📚 AQA Mathematics: Quick Revision via Mind Maps | AQA 数学：思维导图速记法
Mind mapping is a powerful visual tool to organise and memorise complex mathematical concepts for AQA A-level Maths. By structuring topics into branching diagrams, you can see connections, recall formulas faster, and tackle exam questions with confidence. This article walks you through key topics by building a mental map, with concise explanations in both English and Chinese to reinforce your understanding.
思维导图是组织和记忆AQA高等数学复杂概念的强大视觉工具。通过将主题构建成分支图表，你能看清联系，更快回忆公式，并自信地应对考题。本文带你用思维导图方式遍历核心考点，每部分均提供中英双语精简解释，助你高效巩固。
1. Algebra & Functions Mind Map | 代数与函数思维导图
Place “Algebra & Functions” at the centre. The first branch is Quadratics, where you record the three forms: standard y = ax² + bx + c, vertex y = a(x − h)² + k, and factored y = a(x − p)(x − q). Another sub-branch covers the discriminant Δ = b² − 4ac.
将“代数与函数”放在中心。第一个分支是二次函数，记下三种形式：一般式 y = ax² + bx + c、顶点式 y = a(x − h)² + k、因式分解式 y = a(x − p)(x − q)。再分出一个子分支写判别式 Δ = b² − 4ac。
The discriminant tells you the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives no real roots. Completing the square helps find the turning point: x² + bx = (x + b/2)² − (b/2)².
判别式揭示根的性质：Δ > 0 有两个不等实根，Δ = 0 有一个重根，Δ < 0 无实根。配方可用来求转折点：x² + bx = (x + b/2)² − (b/2)²。
Sketch inequalities like ax² + bx + c > 0 by visualising the parabola. The solution is the x-values where the graph lies above the x-axis. Always reverse the inequality sign when multiplying by a negative.
像 ax² + bx + c > 0 这样的不等式，借助抛物线图像求解——解集是图像在 x 轴上方的 x 值。乘以负数时务必反转不等号。
For functions, note domain and range, one-to-one versus many-to-one, inverse function f⁻¹(x) obtained by swapping x and y then solving for y, and composite functions fg(x) = f(g(x)). Remember that f⁻¹(x) exists only if f is one-to-one.
函数分支记录定义域、值域、单射与多射函数、反函数 f⁻¹(x)（交换 x 与 y 后解出 y）以及复合函数 fg(x) = f(g(x))。只有单射函数才存在反函数。
2. Trigonometry & Identities Mind Map | 三角学与恒等式思维导图
Start with “Trigonometry” and branch into basic ratios (sin, cos, tan), exact values for 0°, 30°, 45°, 60°, 90°, and the two key identities: sin²θ + cos²θ ≡ 1 and tan θ ≡ sin θ / cos θ.
以“三角学”为中心，分出基本比（sin、cos、tan）、特殊角（0°、30°、45°、60°、90°）的精确值，以及两个核心恒等式：sin²θ + cos²θ ≡ 1 和 tan θ ≡ sin θ / cos θ。
sin²θ + cos²θ ≡ 1
tan θ ≡ sin θ / cos θ
Use the sine and cosine rules for non-right triangles. The sine rule: a / sin A = b / sin B = c / sin C. The cosine rule: a² = b² + c² − 2bc cos A. Apply the cosine rule when you know two sides and the included angle, or three sides.
对非直角三角形使用正弦定理和余弦定理。正弦定理：a / sin A = b / sin B = c / sin C。余弦定理：a² = b² + c² − 2bc cos A。已知两边及其夹角，或已知三边时，使用余弦定理。
Compound-angle and double-angle formulas often appear. For instance, sin(A ± B) = sinA cosB ± cosA sinB, cos(A ± B) = cosA cosB ∓ sinA sinB, and cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ. Add a branch for solving trig equations, where you find all solutions within a given interval using the CAST diagram or graphs.
和角与倍角公式很常见，如 sin(A ± B) = sinA cosB ± cosA sinB，cos(A ± B) = cosA cosB ∓ sinA sinB，以及 cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ。单独设一个分支解三角方程，在给定区间内用 CAST 图或图像找出所有解。
Radians: 180° = π rad. Arc length s = rθ, sector area A = ½ r²θ. Converting degrees to radians and vice versa is essential.
弧度制：180° = π 弧度。弧长 s = rθ，扇形面积 A = ½ r²θ。熟练转换度与弧度是基本功。
3. Differentiation Mind Map | 微分思维导图
Centre “Differentiation”. The first branches are the basic derivatives: d/dx (xⁿ) = n xⁿ⁻¹, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x, and d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec² x.
将“微分”置于中心，先伸出基本导数分支：d/dx (xⁿ) = n xⁿ⁻¹，d/dx (eˣ) = eˣ，d/dx (ln x) = 1/x，以及 d/dx (sin x) = cos x，d/dx (cos x) = −sin x，d/dx (tan x) = sec² x。
d/dx (xⁿ) = n xⁿ⁻¹
The chain rule: dy/dx = dy/du × du/dx. Use it for composite functions like y = (3x + 2)⁵. The product rule: if y = uv, then dy/dx = u dv/dx + v du/dx. The quotient rule: if y = u/v, then dy/dx = (v du/dx − u dv/dx) / v².
链式法则：dy/dx = dy/du × du/dx，用于复合函数，如 y = (3x + 2)⁵。乘法法则：若 y = uv，则 dy/dx = u dv/dx + v du/dx。除法法则：若 y = u/v，则 dy/dx = (v du/dx − u dv/dx) / v²。
The second derivative d²y/dx² indicates concavity and helps classify stationary points. If f'(x) = 0 and f”(x) > 0, it’s a minimum; if f”(x) < 0, a maximum; if f''(x) = 0, it could be a point of inflection.
二阶导数 d²y/dx² 指示凹凸性，帮助判断驻点类型。f'(x)=0 且 f”(x) > 0 时为极小值点；f”(x) < 0 为极大值点；f''(x)=0 可能为拐点。
Connect differentiation to gradients and tangents. The gradient of a curve at a point is f'(x₁). The equation of the tangent at (x₁, y₁) is y − y₁ = m (x − x₁).
将微分与斜率和切线联系起来：曲线在某点的斜率为 f'(x₁)；在该点 (x₁, y₁) 的切线方程为 y − y₁ = m (x − x₁)。
4. Integration Mind Map | 积分思维导图
Under “Integration”, list the reverse of differentiation: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1), ∫ eˣ dx = eˣ + C, ∫ 1/x dx = ln |x| + C, ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C.
“积分”分支下列出微分的逆运算：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)，∫ eˣ dx = eˣ + C，∫ 1/x dx = ln |x| + C，∫ sin x dx = −cos x + C，∫ cos x dx = sin x + C。
Definite integrals give the area under a curve between limits: ∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a). Remember areas below the x-axis are negative unless you take absolute values for total area.
定积分给出曲线与 x 轴之间的面积：∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)。注意 x 轴下方的面积为负，求总面积时需取绝对值。
Integration by substitution: simplify by letting u = g(x), then du = g'(x) dx. Rewrite the integral in terms of u. For definite integrals, convert the limits too. Integration by parts: ∫ u dv = uv − ∫ v du, useful for products such as x eˣ.
换元积分法：令 u = g(x)，则 du = g'(x) dx，将积分式完全用 u 表示。定积分需同时代换上下限。分部积分法：∫ u dv = uv − ∫ v du，适用于 x eˣ 这类乘积形式。
Trapezium rule for numerical approximation: Area ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ], where h = (b − a)/n. Include its branch as a reminder for cases where exact integration is impossible.
梯形法则用于数值近似：面积 ≈ h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ]，其中 h = (b − a)/n。在无法精确积分时，这条分支可救急。
5. Exponentials & Logarithms Mind Map | 指数与对数思维导图
Exponentials: y = aˣ, with special case y = eˣ. Note that e ≈ 2.718. The derivative of eˣ is eˣ, and the integral is also eˣ + C. The natural logarithm ln x is the inverse of eˣ.
指数函数 y = aˣ，特例 y = eˣ（e ≈ 2.718）。eˣ 的导数仍是 eˣ，积分也是 eˣ + C。自然对数 ln x 是 eˣ 的反函数。
Key log laws: ln(ab) = ln a + ln b; ln(a/b) = ln a − ln b; ln aᵇ = b ln a. The change-of-base formula: logₐ b = ln b / ln a.
核心对数律：ln(ab) = ln a + ln b；ln(a/b) = ln a − ln b；ln aᵇ = b ln a。换底公式：logₐ b = ln b / ln a。
To solve exponential equations, take logs on both sides, e.g., 3ˣ = 20 → x ln 3 = ln 20 → x = ln 20 / ln 3. For log equations, isolate the log term and use the definition: ln A = B ⇒ A = eᴮ.
解指数方程时两边取对数，如 3ˣ = 20 → x ln 3 = ln 20 → x = ln 20 / ln 3。对数方程先分离对数项，再按定义转换：ln A = B ⇒ A = eᴮ。
Real-world applications: exponential growth P = P₀ eᵏᵗ, radioactive decay, and continuous compound interest. Add a modelling branch to your map.
实际应用包括指数增长 P = P₀ eᵏᵗ、放射性衰变以及连续复利。在你的导图上添加一个建模分支。
6. Sequences & Series Mind Map | 数列与级数思维导图
Arithmetic sequences: nᵗʰ term uₙ = a + (n−1)d, sum Sₙ = n/2 [2a + (n−1)d] or n/2 (first + last). Geometric sequences: uₙ = arⁿ⁻¹, sum Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1. Infinite geometric series sum to infinity S∞ = a/(1 − r) provided |r| < 1.
等差数列：第 n 项 uₙ = a + (n−1)d，和 Sₙ = n/2 [2a + (n−1)d] 或 n/2 (首项+末项)。等比数列：uₙ = arⁿ⁻¹，前 n 项和 Sₙ = a(1 − rⁿ)/(1 − r) (r ≠ 1)。无穷等比级数和 S∞ = a/(1 − r)，要求 |r| < 1。
Binomial expansion for (1 + x)ⁿ: (1 + x)ⁿ = 1 + nx + [n(n−1)/2!] x² + … + xⁿ, valid for integer n. For non-integer n, the expansion is infinite and converges for |x| < 1.
二项展开 (1 + x)ⁿ：对于整数 n，有界展开为 1 + nx + [n(n−1)/2!] x² + … + xⁿ。若 n 非整数，则展开为无穷级数，收敛区间为 |x| < 1。
Sigma notation Σ: from r=1 to n of f(r). Use standard results like Σ 1 = n, Σ r = n(n+1)/2, Σ r² = n(n+1)(2n+1)/6 to evaluate sums.
Σ 符号：∑_{r=1}^{n} f(r)。记住标准结果：Σ 1 = n，Σ r = n(n+1)/2，Σ r² = n(n+1)(2n+1)/6，以便计算和式。
Recurrence relations: uₙ₊₁ = f(uₙ). They can generate sequences; link to iteration and limits. Common in AQA pure.
递推关系 uₙ₊₁ = f(uₙ) 可生成数列，和迭代与极限有关，在 AQA 纯数中常见。
7. Vectors Mind Map | 向量思维导图
Vectors in 2D and 3D have both magnitude and direction. Represent a vector as ai + bj or column form. Magnitude |v| = √(a² + b²) in 2D; in 3D add c². A unit vector has magnitude 1: â = v / |v|.
二维和三维向量既有大小又有方向。向量可表示为 ai + bj 或列向量形式。二维下模 |v| = √(a² + b²)；三维加上 c²。单位向量模为 1：â = v / |v|。
The scalar (dot) product: a · b = |a||b| cos θ, or a·b = a₁b₁ + a₂b₂ (+ a₃b₃). It is useful to find the angle between vectors: cos θ = (a·b)/(|a||b|). If a·b = 0, the vectors are perpendicular.
数量积（点积）：a · b = |a||b| cos θ，或 a·b = a₁b₁ + a₂b₂ (+ a₃b₃)。用它求向量间夹角：cos θ = (a·b)/(|a||b|)。若 a·b = 0，两向量垂直。
Position vectors and vector equations of lines: a line can be written as r = a + t d, where a is a point on the line and d is the direction vector. For two lines, determine if they intersect, are parallel, or skew.
位置向量和直线的向量方程：直线可写为 r = a + t d，a 是直线上一点，d 是方向向量。两条直线可判断是相交、平行还是异面。
8. Coordinate Geometry & Parametric Equations Mind Map | 坐标几何与参数方程思维导图
Straight lines: gradient m = (y₂ − y₁)/(x₂ − x₁). Equation forms: y = mx + c, y − y₁ = m(x − x₁), and ax + by + c = 0. Distance = √[(x₂−x₁)² + (y₂−y₁)²]. Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).
直线：斜率 m = (y₂ − y₁)/(x₂ − x₁)。方程形式有 y = mx + c，y − y₁ = m(x − x₁) 和 ax + by + c = 0。距离 = √[(x₂−x₁)² + (y₂−y₁)²]，中点 = ((x₁+x₂)/2, (y₁+y₂)/2)。
Circles: equation (x − a)² + (y − b)² = r², centre (a,b) radius r. Expand to x² + y² + 2gx + 2fy + c = 0 where center = (−g, −f) and radius = √(g² + f² − c). Tangent meets radius at 90°.
圆：方程 (x − a)² + (y − b)² = r²，圆心 (a,b) 半径 r。展开可得一般式 x² + y² + 2gx + 2fy + c = 0，圆心 (−g, −f
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