📚 Further Maths Core Pure 2 Key Concepts Explained | 进阶数学核心纯数2知识点精讲
Core Pure Mathematics 2 extends the techniques from Core Pure 1, introducing more advanced topics such as complex number transformations, hyperbolic functions, polar coordinates, second-order differential equations and matrix diagonalisation. This article provides a structured revision walkthrough of each major area, with clear explanations and practical examples that reflect typical exam-style applications.
核心纯数2 在核心纯数1 的基础上进一步拓展，引入了复数变换、双曲函数、极坐标、二阶微分方程和矩阵对角化等更高级的主题。本文将对每一个主要领域进行结构化复习讲解，配以清晰的解释和贴近真实考试情境的实例。
1. Complex Numbers and de Moivre’s Theorem | 复数与棣莫弗定理
Express a complex number in modulus-argument form z = r(cos θ + i sin θ), where r = |z| and θ = arg(z). De Moivre’s theorem states (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ), for any integer n. This allows efficient computation of powers and roots of complex numbers.
将复数表示为模-幅角形式 z = r(cos θ + i sin θ)，其中 r = |z|，θ = arg(z)。棣莫弗定理指出 (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)，对任意整数 n 成立。这可以高效地计算复数的幂与方根。
To find the n-th roots of a complex number w, express w in polar form, then the k-th root is zₖ = r^(1/n)[cos((θ+2πk)/n) + i sin((θ+2πk)/n)], k = 0,1,…,n-1. The roots lie on a circle of radius r^(1/n) and are equally spaced in argument.
求复数 w 的 n 次方根时，先将 w 写成极坐标形式，然后第 k 个根为 zₖ = r^(1/n)[cos((θ+2πk)/n) + i sin((θ+2πk)/n)]，k = 0,1,…,n-1。这些根分布在半径为 r^(1/n) 的圆上，且幅角等距分布。
Complex number transformations such as w = z + α (translation), w = kz (enlargement/rotation), and w = 1/z (inversion) are examined geometrically. The mapping w = z² transforms a circle into a cardioid-like shape, useful for understanding conformal mappings.
复数变换如 w = z + α（平移）、w = kz（缩放/旋转）和 w = 1/z（反演）需要从几何上分析。映射 w = z² 会将圆变为类心形线，有助于理解共形映射。
2. Loci in the Complex Plane | 复平面上的轨迹
The equation |z – a| = r describes a circle centre a, radius r. The locus |z – a| = |z – b| represents the perpendicular bisector of the line segment joining a and b. Arg(z – a) = θ represents a ray from a, making angle θ with the positive real axis.
方程 |z – a| = r 描述以 a 为圆心、r 为半径的圆。轨迹 |z – a| = |z – b| 表示连接 a 与 b 的线段的垂直平分线。Arg(z – a) = θ 表示从 a 出发的射线，与正实轴夹角为 θ。
Regions defined by inequalities like |z – 3| < 2 or Arg(z) > π/4 should be sketched and their intersections identified. Combining modulus and argument constraints creates sectors, annuli and other standard regions.
由不等式定义的区域如 |z – 3| < 2 或 Arg(z) > π/4 需要画出并求出交集。将模和幅角条件结合可以构成扇形、圆环等标准区域。
3. Hyperbolic Functions | 双曲函数
The hyperbolic sine and cosine are defined by cosh x = (eˣ + e⁻ˣ)/2, sinh x = (eˣ – e⁻ˣ)/2. They satisfy cosh² x – sinh² x = 1. The derivative of sinh x is cosh x, and derivative of cosh x is sinh x.
双曲正弦与双曲余弦定义为 cosh x = (eˣ + e⁻ˣ)/2，sinh x = (eˣ – e⁻ˣ)/2。它们满足 cosh² x – sinh² x = 1。sinh x 的导数是 cosh x，cosh x 的导数是 sinh x。
Inverse hyperbolic functions can be expressed in logarithmic form: arsinh x = ln(x + √(x² + 1)), arcosh x = ln(x + √(x² – 1)), artanh x = (1/2)ln((1+x)/(1-x)), |x| < 1. These are derived by solving quadratic equations in eⁱ.
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 = (1/2)ln((1+x)/(1-x))，|x| < 1。这些可以通过解关于 eⁱ 的二次方程推导。
Identities linking hyperbolic to trigonometric functions: cosh(ix) = cos x, sinh(ix) = i sin x. This connection helps integrate certain forms and solve differential equations with constant coefficients.
双曲函数与三角函数的恒等式：cosh(ix) = cos x，sinh(ix) = i sin x。这一联系有助于积分某些形式并求解常系数微分方程。
4. Polar Coordinates | 极坐标
In polar coordinates, a curve is given by r = f(θ). The area enclosed by a polar curve from θ = α to θ = β is A = ½ ∫[α,β] r² dθ. The arc length is s = ∫[α,β] √(r² + (dr/dθ)²) dθ.
在极坐标中，曲线由 r = f(θ) 给出。极曲线从 θ = α 到 θ = β 围成的面积为 A = ½ ∫[α,β] r² dθ。弧长为 s = ∫[α,β] √(r² + (dr/dθ)²) dθ。
Common polar curves include cardioid r = a(1 + cos θ), circle r = a, rose curves r = a sin(nθ), and lemniscate r² = a² cos 2θ. Symmetry tests (about θ = 0, θ = π/2, pole) help reduce computation.
常见极曲线包括心形线 r = a(1 + cos θ)、圆 r = a、玫瑰线 r = a sin(nθ) 和双纽线 r² = a² cos 2θ。对称性测试（关于 θ = 0、θ = π/2、极点）有助于减少计算量。
Tangents to polar curves: the gradient is given by dy/dx = (r’ sin θ + r cos θ)/(r’ cos θ – r sin θ), where r’ = dr/dθ. Parallel and perpendicular tangents are found by setting numerator or denominator to zero.
极曲线的切线：梯度由 dy/dx = (r’ sin θ + r cos θ)/(r’ cos θ – r sin θ) 给出，其中 r’ = dr/dθ。平行或垂直切线可通过令分子或分母为零求出。
5. Series and the Method of Differences | 级数与差分法
The method of differences simplifies sums of the form Σ(uᵣ – uᵣ₊₁) or similar. When terms cancel in successive pairs, the sum reduces to the first few and last few terms. Standard forms like Σ 1/(r(r+1)) can be split using partial fractions and then summed.
差分法可简化形如 Σ(uᵣ – uᵣ₊₁) 的和。当相邻项成对抵消时，总和简化为前几项与最后几项。标准形式如 Σ 1/(r(r+1)) 可通过部分分式分解再求和。
Summations for r from 1 to n: Σ r = n(n+1)/2, Σ r² = n(n+1)(2n+1)/6, Σ r³ = [n(n+1)/2]². These are used alongside method of differences to evaluate finite series.
从 r = 1 到 n 的求和公式：Σ r = n(n+1)/2，Σ r² = n(n+1)(2n+1)/6，Σ r³ = [n(n+1)/2]²。这些与差分法结合使用来计算有限级数。
6. Maclaurin Series | 麦克劳林级数
A function f(x) can be expanded as a Maclaurin series: f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … . Common series include eˣ = Σ x^r/r!, sin x = Σ (-1)^r x^(2r+1)/(2r+1)!, cos x = Σ (-1)^r x^(2r)/(2r)!, ln(1+x) = Σ (-1)^(r-1) x^r/r.
函数 f(x) 可展开为麦克劳林级数：f(x) = f(0) + f'(0)x + f”(0)x²/2! + f”'(0)x³/3! + … 。常见级数包括 eˣ = Σ x^r/r!，sin x = Σ (-1)^r x^(2r+1)/(2r+1)!，cos x = Σ (-1)^r x^(2r)/(2r)!，ln(1+x) = Σ (-1)^(r-1) x^r/r。
Composite functions, such as e^(sin x) or tan⁻¹ x, require repeated differentiation. Alternatively, series can be combined: for example, the series for (1+x)^k (binomial) is valid for |x| < 1, giving (1+x)^k = 1 + kx + k(k-1)x²/2! + … .
复合函数如 e^(sin x) 或 tan⁻¹ x 需要反复求导。或者可以组合级数：例如 (1+x)^k（二项式）的级数在 |x| < 1 时有效，即 (1+x)^k = 1 + kx + k(k-1)x²/2! + … 。
7. First- and Second-Order Differential Equations | 一阶与二阶微分方程
First-order linear differential equations of the form dy/dx + P(x)y = Q(x) are solved using an integrating factor I = e^(∫P dx). Multiply through by I, then the left side becomes d/dx(Iy).
形如 dy/dx + P(x)y = Q(x) 的一阶线性微分方程使用积分因子 I = e^(∫P dx) 求解。方程两边乘以 I 后，左边变为 d/dx(Iy)。
Second-order homogeneous linear equations with constant coefficients: a d²y/dx² + b dy/dx + c y = 0. The auxiliary equation a m² + b m + c = 0 gives roots m₁, m₂. If real distinct, y = A e^(m₁ x) + B e^(m₂ x); if repeated, y = (A + Bx)e^(m x); if complex m = α ± iβ, y = e^(αx)(A cos βx + B sin βx).
常系数二阶齐次线性方程：a d²y/dx² + b dy/dx + c y = 0。辅助方程 a m² + b m + c = 0 的根为 m₁, m₂。若为不等实根，y = A e^(m₁ x) + B e^(m₂ x)；重根时 y = (A + Bx)e^(m x)；若为复根 m = α ± iβ，y = e^(αx)(A cos βx + B sin βx)。
Non-homogeneous equations: find the complementary function (CF) and a particular integral (PI). For PI, try a form based on the RHS function: polynomial, exponential, trigonometric, or a combination. Substitute to determine coefficients.
非齐次方程：求余函数（CF）和特解（PI）。PI 的形式尝试取决于右侧函数：多项式、指数、三角或组合。代入确定系数。
8. Eigenvalues and Eigenvectors | 特征值与特征向量
For a square matrix A, an eigenvector v satisfies A v = λ v, where λ is the eigenvalue. Solve det(A – λ I) = 0 to find eigenvalues. For each λ, solve (A – λ I)v = 0 to find the corresponding eigenvectors.
对于方阵 A，特征向量 v 满足 A v = λ v，其中 λ 是特征值。解 det(A – λ I) = 0 求特征值。对每一个 λ，解 (A – λ I)v = 0 得到对应的特征向量。
When a 3×3 matrix has three distinct eigenvalues, it is diagonalisable: P⁻¹ A P = D, where D is diagonal matrix of eigenvalues and P is the matrix of corresponding eigenvectors as columns. This diagonal form simplifies powers of A.
当一个 3×3 矩阵有三个相异的特征值时，它可对角化：P⁻¹ A P = D，其中 D 是对角特征值矩阵，P 是由对应的特征向量作为列构成的矩阵。这种对角形式可简化 A 的幂计算。
For symmetric matrices, eigenvectors corresponding to distinct eigenvalues are orthogonal. This property is essential in quadratic forms and principal axis transformations.
对于对称矩阵，对应于不同特征值的特征向量互相正交。这一性质在二次型和主轴变换中至关重要。
9. Matrix Transformation and Invariant Lines | 矩阵变换与不变线
A linear transformation in the plane is represented by a 2×2 matrix M. A line through the origin is invariant if points on it map to points on the same line. This occurs when a direction vector v satisfies M v = μ v for some scalar μ, i.e., v is an eigenvector.
平面上的线性变换由一个 2×2 矩阵 M 表示。如果一条过原点的直线上的点映射后仍在该直线上，则这条线是不变的。这发生在方向向量 v 满足 M v = μ v（μ 为标量）时，即 v 是一个特征向量。
Lines of invariant points satisfy M x = x for all x on the line. That means every point on the line is fixed, which requires (M – I)x = 0 to have a line of solutions.
不变点线满足对线上所有 x 有 M x = x。这意味着线上每一点都是固定点，这就要求 (M – I)x = 0 有一整条线的解。
To find invariant lines of the form y = mx, substitute (x, mx) into the transformation and require the image coordinates to satisfy the same ratio y’/x’ = m. Solve the resulting quadratic for m.
求形如 y = mx 的不变线时，将 (x, mx) 代入变换，并要求像坐标满足相同的比值 y’/x’ = m。解所得关于 m 的二次方程。
10. Volumes of Revolution and Further Integration | 旋转体体积与进阶积分
Volume of revolution about x-axis: V = π ∫[a,b] y² dx. About y-axis: V = π ∫[c,d] x² dy. For polar curves, the volume generated by rotating the area about the initial line is V = (2π/3) ∫ r³ sin θ dθ (special formula).
绕 x 轴旋转体积：V = π ∫[a,b] y² dx。绕 y 轴：V = π ∫[c,d] x² dy。对于极曲线，将面积绕初始线旋转生成的体积有专门公式 V = (2π/3) ∫ r³ sin θ dθ。
Mean value of a function f(x) over [a,b] is M = (1/(b-a)) ∫[a,b] f(x) dx. Root mean square (RMS) value is √[ (1/(b-a)) ∫[a,b] f(x)² dx ]. These appear in applied contexts.
函数 f(x) 在区间 [a,b] 上的平均值为 M = (1/(b-a)) ∫[a,b] f(x) dx。均方根值（RMS）为 √[ (1/(b-a)) ∫[a,b] f(x)² dx ]。这在应用题中常见。
Integration using the substitution t = tan(x/2) converts trigonometric integrals with sin x and cos x into rational functions. Let sin x = 2t/(1+t²), cos x = (1-t²)/(1+t²), dx = 2 dt/(1+t²).
利用替换 t = tan(x/2) 可将含有 sin x 和 cos x 的三角积分化为有理函数。令 sin x = 2t/(1+t²)，cos x = (1-t²)/(1+t²)，dx = 2 dt/(1+t²)。
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