📚 Year 13 Edexcel Maths: Key Terminology Memorisation Guide | Edexcel A Level 数学核心术语速记指南
This guide delivers a rapid and structured overview of the must-know mathematical vocabulary for Year 13 Edexcel Mathematics. Covering Pure, Statistics and Mechanics, the terms are paired with memorable shortcuts to help you recall definitions, formulas and applications under exam pressure. Read through each section to lock in the language that examiners expect.
本指南为你快速梳理 Edexcel A Level 数学（Year 13）必备术语，涵盖纯数、统计和力学三大模块。每个术语都配有简洁的记忆口诀，帮助你在考试中准确回忆定义、公式与应用场景，全面提升读题和答题效率。
1. Differentiation Terminology | 导数术语
Derivative – written as f'(x) or dy/dx, it gives the instantaneous rate of change or gradient of the tangent. Think: ‘prime’ pushes the function forward into its slope.
导数 – 记作 f'(x) 或 dy/dx，表示函数在某点的瞬时变化率，也是切线的斜率。记忆：加一撇 ‘prime’ 就是求斜率。
Chain rule – used for composite functions: dy/dx = dy/du × du/dx. Imagine peeling an onion, layer by layer, differentiating the outer function first then multiplying by the derivative of the inside.
链式法则 – 用于复合函数求导：dy/dx = dy/du × du/dx。像剥洋葱一样，先对外层求导，再乘以内层的导数。
Product rule – if y = uv then dy/dx = u’v + uv’. Memorise as ‘left prime times right plus left times right prime’.
乘法法则 – 若 y = uv，则 dy/dx = u’v + uv’。记忆口诀：’左导乘右加左乘右导’。
Quotient rule – for y = u/v, dy/dx = (vu’ – uv’)/v². A popular mnemonic: ‘low d-high minus high d-low, square the bottom and away you go’.
除法法则 – 对于 y = u/v，dy/dx = (vu’ – uv’)/v²。口诀：’分母导分子减分子导分母，除以分母的平方’。
Implicit differentiation – when y is not given explicitly, differentiate both sides with respect to x, remembering to multiply by dy/dx every time you differentiate a y term. The catchphrase: ‘when you see y, stick a dy/dx on it’.
隐函数求导 – 当 y 不能用 x 明显表达时，对方程两边关于 x 求导，每次对含 y 的项求导都要乘上 dy/dx。口诀：’见到 y 就挂一个 dy/dx’。
Stationary points – where dy/dx = 0; use second derivative test: if d²y/dx² > 0 it is a minimum, if d²y/dx² < 0 it is a maximum, and if = 0 check gradient either side for a point of inflection.
驻点 – dy/dx = 0 的点；用二阶导数检验：d²y/dx² > 0 则为极小值，< 0 则为极大值，若等于 0 需检查两侧斜率确认是否为拐点。
Connected rates of change – use the chain rule to link rates, e.g. dV/dt = dV/dr × dr/dt. Treat derivatives as fractions that can be ‘chained’.
相关变化率 – 用链式法则将变化率联系起来，如 dV/dt = dV/dr × dr/dt。把导数看作可以“链接”的分数即可。
2. Integration Terminology | 积分术语
Integration – the reverse process of differentiation, also called antidifferentiation. The indefinite integral yields a family of functions plus a constant C; the definite integral ∫ₐᵇ f(x) dx computes the signed area under the curve. Memory aid: ‘integrate to accumulate’.
积分 – 求导的逆运算，也称为反导。不定积分得到一族函数加上常数 C；定积分 ∫ₐᵇ f(x) dx 计算出曲线下的带号面积。记忆：’积分就是累加’。
Integration by substitution – choose u = g(x), find du = g'(x) dx and rewrite the integral entirely in terms of u. Think: ‘replace the mess with a single letter’.
换元积分法 – 令 u = g(x)，求出 du = g'(x) dx，把整个积分换成关于 u 的表达式。思路：’用单个字母替换一团乱麻’。
Integration by parts – from the product rule: ∫ u dv = uv – ∫ v du. Use the LIATE rule to pick u: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. This order helps you decide which part to differentiate.
分部积分法 – 源自乘法法则：∫ u dv = uv − ∫ v du。用 LIATE 顺序选择 u：对数、反三角、代数、三角、指数。这个顺序帮你决定对哪部分求导。
Partial fractions in integration – when integrating a rational expression, split it into simpler fractions first. Remember: for repeated linear factors (ax+b)ⁿ, you need fractions with denominators (ax+b), (ax+b)², …, up to (ax+b)ⁿ.
有理分式积分中的部分分式 – 对有理式积分前先拆成简单的分式。注意：若有重线性因子 (ax+b)ⁿ，部分分式中必须包含分母为 (ax+b), (ax+b)², …, (ax+b)ⁿ 的各项。
Trapezium rule – an approximate numerical integration: Area ≈ h/2[y₀ + 2(y₁+y₂+…) + yₙ], where h = (b−a)/n. Visualise fitting trapezia under the curve.
梯形法则 – 数值积分近似公式：面积 ≈ h/2[y₀ + 2(y₁ + y₂ + …) + yₙ]，其中 h = (b−a)/n。想象用一个个梯形逼近曲线下的区域。
Differential equations – separate variables, integrate both sides, apply initial conditions to find the constant. The phrase ‘shuffle dx, dy, then integrate’ captures the method.
微分方程 – 分离变量，两边积分，代入初始条件求出常数。方法可概括为：’倒腾 dx, dy，然后积分’。
3. Trigonometry and Radian Terms | 三角函数与弧度术语
Radian – 1 radian is the angle subtended by an arc equal to the radius. With radians, arc length s = rθ, sector area A = ½r²θ. Remember: sector area looks like an isosceles triangle (½ × base × height), but with r²θ.
弧度 – 1 弧度是弧长等于半径时所对的圆心角。使用弧度时，弧长 s = rθ，扇形面积 A = ½r²θ。记忆：扇形面积公式像等腰三角形面积 (½ × 底 × 高)，只不过用了 r²θ。
Reciprocal trig functions – sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ = cosθ/sinθ. The identity 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ are sisters of sin²θ + cos²θ = 1.
倒数三角函数 – sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = 1/tan θ。恒等式 1+tan²θ = sec²θ 和 1+cot²θ = cosec²θ 是 sin²θ+cos²θ=1 的姊妹形式。
Compound angle formulas – sin(A±B) = sinA cosB ± cosA sinB; cos(A±B) = cosA cosB ∓ sinA sinB; tan(A±B) = (tanA ± tanB)/(1 ∓ tanA tanB). The sign flips for cos when it’s a minus: ‘cos is choosy’.
和角公式 – sin(A±B)=sinA cosB ± cosA sinB；cos(A±B)=cosA cosB ∓ sinA sinB；tan(A±B)=(tanA ± tanB)/(1 ∓ tanA tanB)。cos 的符号在减号时变化，口诀：’cos 很挑剔’。
Double angle formulas – derived from compound formulas: sin 2θ = 2 sinθ cosθ; cos 2θ = cos²θ − sin²θ = 1 − 2 sin²θ = 2 cos²θ − 1; tan 2θ = 2 tanθ/(1 − tan²θ). Choose the cos version that matches the rest of your problem.
二倍角公式 – sin 2θ=2 sinθ cosθ；cos 2θ = cos²θ − sin²θ = 1−2 sin²θ = 2 cos²θ−1；tan 2θ = 2 tanθ/(1−tan²θ)。根据题目形式灵活选用 cos 二倍角的版本。
R-formula – a sinθ + b cosθ can be written as R sin(θ±α) or R cos(θ±α), with R = √(a²+b²) and α = arctan(b/a) or appropriate. ‘Combined wave’ trick: R is the amplitude of the new single wave.
R-公式 – 形如 a sinθ + b cosθ 可化为 R sin(θ±α) 或 R cos(θ±α)，其中 R = √(a²+b²)，α = arctan(b/a) 适当选取。记忆：R 是合成后单一波形的振幅。
4. Vectors in 3D | 空间向量
Position vector – the vector from the origin to a point, often written as r = xi + yj + zk. Think ‘origin to point’.
位置向量 – 从原点到某一点的向量，常写作 r = xi + yj + zk。方向：’从原点出发’。
Magnitude – for a vector a = xi + yj + zk, |a| = √(x² + y² + z²). It is the 3D version of Pythagoras.
模长 – 向量 a = xi + yj + zk 的模长为 √(x²+y²+z²)，就是三维空间的勾股定理。
Unit vector – a vector of length 1 in a given direction: â = a/|a|. Used to specify direction cleanly.
单位向量 – 长度为 1 的方向向量，â = a/|a|。常用来干净地表示方向。
Scalar (dot) product – a·b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂. Use to find angles between vectors. If a·b = 0, vectors are perpendicular. ‘Dot gives the angle’.
数量积（点乘） – a·b = |a||b| cos θ = x₁x₂ + y₁y₂ + z₁z₂。用于求两向量夹角；若 a·b = 0，则两向量垂直。’点乘出角度’。
Vector equation of a line – r = a + λb, where a is a point on the line and b is the direction vector. To find the intersection of two lines, equate the position vectors and solve for the parameters.
直线向量方程 – r = a + λb，a 为直线上一点，b 为方向向量。求两直线交点时，令位置向量相等并解参数即可。
Skew lines – in 3D, lines that do not intersect and are not parallel. Check if the system for intersection has no solution and direction vectors are not multiples.
异面直线 – 三维空间中既不平行也不相交的两条直线。通过检验联立方程无解且方向向量不成比例来判断。
5. Numerical Methods | 数值方法
Locating roots – show f(a) and f(b) have opposite signs; by the sign-change rule there is a root in [a, b]. ‘Sign change guarantees at least one root’ for continuous functions.
定位根 – 证明 f(a) 与 f(b) 异号，根据变号法则连续函数在区间 [a,b] 内至少有一根。’一变号，根就到’。
Iteration – rearrange equation f(x)=0 to x = g(x), then use xₙ₊₁ = g(xₙ). Draw cobweb or staircase diagrams to visualise convergence. Tip: a diagram helps you see whether the iteration converges.
迭代法 – 将 f(x)=0 改写为 x = g(x)，然后使用 xₙ₊₁ = g(xₙ)。画蛛网图或阶梯图可直观判断收敛性。建议：画个草图秒懂收敛行为。
Newton-Raphson method – faster iteration using tangents: xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ). Formula memory: current x minus the ratio of function to slope; ‘tangent chops down to the root’.
牛顿-拉夫逊法 – 用切线更快速地迭代：xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ)。记法：当前 x 减去函数值与斜率的比值，’切线砍向根’。
Numerical integration – when an integral cannot be found exactly, use the trapezium rule or, at a further level, Simpson’s rule. Edexcel focuses on trapezium rule; remember to state whether it overestimates or underestimates by looking at the curve’s concavity.
数值积分 – 当无法求得精确积分时，使用梯形法则（Edexcel 要求）。通过观察曲线凹凸性判断估算值是偏高还是偏低。
6. Functions, Mappings and Transformations | 函数、映射与变换
Domain and range – domain is the set of input values (x), range is the set of output values (f(x)). ‘Domain goes in, range comes out’.
定义域与值域 – 定义域是所有输入 x 的集合，值域是所有输出 f(x) 的集合。’进的是定义域，出的是值域’。
One-to-one and many-to-one – a function must be single-valued, but can be many-to-one. Only one-to-one functions have inverses that are also functions. The horizontal line test checks one-to-one: any horizontal line cuts the graph at most once.
一一映射与多一映射 – 函数必须是一对多的，但可以是多对一。只有一一映射才存在反函数。水平线检验：任意水平线与图像至多一个交点即为一对一。
Inverse function – f⁻¹(x) is the reflection of f(x) in the line y = x. To find it, write y = f(x), swap x and y, then solve for y. Domain of f⁻¹ equals the range of f.
反函数 – f⁻¹(x) 是 f(x) 关于直线 y=x 的对称图形。求法：令 y=f(x)，交换 x 和 y，解出 y。f⁻¹ 的定义域就是原函数的值域。
Composite functions – fg(x) means first apply g, then f. Read from right to left: ‘g then f’. The domain of fg is those x in the domain of g for which g(x) is in the domain of f.
复合函数 – fg(x) 表示先作用 g，再作用 f。从右往左读：’先 g 后 f’。定义域为满足 g(x) 在 f 定义域内的所有 x。
Modulus function – |x| = x if x ≥ 0, −x if x < 0. To solve equations like |ax+b| = c, split into ax+b = c and ax+b = −c. Graph is a V-shape.
绝对值函数 – |x|：非负不变，负变正。解 |ax+b| = c 时拆成 ax+b = c 和 ax+b = −c。图像是 V 形。
Transformations – f(x+a) shifts left by a; f(x−a) shifts right; f(x)+a shifts up; f(ax) stretches horizontally by factor 1/a; af(x) stretches vertically by factor a. Mnemonic: ‘inside the bracket affects x and does the opposite’.
函数变换 – f(x+a) 左移 a；f(x−a) 右移 a；f(x)+a 上移 a；f(ax) 水平方向伸缩 1/a 倍；af(x) 竖直方向伸缩 a 倍。口诀：’括号内管 x，作用相反’。
7. Sequences, Series and Binomial Expansion | 序列、级数与二项式展开
Arithmetic sequence – n-th term uₙ = a + (n−1)d; sum Sₙ = n/2[2a + (n−1)d] or n/2(first+last). ‘Arithmetic means constant difference’.
等差数列 – 通项 uₙ = a + (n−1)d；求和 Sₙ = n/2[2a + (n−1)d] 或 n/2(首项+末项)。’等差就是差不变’。
Geometric sequence – uₙ = arⁿ⁻¹; sum to n terms Sₙ = a(1−rⁿ)/(1−r) for |r|≠1. Infinite sum S∞ = a/(1−r) converges only when |r| < 1. ‘Geometric means common ratio’.
等比数列 – uₙ = arⁿ⁻¹；有限和 Sₙ = a(1−rⁿ)/(1−r) (|r|≠1)；无穷和 S∞ = a/(1−r) 只在 |r|<1 时收敛。’等比就是比不变’。
Binomial expansion for (1+x)ⁿ – valid for any rational n, provided |x| < 1: (1+x)ⁿ = 1 + nx + n(n−1)/2! x² + n(n−1)(n−2)/3! x³ + … Note that this is an infinite series when n is not a positive integer; you
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