AQA Year 13 Mathematics: Key Terminology Quick Reference | AQA 13年级数学：关键术语速记指南

📚 AQA Year 13 Mathematics: Key Terminology Quick Reference | AQA 13年级数学：关键术语速记指南

Mastering the specialised vocabulary of A-level Mathematics is essential for understanding exam questions and expressing solutions accurately. This guide provides a concise collection of key terms across core, mechanics and statistics topics for the AQA Year 13 specification, with clear definitions in both English and Chinese to support bilingual learners.

掌握A-level数学的专业词汇对于理解考试题目和准确表达解题过程至关重要。本指南为AQA 13年级的核心数学、力学与统计专题提供了一套简明关键术语，配有中英文清晰定义，以支持双语学习者的需求。

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1. Functions and Graphs | 函数与图像

Function – a relation where each input (x-value) is mapped to exactly one output (y-value). Denoted as f(x).

函数 – 每个输入（x值）恰好对应一个输出（y值）的关系。记作f(x)。

Domain – the set of all possible input values (x-values) for which a function is defined.

定义域 – 函数所有可能输入值（x值）的集合，在该集合上函数有定义。

Range – the set of all possible output values (y-values) produced by the function.

值域 – 函数产生的所有可能输出值（y值）的集合。

One-to-one function – a function where each y-value corresponds to exactly one x-value; passes both the vertical and horizontal line tests.

一一函数 – 每个y值恰好对应一个x值的函数；同时满足垂直线与水平线测试。

Inverse function f⁻¹(x) – ; the graph is a reflection of y = f(x) in the line y = x. Only exists if f is one-to-one.

反函数 f⁻¹(x) – 逆转f(x)的作用；其图像是y = f(x)关于直线y = x的反射。仅当f为一一函数时存在。

Modulus function |x| – gives the absolute value, defined as |x| = x for x ≥ 0 and |x| = −x for x < 0.

模函数 |x| – 表示绝对值，定义为|x| = x（x ≥ 0）及|x| = −x（x < 0）。

Transformation – a change to the graph of f(x), such as translation f(x + a), stretch a f(x) or reflection −f(x).

变换 – 对f(x)图像的改变，例如平移f(x + a)、伸缩a f(x)或反射−f(x)。

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2. Differentiation | 微分

Derivative (gradient function) – the rate of change of a function, written as f'(x) or dy/dx.

导数（梯度函数） – 函数的变化率，写作f'(x)或dy/dx。

Differentiation from first principles – using the limit definition f'(x) = lim(h→0) [f(x+h)−f(x)]/h.

从基本原理求导 – 使用极限定义 f'(x) = lim(h→0) [f(x+h)−f(x)]/h。

Chain rule – dy/dx = dy/du × du/dx, used for composite functions such as f(g(x)).

链式法则 – dy/dx = dy/du × du/dx，用于复合函数如f(g(x))。

Product rule – if y = u v, then dy/dx = u dv/dx + v du/dx.

积法则 – 若 y = u v，则 dy/dx = u dv/dx + v du/dx。

Quotient rule – if y = u/v, then dy/dx = (v du/dx − u dv/dx) / v².

商法则 – 若 y = u/v，则 dy/dx = (v du/dx − u dv/dx) / v²。

Stationary point – a point where f'(x) = 0; can be a local maximum, local minimum or point of inflection.

驻点 – 满足f'(x) = 0的点；可以是局部极大值、局部极小值或拐点。

Second derivative d²y/dx² – the rate of change of the derivative; used to determine concavity and classify stationary points.

二阶导数 d²y/dx² – 导数的变化率；用于判断凹凸性并对驻点进行分类。

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3. Integration | 积分

Indefinite integral ∫ f(x) dx – the reverse process of differentiation, includes a constant of integration, +C.

不定积分 ∫ f(x) dx – 微分的逆运算，包含积分常数+C。

Definite integral ∫ₐᵇ f(x) dx – the signed area under the curve y = f(x) from x=a to x=b.

定积分 ∫ₐᵇ f(x) dx – 曲线y = f(x)下方从x=a到x=b的有向面积。

Reverse chain rule – integrates expressions of the form k f'(g(x)) g'(x) to k f(g(x)) + C.

反链式法则 – 将形如 k f'(g(x)) g'(x) 的表达式积分为 k f(g(x)) + C。

Integration by substitution – uses u = g(x) and du = g'(x) dx to simplify the integral.

换元积分法 – 用 u = g(x) 和 du = g'(x) dx 简化被积函数。

Integration by parts – formula: ∫ u dv = u v − ∫ v du; based on the product rule.

分部积分法 – 公式：∫ u dv = u v − ∫ v du；基于积法则。

Volume of revolution – V = π ∫ₐᵇ y² dx (about x-axis) or V = π ∫ₐᵇ x² dy (about y-axis).

旋转体体积 – V = π ∫ₐᵇ y² dx（绕x轴）或 V = π ∫ₐᵇ x² dy（绕y轴）。

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4. Sequences and Series | 数列与级数

Sequence – an ordered list of numbers defined by an nth term formula uₙ.

数列 – 由通项公式uₙ定义的有序数字列表。

Arithmetic progression – sequence with a constant common difference d; uₙ = a + (n−1)d, sum Sₙ = n/2 [2a + (n−1)d].

等差数列 – 具有常数公差d的数列；uₙ = a + (n−1)d，和 Sₙ = n/2 [2a + (n−1)d]。

Geometric progression – a sequence with a constant common ratio r; uₙ = a rⁿ⁻¹, sum Sₙ = a(1−rⁿ)/(1−r) for r ≠ 1.

等比数列 – 具有常数公比r的数列；uₙ = a rⁿ⁻¹，和 Sₙ = a(1−rⁿ)/(1−r) 当 r ≠ 1。

Sum to infinity S∞ – converges only if |r| < 1, given by S∞ = a/(1−r).

无穷和 S∞ – 仅当 |r| < 1 时收敛，公式为 S∞ = a/(1−r)。

Sigma notation Σ – compact way to write a sum, e.g. Σ (r=1 to n) uᵣ.

求和符号 Σ – 表示求和的简洁写法，例如 Σ (r=1 to n) uᵣ。

Binomial expansion (1+x)ⁿ – (1+x)ⁿ = 1 + n x + [n(n−1)/2!] x² + ... valid for |x| < 1 and rational n.

二项式展开 (1+x)ⁿ – (1+x)ⁿ = 1 + n x + [n(n−1)/2!] x² + ... 对 |x| < 1 且有理数 n 有效。

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5. Trigonometry | 三角学

Radian measure – the angle subtended by an arc of length r on a circle of radius r; π rad = 180°.

弧度制 – 半径为r的圆上弧长为 r 的弧所对的圆心角；π rad = 180°。

Arc length and sector area – arc length s = rθ, sector area A = ½ r²θ (θ in radians).

弧长与扇形面积 – 弧长 s = rθ，扇形面积 A = ½ r²θ（θ单位为弧度）。

Sine rule – a/sin A = b/sin B = c/sin C, used for non-right-angled triangles.

正弦定理 – a/sin A = b/sin B = c/sin C，用于非直角三角形。

Cosine rule – a² = b² + c² − 2bc cos A, applies to any triangle.

余弦定理 – a² = b² + c² − 2bc cos A，适用于任意三角形。

Double angle formulae – sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.

倍角公式 – sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ。

Secant, cosecant, cotangent – sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ = cos θ/sin θ.

正割、余割、余切 – sec θ = 1/cos θ，cosec θ = 1/sin θ，cot θ = 1/tan θ = cos θ/sin θ。

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6. Exponentials and Logarithms | 指数与对数

Exponential function eˣ – the function whose derivative is itself; d/dx (eˣ) = eˣ.

指数函数 eˣ – 导数等于自身的函数；d/dx (eˣ) = eˣ。

Natural logarithm ln x – the inverse of eˣ; defined for x > 0, d/dx (ln x) = 1/x.

自然对数 ln x – eˣ的反函数；定义在 x > 0，d/dx (ln x) = 1/x。

Laws of logarithms – ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, ln(aⁿ) = n ln a.

对数运算法则 – ln(ab) = ln a + ln b，ln(a/b) = ln a − ln b，ln(aⁿ) = n ln a。

Exponential growth and decay – modelled by y = A eᵏᵗ; k > 0 for growth, k < 0 for decay.

指数增长与衰减 – 模型为 y = A eᵏᵗ；k > 0 表示增长，k < 0 表示衰减。

Derivative of aˣ – d/dx (aˣ) = aˣ ln a, for a > 0.

aˣ的导数 – d/dx (aˣ) = aˣ ln a，其中 a > 0。

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7. Vectors | 向量

Vector quantity – has both magnitude and direction, often represented in bold or with an arrow (e.g., a, a⃗).

向量 – 既有大小又有方向，常用粗体或箭头表示（如 a、a⃗）。

Magnitude |a| – the length (size) of a vector; for a = x i + y j, |a| = √x² + y².

模 |a| – 向量的长度（大小）；对于 a = x i + y j，|a| = √x² + y²。

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

单位向量 – 模为1的向量；常写作 â = a / |a|。

Scalar (dot) product a·b – a·b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂; if a·b = 0, vectors are perpendicular.

数量积（点积）a·b – a·b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂；若 a·b = 0，两向量垂直。

Vector equation of a line – r = a + t b, where a is a point on the line and b is the direction vector.

直线的向量方程 – r = a + t b，其中 a 为线上一点，b 为方向向量。

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8. Numerical Methods | 数值方法

Change of sign method – if f(a) and f(b) have opposite signs, a root lies in [a, b]; used to locate roots.

符号变化法 – 若 f(a) 与 f(b) 符号相反，则在 [a, b] 内存在一个根；用于确定根的位置。

Iteration xₙ₊₁ = g(xₙ) – a sequence from a starting value x₀; if it converges, the limit is a solution to x = g(x).

迭代公式 xₙ₊₁ = g(xₙ) – 从初值 x₀ 生成序列；若收敛，极限即为 x = g(x) 的解。

Newton-Raphson method – xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ); rapid convergence if the initial estimate is close to the root.

牛顿-拉弗森法 – xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ)；若初值靠近根，收敛速度很快。

Cobweb and staircase diagrams – hical representations of iteration convergence or divergence, plotting y = x and y = g(x).

蛛网图与阶梯图 – 迭代收敛或发散的图形表示，绘制 y = x 与 y = g(x)。

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9. Mechanics: Kinematics | 力学：运动学

Displacement s – the distance from a fixed origin in a specified direction (vector).

位移 s – 从固定原点在指定方向上的距离（向量）。

Velocity v – the rate of change of displacement; v = ds/dt.

速度 v – 位移的变化率；v = ds/dt。

Acceleration a – the rate of change of velocity; a = dv/dt = d²s/dt².

加速度 a – 速度的变化率；a = dv/dt = d²s/dt²。

SUVAT equations – nstant acceleration: v = u + at, s = ut + ½at², s = ½(u+v)t, v² = u² + 2as, s = vt − ½at².

SUVAT方程 – 适用于匀加速度：v = u + at，s = ut + ½at²，s = ½(u+v)t，v² = u² + 2as，s = vt − ½at²。

Variable acceleration – use calculus: v = ∫ a dt, s = ∫ v dt; also a = v dv/ds.

变加速度 – 使用微积分：v = ∫ a dt，s = ∫ v dt；另有 a = v dv/ds。

Distance travelled – total length of the path; found by integrating |v| over time when direction changes.

路程 – 路径的总长度；当运动方向改变时，需对 |v| 关于时间积分求得。

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10. Mechanics: Forces and Newton's Laws | 力学：力与牛顿定律

Resultant force – the single force that has the same effect as all the forces acting on a particle combined.

合力 – 效果等同于作用在质点上所有力之和的单一力。

Newton's first law – an object remains at rest or in uniform motion unless acted upon by a resultant force.

牛顿第一定律 – 物体将保持静止或匀速直线运动状态，除非有合力作用于它。

Newton's second law – F = m a, where F is the resultant force, m is mass and a is acceleration.

牛顿第二定律 – F = m a，其中 F 为合力，m 为质量，a 为加速度。

Newton's third law – for every action there is an equal and opposite reaction; forces always occur in pairs.

牛顿第三定律 – 每个作用力都有一个大小相等、方向相反的反作用力；力总是成对出现。

Friction – friction Fₘₐₓ = μ R, where μ is the coefficient of friction and R is the normal reaction.

摩擦力 – 阻碍运动的力；最大静摩擦力 Fₘₐₓ = μ R，其中 μ 为摩擦系数，R 为法向反作用力。

Tension – the pulling force transmitted through a string, cable or chain when it is pulled tight by forces acting from opposite ends.

张力 – 通过绳子、缆索或链条传递的拉力，当两端受力而被拉紧时产生。

Connected particles – or a light rod; modelling requires resolving forces and applying F = m a to each particle.

连接质点 – 通过轻质不可伸长的绳或轻杆连接的系统；建模需对每个质点分解力并应用 F = m a。

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11. Statistics: Probability Distributions | 统计：概率分布

Random variable X – a variable whose value depends on the outcome of a random event; can be discrete or continuous.

随机变量 X – 其取值依赖于随机事件结果的变量；可以是离散或连续的。

Expectation E(X) and variance Var(X) – E(X) = Σ xₚ P(X=x) for discrete, or ∫ x f(x) dx for continuous; Var(X) = E(X²) − [E(X)]².

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