣ − e⁻ˣ)/2. It is an odd function, unbounded, with derivative cosh x.
双曲正弦 sinh x = (eˣ − e⁻ˣ)/2，奇函数，无界，导数为 cosh x。
Hyperbolic Cosine cosh x = (eˣ + e⁻ˣ)/2. It is an even function, always ≥ 1, with minimum at x=0 where cosh 0 = 1. Derivative is sinh x.
双曲余弦 cosh x = (eˣ + e⁻ˣ)/2，偶函数，恒 ≥ 1，在 x=0 取最小值 cosh 0 = 1，导数为 sinh x。
Hyperbolic Tangent tanh x = sinh x / cosh x = (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ). Range (−1, 1); odd function; horizontal asymptotes as x → ±∞.
双曲正切 tanh x = sinh x / cosh x = (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ)，值域 (−1, 1)，奇函数，有水平渐近线。
Osborn’s Rule To convert a trig identity into a hyperbolic identity, replace cos with cosh and sin with i sinh, then change the sign of any product of two sines. Equivalently, cos²θ + sin²θ = 1 becomes cosh²x − sinh²x = 1.
奥斯本法则 将三角恒等式转换为双曲恒等式：用 cosh 代替 cos，i sinh 代替 sin，然后改变两个正弦乘积的符号。如 cos²θ + sin²θ = 1 变为 cosh²x − sinh²x = 1。
Reciprocal Hyperbolic Functions sech x = 1/cosh x, cosech x = 1/sinh x, coth x = 1/tanh x. Their domains, ranges and graphs mirror trig reciprocals.
倒数双曲函数 sech x = 1/cosh x，cosech x = 1/sinh x，coth x = 1/tanh x。定义域、值域和图像与三角倒数函数类似。
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. They differentiate to standard forms.
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）。它们的导数有标准公式。
4. Polar Coordinates | 极坐标
Pole and Polar Axis The origin O (pole) and a fixed horizontal ray (polar axis). A point P is represented by (r, θ) where r ≥ 0 is the distance from O, and θ is the angle from the axis.
极点与极轴 原点 O（极点）和固定的水平射线（极轴）。点 P 用 (r, θ) 表示，r ≥ 0 为到 O 的距离，θ 为与极轴的夹角。
Polar Equation A curve defined by r = f(θ). Many symmetric curves (cardioids, limacons, roses) are best expressed this way. Using negative r interprets the point in the opposite direction.
极坐标方程 由 r = f(θ) 定义的曲线。许多对称曲线（心形线、蜗线、玫瑰线）最适合用此表示。负 r 对应反方向上的点。
Area of a Sector The area bounded by the curve r = f(θ) and the rays θ = α, θ = β is A = ½ ∫_α^β r² dθ. Remember to square r first.
扇形面积 曲线 r = f(θ) 与射线 θ = α、θ = β 围成的面积为 A = ½ ∫_α^β r² dθ。务必先平方 r。
Tangent to a Polar Curve
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