WJEC A-Level Further Maths Year 13 Formula & Theorem Quick Reference Handbook | WJEC A-Level 进阶数学 Year 13 公式定理速查手册

📚 WJEC A-Level Further Maths Year 13 Formula & Theorem Quick Reference Handbook | WJEC A-Level 进阶数学 Year 13 公式定理速查手册

This comprehensive revision guide brings together all the essential formulae, theorems, and key results you will need for the Year 13 WJEC Further Mathematics examination. Designed for rapid reference, each section pairs concise English explanations with Chinese translations, covering Further Pure Mathematics core topics as well as some key results from optional applications. All mathematical expressions are presented using standard HTML and Unicode characters, with no reliance on LaTeX, ensuring clear and accurate display in any digital context.

这本全面的复习速查手册汇集了你在 Year 13 WJEC 进阶数学考试中所需的所有重要公式、定理和关键结论。手册专为快速查阅而设计，每个部分都配有简洁的英文解释与中文翻译，内容涵盖进阶纯数学的核心主题以及部分选修应用的关键结果。所有数学表达式均使用标准 HTML 和 Unicode 字符呈现，完全不依赖 LaTeX，确保在任何数字环境中都能清晰准确地显示。

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1. Complex Numbers | 复数

Cartesian form z = x + iy, polar form z = r(cos θ + i sin θ), or exponential form z = r e^iθ, where r = |z| and θ = arg(z).

复数可以用笛卡儿形式 z = x + iy、极坐标形式 z = r(cos θ + i sin θ) 或指数形式 z = r e^iθ 表示，其中 r = |z|，θ = arg(z)。

De Moivre's Theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)

棣莫弗定理: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)

Euler's formula links the exponential and trigonometric functions: e^iθ cos θ + i sin θ, which allows compact representation of complex multiplication and powers.

欧拉公式将指数函数与三角函数联系起来：e^iθ = cos θ + i sin θ，这使得复数的乘法和幂运算可以用紧凑的形式表示。

The n^th roots of a complex number z = r(cos θ + i sin θ) are given by z_k = r^1/n[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)] for k = 0, 1, …, n−1.

复数 z = r(cos θ + i sin θ) 的 n 次方根由 z_k = r^1/n[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)] 给出，其中 k = 0, 1, …, n−1。

The conjugate of z = x + iy is z* = x − iy, and |z|² = z z*.

复数 z = x + iy 的共轭为 z* = x − iy，且 |z|² = z z*。

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2. Hyperbolic Functions | 双曲函数

The fundamental hyperbolic functions are defined by sinh x = (e^x − e^−x)/2, cosh x = (e^x + e^−x)/2, and tanh x = sinh x / cosh x.

基本的双曲函数定义为 sinh x = (e^x − e^−x)/2，cosh x = (e^x + e^−x)/2，tanh x = sinh x / cosh x。

Osborn's Rule: To convert a trigonometric identity, replace cos with cosh and sin with i sinh, then flip the sign of any term containing sinh².

奥斯本法则：转换三角恒等式时，将 cos 替换为 cosh，sin 替换为 i sinh，并翻转任何含有 sinh² 的项的符号。

inh 2x = 2 sinh x cosh x, and cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x.

关键的恒等式包括 cosh² x − sinh² x = 1，sinh 2x = 2 sinh x cosh x，以及 cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x。

Inverse hyperbolic functions can be expressed in logarithmic form: arsinh x = ln(x + √x² + 1), arcosh x = ln(x + √x² − 1) for x≥1, artanh x = ½ ln((1+x)/(1−x)) for |x|<1.

反双曲函数可以用对数形式表示：arsinh x = ln(x + √x² + 1)，arcosh x = ln(x + √x² − 1)（x≥1），artanh x = ½ ln((1+x)/(1−x))（|x|<1）。

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3. Polar Coordinates | 极坐标

pole O and θ is the angle from the initial line. To convert: x = r cos θ, y = r sin θ, r = √x² + y².

点 P 用 (r, θ) 描述，其中 r 是到极点 O 的距离，θ 是从极轴测量的角度。坐标转换：x = r cos θ，y = r sin θ，r = √x² + y²。

Area in polar form: A = ½ ∫_α^β r² dθ

极坐标下的面积: A = ½ ∫_α^β r² dθ

The arc length of a curve r = f(θ) from θ = α to θ = β is s = ∫_α^β √(r² + (dr/dθ)²) dθ.

曲线 r = f(θ) 从 θ = α 到 θ = β 的弧长为 s = ∫_α^β √(r² + (dr/dθ)²) dθ。

For a surface of revolution about the initial line (θ = 0), the surface area is S = 2π ∫_α^β r sin θ √(r² + (dr/dθ)²) dθ.

曲线绕极轴旋转生成的曲面面积为 S = 2π ∫_α^β r sin θ √(r² + (dr/dθ)²) dθ。

d r = a(1+cos θ), limacon, rose curves r = a cos(nθ) or r = a sin(nθ), and spirals r = aθ.

常见的极坐标曲线包括心形线 r = a(1+cos θ)、蚶线、玫瑰线 r = a cos(nθ) 或 r = a sin(nθ)，以及螺线 r = aθ。

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4. Matrices and Linear Transformations | 矩阵与线性变换

= ad − bc. A is singular if det A = 0; the inverse is A⁻¹ = (1/det A) [[d, −b], [−c, a]].

对于 2×2 矩阵 A = [[a, b], [c, d]]，行列式为 det A = ad − bc。若 det A = 0，则 A 是奇异的；逆矩阵为 A⁻¹ = (1/det A) [[d, −b], [−c, a]]。

Eigenvalues: solve det(A − λI) = 0. For each λ, eigenvectors v satisfy (A − λI)v = 0.

特征值：解 det(A − λI) = 0。对于每个 λ，特征向量 v 满足 (A − λI)v = 0。

[1,0]]), and stretches. The product of two matrices represents their combined transformation.

矩阵变换包括旋转 [[cos θ, −sin θ], [sin θ, cos θ]]、反射（例如关于 y = x 的反射：[[0,1],[1,0]]）和伸缩。两个矩阵的乘积表示其复合变换。

For 3×3 matrices, determinants can be evaluated by Sarrus' rule or by expanding a row/column. The vector product can be written using a determinant.

对于 3×3 矩阵，行列式可用萨鲁斯法则或按行/列展开计算。向量积可用行列式表示。

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5. Further Sequences and Series | 进阶数列与级数

The Maclaurin series expansions: e^x = Σ xⁿ/n!, sin x = Σ (−1)ⁿ x²ⁿ⁺¹/(2n+1)!, cos x = Σ (−1)ⁿ x²ⁿ/(2n)!, ln(1+x) = Σ (−1)ⁿ⁻¹ xⁿ/n (|x|<1).

麦克劳林级数展开：e^x = Σ xⁿ/n!，sin x = Σ (−1)ⁿ x²ⁿ⁺¹/(2n+1)!，cos x = Σ (−1)ⁿ x²ⁿ/(2n)!，ln(1+x) = Σ (−1)ⁿ⁻¹ xⁿ/n（|x|<1）。

General Maclaurin series: f(x) = Σ f⁽ⁿ⁾(0) xⁿ/n!

一般麦克劳林级数：f(x) = Σ f⁽ⁿ⁾(0) xⁿ/n!

Method of differences: to sum Σ u_r, express u_r = v_r − v_r+1 or similar, so that the sum telescopes to v₁ − v_n+1.

差分法：求和 Σ u_r 时，将 u_r 表示为 v_r − v_r+1 等形式，使得求和可裂项化简为 v₁ − v_n+1。

Sum of natural numbers: Σ r = n(n+1)/2, Σ r² = n(n+1)(2n+1)/6, Σ r³ = n²(n+1)²/4.

自然数求和公式：Σ r = n(n+1)/2，Σ r² = n(n+1)(2n+1)/6，Σ r³ = n²(n+1)²/4。

in WJEC applied modules; basic forms are f(x) = a₀/2 + Σ [aₙ cos(nπx/L) + bₙ sin(nπx/L)].

部分应用模块可能需要傅里叶正弦和余弦级数；其基本形式为 f(x) = a₀/2 + Σ [aₙ cos(nπx/L) + bₙ sin(nπx/L)]。

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6. Proof by Induction | 数学归纳法

how (i) base case P(n₀) true, (ii) inductive step: assume P(k) true and prove P(k+1) true.

原理：要证明对所有整数 n ≥ n₀ 命题 P(n) 成立，需证 (i) 基础情形 P(n₀) 为真，(ii) 归纳步骤：假设 P(k) 为真，证明 P(k+1) 为真。

Key types: summation, divisibility, matrices (powers), recurrence relations, inequalities.

常见类型：求和、整除性、矩阵幂、递推关系、不等式。

a·f(k) + something clearly divisible, or use modulo arithmetic within the induction step.

对于整除性，通常需证明 f(k+1) = a·f(k) + 另一个显然可整除的部分，或在归纳步骤中使用模运算。

The inductive step for inequalities often requires showing that if P(k) holds, then an inequality for P(k+1) follows by logical chain, e.g., using the assumption to bound expressions.

不等式的归纳步骤通常要求：若 P(k) 成立，则通过逻辑链（例如利用归纳假设对表达式进行放缩）推导出 P(k+1) 的不等式。

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7. Differential Equations | 微分方程

First-order separable: dy/dx = g(x)h(y) ⇒ ∫ 1/h(y) dy = ∫ g(x) dx. Do not forget the constant of integration.

一阶可分离变量方程：dy/dx = g(x)h(y) ⇒ ∫ 1/h(y) dy = ∫ g(x) dx。不要忘记积分常数。

Integrating factor for y' + P(x)y = Q(x): I.F. = e^{∫ P(x) dx}, then d/dx (y · I.F.) = Q(x) · I.F.

一阶线性方程 y' + P(x)y = Q(x) 的积分因子：I.F. = e^{∫ P(x) dx}，然后 d/dx (y · I.F.) = Q(x) · I.F.

+ b dy/dx + c y = 0. Solve auxiliary equation am² + bm + c = 0. Real distinct roots m₁, m₂ ⇒ y = A e^m₁x + B e^m₂x; repeated root m ⇒ y = (A + Bx) e^mx; complex roots α ± iβ ⇒ y = e^αx(A cos βx + B sin βx).

二阶常系数齐次线性方程：a d²y/dx² + b dy/dx + c y = 0。解辅助方程 am² + bm + c = 0。相异实根 m₁, m₂ ⇒ y = A e^m₁x + B e^m₂x；重根 m ⇒ y = (A + Bx) e^mx；复根 α ± iβ ⇒ y = e^αx(A cos βx + B sin βx)。

integral. Standard P.I. forms: for f(x) = polynomial, try polynomial of same degree; for e^kx, try Ce^kx (or C x e^kx if k is root of A.E.); for sin/ cos, try p sin ωx + q cos ωx.

对于非齐次方程 a d²y/dx² + b dy/dx + c y = f(x)，通解为 y = 补充函数 + 特解。标准特解形式：若 f(x) 为多项式，则尝试同次多项式；对于 e^kx，尝试 Ce^kx（若 k 是辅助方程的单根，则乘 x）；对于 sin/ cos，尝试 p sin ωx + q cos ωx。

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8. Further Calculus | 进阶微积分

Reduction formulae: To integrate I_n = ∫ f_n(x) dx, relate I_n to I_n−1 or I_n−2 using integration by parts. Example: I_n = ∫ sinⁿ x dx = −(1/n) sinⁿ⁻¹ x cos x + ((n−1)/n) I_n−2.

递推公式：对于积分 I_n = ∫ f_n(x) dx，利用分部积分法将 I_n 与 I_n−1 或 I_n−2 联系起来。例如 I_n = ∫ sinⁿ x dx = −(1/n) sinⁿ⁻¹ x cos x + ((n−1)/n) I_n−2。

Arc length for y = f(x): s = ∫_a^b √(1 + (dy/dx)²) dx. For parametric x(t), y(t): s = ∫ √( (dx/dt)² + (dy/dt)² ) dt.

弧长：y = f(x) 时 s = ∫_a^b √(1 + (dy/dx)²) dx；参数形式 x(t), y(t) 时 s = ∫ √( (dx/dt)² + (dy/dt)² ) dt。

Volume of revolution about x-axis: V = π ∫_a^b (dx/dt) dt. Surface area of revolution: S = 2π ∫ y ds, where ds is the arc length element.

绕 x 轴旋转体的体积：V = π ∫_a^b y² dx；参数形式下 V = π ∫ y² (dx/dt) dt。旋转体表面积：S = 2π ∫ y ds，其中 ds 为弧长微元。

Improper integrals with infinite limit: if lim_R→∞ ∫_a^R f(x) dx exists finite, the integral converges; otherwise diverges. Use similar limiting process for unbounded integrands.

无穷限的反常积分：若 lim_R→∞ ∫_a^R f(x) dx 存在有限值，则积分收敛；否则发散。对于无界被积函数也用极限类似处理。

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9. Vectors | 向量

Scalar (dot) product: a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃. Used to find angles and projections.

数量积（点积）：a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。用于求夹角和投影。

Vector (cross) product: a × b = |a||b| sin θ n̂, orthogonal to both a and b; magnitude gives area of parallelogram.

向量积（叉积）：a × b = |a||b| sin θ n̂，垂直于 a 和 b；其模等于以 a、b 为边的平行四边形面积。

ction vector. For a plane: r · n = d (normal form) or r = a + λ u + μ v (parametric form).

直线的向量方程：r = a + λ d，其中 a 是直线上一点，d 是方向向量。平面的向量方程：r · n = d（法线式）或 r = a + λ u + μ v（参数形式）。

Shortest distance from a point P with position vector p to a plane r · n̂ = k is |p · n̂ − k|. From a point to a line r = a + λ d, distance = |(p − a) × d| / |d|.

点 P（位置向量 p）到平面 r · n̂ = k 的最短距离为 |p · n̂ − k|。点到直线 r = a + λ d 的距离 = |(p − a) × d| / |d|。

kew. Intersection of two planes: solve simultaneously; if parallel, they do not intersect.

两直线的交点：解 r = a + λ d = b + μ e，若无解则两直线异面。两平面的交线：联立方程求解；若平行，则不相交。

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10. Further Trigonometric Identities and Complex Applications | 进阶三角恒等式与复数应用

multiple-angle formulas: sin(A ± B) = sin A cos B ± cos A sin B, cos(A ± B) = cos A cos B ∓ sin A sin B, tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B).

复合角与倍角公式：sin(A ± B) = sin A cos B ± cos A sin B，cos(A ± B) = cos A cos B ∓ sin A sin B，tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)。

t²), tan θ = 2t/(1−t²) where t = tan(θ/2). Use for integration of rational trig functions.

万能代换：令 t = tan(θ/2)，则 sin θ = 2t/(1+t²)，cos θ = (1−t²)/(1+t²)，tan θ = 2t/(1−t²)。用于有理三角函数的积分。

cos P + cos Q = 2 cos((P+Q)/2) cos((P−Q)/2), cos P − cos Q = −2 sin((P+Q)/2) sin((P−Q)/2).

和差化积公式：sin P + sin Q = 2 sin((P+Q)/2) cos((P−Q)/2)，sin P − sin Q = 2 cos((

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