AS Further Maths: Mind Map Quick Revision | AS 进阶数学：思维导图速记

📚 AS Further Maths: Mind Map Quick Revision | AS 进阶数学：思维导图速记

This mind map revision guide walks you through the core topics of AS Level Further Mathematics. Each section summarises key definitions, essential formulae, and typical problem-solving steps in a clear, bilingual format. Use these interconnected concept nodes to strengthen your revision and build deeper understanding.

这篇思维导图速记带你梳理 AS 进阶数学的核心主题。每一节都以清晰的中英双语形式总结关键定义、必备公式和典型解题步骤。利用这些相互关联的概念节点，强化你的复习并建立更深层的理解。

1. Complex Numbers – Basics & Argand Diagram | 复数基础与 Argand 图

A complex number is written as z = x + iy, where x, y are real numbers and i = √(-1). The real part is Re(z) = x and the imaginary part is Im(z) = y. Every complex number can be plotted on an Argand diagram with the x-axis representing the real part and the y-axis representing the imaginary part.

复数写作 z = x + iy，其中 x, y 为实数，i = √(-1)。实部 Re(z) = x，虚部 Im(z) = y。每个复数都可以绘制在 Argand 图上，x 轴表示实部，y 轴表示虚部。

The modulus of z is r = |z| = √(x² + y²), representing the distance from the origin. The argument θ = arg(z) satisfies tanθ = y/x, taking care of the quadrant. The complex conjugate is z̄ = x – iy; geometrically it is a reflection across the real axis.

z 的模为 r = |z| = √(x² + y²)，表示到原点的距离。辐角 θ = arg(z) 满足 tanθ = y/x，注意所在象限。复共轭 z̄ = x – iy；几何上它是关于实轴的反射。

• Key fact: z × z̄ = |z|² (always real and non‑negative).
• 关键事实：z × z̄ = |z|²（总是实数且非负）。

2. Modulus–Argument Form & de Moivre’s Theorem | 模-辐角形式与棣莫弗定理

The modulus–argument form is z = r(cosθ + i sinθ). This representation simplifies multiplication and division: for z₁ = r₁(cosθ₁ + i sinθ₁) and z₂ = r₂(cosθ₂ + i sinθ₂), we have z₁z₂ = r₁r₂[cos(θ₁+θ₂) + i sin(θ₁+θ₂)] and z₁/z₂ = (r₁/r₂)[cos(θ₁-θ₂) + i sin(θ₁-θ₂)].

模-辐角形式为 z = r(cosθ + i sinθ)。这种表示法简化了乘法和除法：对于 z₁ = r₁(cosθ₁ + i sinθ₁) 和 z₂ = r₂(cosθ₂ + i sinθ₂)，有 z₁z₂ = r₁r₂[cos(θ₁+θ₂) + i sin(θ₁+θ₂)]，z₁/z₂ = (r₁/r₂)[cos(θ₁-θ₂) + i sin(θ₁-θ₂)]。

de Moivre’s theorem: [r(cosθ + i sinθ)]ⁿ = rⁿ[cos(nθ) + i sin(nθ)] for any integer n. This is especially useful for finding powers and roots of complex numbers.

棣莫弗定理：[r(cosθ + i sinθ)]ⁿ = rⁿ[cos(nθ) + i sin(nθ)] 对任意整数 n 成立。这对计算复数的幂和根特别有用。

To find the nᵗʰ roots of a complex number w: write w in modulus–argument form, then the roots are given by zₖ = r^(1/n)[cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)], k = 0, 1, …, n-1. The roots lie evenly spaced on a circle of radius r^(1/n).

求复数 w 的 n 次方根：将 w 写成模-辐角形式，则根为 zₖ = r^(1/n)[cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)]，k = 0, 1, …, n-1。这些根均匀分布在半径为 r^(1/n) 的圆周上。

3. Roots of Polynomials | 多项式根的关系

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum α+β = -b/a and the product αβ = c/a. For a cubic ax³ + bx² + cx + d = 0 with roots α, β, γ, we have Σα = -b/a, Σαβ = c/a, and αβγ = -d/a. These relationships hold even when roots are complex.

对于二次方程 ax² + bx + c = 0，根为 α 和 β，和 α+β = -b/a，积 αβ = c/a。对于三次方程 ax³ + bx² + cx + d = 0，根为 α, β, γ，有 Σα = -b/a，Σαβ = c/a，αβγ = -d/a。这些关系即使根是复数时也成立。

Similar symmetric sums can be written for quartic equations. Typical AS problems ask you to find expressions like α²+β², (α-β)², or to construct a new polynomial with roots such as α², 1/α, α+β, etc. Use the known sums and products to avoid finding the roots explicitly.

对于四次方程也能写出类似的对称和。典型的 AS 考题要求你求出诸如 α²+β²、(α-β)² 的表达式，或者构造以 α²、1/α、α+β 等为新根的多项式。利用已知的和与积，避免显式求出根。

4. Matrices – Operations & Determinants | 矩阵运算与行列式

A matrix is a rectangular array of numbers. For square matrices, the determinant det(A) = |A| is a scalar that tells us if a matrix is invertible. For a 2×2 matrix A = [[a, b], [c, d]], det(A) = ad – bc.

矩阵是一个矩形的数字阵列。对于方阵，行列式 det(A) = |A| 是一个标量，用来判断矩阵是否可逆。对于 2×2 矩阵 A = [[a, b], [c, d]]，det(A) = ad – bc。

For a 3×3 matrix, use expansion by minors or the Sarrus pattern. Matrix addition and scalar multiplication are entry‑wise. Matrix multiplication AB is defined when the number of columns of A equals the number of rows of B; the product is calculated by taking the dot products of rows of A with columns of B.

对于 3×3 矩阵，使用子式展开或 Sarrus 法则。矩阵加法和数乘按元素进行。矩阵乘法 AB 在 A 的列数等于 B 的行数时有定义；乘积通过 A 的行与 B 的列点乘计算。

• Note: AB ≠ BA in general; matrix multiplication is not commutative.
• 注意：通常 AB ≠ BA；矩阵乘法不满足交换律。

The identity matrix I satisfies AI = IA = A.

单位矩阵 I 满足 AI = IA = A。

5. Solving Linear Systems & Inverse Matrices | 解线性方程组与逆矩阵

A system of linear equations can be written in matrix form Ax = b. If det(A) ≠ 0, the unique solution is x = A⁻¹b. The inverse A⁻¹ of a 2×2 matrix is (1/det(A))[[d, -b], [-c, a]]. For 3×3 matrices, the inverse can be calculated using the adjugate (transpose of the cofactor matrix) divided by the determinant.

线性方程组可以写成矩阵形式 Ax = b。若 det(A) ≠ 0，唯一解为 x = A⁻¹b。2×2 矩阵的逆为 (1/det(A))[[d, -b], [-c, a]]。对于 3×3 矩阵，逆可通过伴随矩阵（余子式矩阵的转置）除以行列式计算。

A singular matrix has det = 0 and either no solution or infinitely many solutions depending on the consistency of the system. Row operations (Gaussian elimination) provide an alternative method without finding the inverse.

奇异矩阵行列式为 0，方程组要么无解要么有无穷多解，取决于相容性。行变换（高斯消元法）提供了一种无需计算逆矩阵的替代方法。

6. Vectors – Scalar & Vector Products | 向量 – 点乘与叉乘

In 3-dimensional space, vectors can be written as a = a₁i + a₂j + a₃k. The scalar (dot) product a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cosθ, where θ is the angle between the vectors. It is used to test perpendicularity: a·b = 0 ⇔ vectors are orthogonal.

在三维空间中，向量可写为 a = a₁i + a₂j + a₃k。标量积（点乘）a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cosθ，其中 θ 为向量的夹角。它用于检验垂直：a·b = 0 ⇔ 向量正交。

The vector (cross) product a × b produces a vector perpendicular to both a and b. Its magnitude |a × b| = |a||b|sinθ gives the area of the parallelogram spanned by a and b. In component form: a × b = (a₂b₃ – a₃b₂)i – (a₁b₃ – a₃b₁)j + (a₁b₂ – a₂b₁)k.

向量积（叉乘）a × b 产生一个同时垂直于 a 和 b 的向量。其大小 |a × b| = |a||b|sinθ 给出由 a 和 b 张成的平行四边形的面积。分量形式：a × b = (a₂b₃ – a₃b₂)i – (a₁b₃ – a₃b₁)j + (a₁b₂ – a₂b₁)k。

7. Equations of Lines & Planes | 直线与平面方程

A line in 3D can be given in vector form r = a + λd, where a is a point on the line and d is a direction vector. Two lines are parallel if their direction vectors are scalar multiples; they intersect if we can solve for λ and μ when equating their parametric forms.

空间中的直线可用向量形式表示 r = a + λd，其中 a 是直线上一点，d 是方向向量。若两条直线的方向向量成比例，则它们平行；若令它们的参数形式相等可解出 λ 和 μ，则它们相交。

A plane can be written as r·n = p (scalar product form), where n is a normal vector perpendicular to the plane. Alternatively, r = a + λb + μc gives the plane in parametric form using two non‑parallel direction vectors b and c that lie in the plane.

平面可写为 r·n = p（标量积形式），其中 n 是垂直于平面的法向量。或者，r = a + λb + μc 给出平面的参数形式，其中 b 和 c 是平面内两个不平行的方向向量。

The angle between two planes equals the angle between their normal vectors; the angle between a line and a plane is the complement of the angle between the direction vector and the normal.

两个平面之间的夹角等于它们法向量之间的夹角；直线与平面之间的夹角是方向向量与法向量夹角的余角。

8. Summation of Series | 数列求和

Standard summations to memorise for AS Further Maths: Σ r = n(n+1)/2, Σ r² = n(n+1)(2n+1)/6, Σ r³ = n²(n+1)²/4. These are used to find sums of polynomial series, often combined with algebraic manipulation.

AS 进阶数学需记忆的标准求和公式：Σ r = n(n+1)/2，Σ r² = n(n+1)(2n+1)/6，Σ r³ = n²(n+1)²/4。这些用于求多项式级数的和，常结合代数变形。

For series like Σ (r+1)(r-3), expand the term and split the sum into separate sums using the standard results. The method of differences is also frequently tested: if a term can be written as f(r) – f(r+1) or similar, the series telescopes and an explicit expression for the sum to n terms can be found easily.

对于形如 Σ (r+1)(r-3) 的级数，展开项并利用标准结果将求和拆分成多个求和式。差分法也经常考查：若一项可写成 f(r) – f(r+1) 或类似形式，则级数可裂项相消，轻松求出前 n 项和的显式表达式。

9. Proof by Induction | 数学归纳法

Proof by induction is a powerful method for establishing that a statement P(n) holds for all positive integers n. The four standard steps: (1) Basis step – verify P(1) is true; (2) Inductive hypothesis – assume P(k) is true for some arbitrary k ≥ 1; (3) Inductive step – show that P(k+1) follows from P(k); (4) Conclusion – hence P(n) is true for all n by mathematical induction.

数学归纳法是一种证明命题 P(n) 对所有正整数 n 成立的强大方法。标准四步：(1) 基础步骤 – 验证 P(1) 为真；(2) 归纳假设 – 假设对于某个任意的 k ≥ 1，P(k) 成立；(3) 归纳递推 – 由 P(k) 推出 P(k+1) 成立；(4) 结论 – 因此由数学归纳法，P(n) 对所有 n 成立。

Typical AS induction problems involve summation of series, divisibility, matrix powers, and recurrence relations. For divisibility, show that if f(k) is divisible by some integer d, then f(k+1) is also divisible by d, often by expressing f(k+1) in terms of f(k) plus a multiple of d.

典型的 AS 归纳问题涉及级数求和、整除性、矩阵幂和递推关系。对于整除性，证明若 f(k) 能被某整数 d 整除，则 f(k+1) 也能被 d 整除，通常通过将 f(k+1) 表示为 f(k) 加上 d 的倍数来实现。

10. Hyperbolic Functions – Definitions & Identities | 双曲函数 – 定义与恒等式

Hyperbolic functions are defined using exponentials: sinh x = (eˣ – e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x. Unlike trigonometric functions, cosh²x – sinh²x = 1, not a sum.

双曲函数用指数定义：sinh x = (eˣ – e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x。与三角函数不同，恒等式为 cosh²x – sinh²x = 1，而非相加。

Key identities to remember: sinh(2x) = 2 sinh x cosh x, cosh(2x) = cosh²x + sinh²x = 2cosh²x – 1 = 2sinh²x + 1. The inverse hyperbolic functions can be expressed in log form: arsinh x = ln(x + √(x²+1)), arcosh x = ln(x + √(x²-1)), artanh x = ½ln((1+x)/(1-x)).

需牢记的关键恒等式：sinh(2x) = 2 sinh x cosh x，cosh(2x) = cosh²x + sinh²x = 2cosh²x – 1 = 2sinh²x + 1。反双曲函数可用对数形式表示：arsinh x = ln(x + √(x²+1))，arcosh x = ln(x + √(x²-1))，artanh x = ½ln((1+x)/(1-x))。

11. Differentiation & Integration of Hyperbolics | 双曲函数的微积分

Derivatives: d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x, d/dx (tanh x) = sech²x. Note that the derivative of cosh is sinh without a sign change, unlike the trig counterpart.

导数：d/dx (sinh x) = cosh x，d/dx (cosh x) = sinh x，d/dx (tanh x) = sech²x。注意 cosh 的导数是 sinh，符号不变，与三角函数不同。

Integrals mirror the derivatives: ∫ sinh x dx = cosh x + C, ∫ cosh x dx = sinh x + C. For integrals involving inverse hyperbolic functions, the recognition forms are often examined: ∫ 1/√(x²+a²) dx = arsinh(x/a) + C, ∫ 1/√(x²-a²) dx = arcosh(x/a) + C (x > a). These are alternatives to the trig substitution forms found in ordinary A‑level Maths.

积分与导数互逆：∫ sinh x dx = cosh x + C，∫ cosh x dx = sinh x + C。涉及反双曲函数的积分常考识别形式：∫ 1/√(x²+a²) dx = arsinh(x/a) + C，∫ 1/√(x²-a²) dx = arcosh(x/a) + C（x > a）。这些是普通 A‑level 数学中三角代换形式的替代方法。

12. Maclaurin Series & Further Techniques | 麦克劳林级数与进阶技巧

The Maclaurin series expansion of a function f(x) is given by f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … . Standard expansions to learn: eˣ = 1 + x + x²/2! + x³/3! + … ; sin x = x – x³/3! + x⁵/5! – … ; cos x = 1 – x²/2! + x⁴/4! – … ; ln(1+x) = x – x²/2 + x³/3 – … (valid for -1 < x ≤ 1).

函数 f(x) 的麦克劳林级数展开为 f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + …。需记住的标准展开：eˣ = 1 + x + x²/2! + x³/3! + …；sin x = x – x³/3! + x⁵/5! – …；cos x = 1 – x²/2! + x⁴/4! – …；ln(1+x) = x – x²/2 + x³/3 – …（有效范围 -1 < x ≤ 1）。

To find series for composite functions (e.g., sin²x, e⁻²ˣ), substitute into the standard expansions and expand up to the required power. The series can also be used to evaluate limits and approximate values.

求复合函数（如 sin²x, e⁻²ˣ）的级数时，代入标准展开式并按所需幂次展开。级数还可用于计算极限和近似值。

Another cross‑topic skill is combining series expansions with differential equations or integrals to obtain approximate solutions where an exact closed form is not required.

另一个跨主题技能是将级数展开与微分方程或积分相结合，在不需要精确闭合形式时获得近似解。

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