📚 A-Level CIE Further Mathematics: Quick Formula & Theorem Handbook | A-Level CIE 进阶数学：公式定理速查手册
This quick reference handbook compiles the essential formulas and theorems required for the CIE A-Level Further Mathematics syllabus. Use it as a handy revision tool to reinforce core concepts before examinations.
这本速查手册汇集了CIE A-Level进阶数学课程所需的核心公式和定理。它可以作为考前强化核心概念的便捷复习工具。
1. Complex Numbers | 复数
unit i, defined such that i² = −1. They are fundamental in advanced algebra and analysis.
复数通过引入虚数单位 i（满足 i² = −1）扩展了实数系统。它们在高等代数和分析中十分基础。
z = x + iy, Re(z) = x, Im(z) = y
The modulus of z is |z| = √x² + y² and the argument is θ = arg(z), defined by tan θ = y/x.
z 的模为 |z| = √x² + y²，辐角为 θ = arg(z)，由 tan θ = y/x 定义。
z = r(cos θ + i sin θ) = r eiθ
De Moivre' )ⁿ = cos(nθ) + i sin(nθ), a powerful tool for finding powers and roots of complex numbers.
棣莫弗定理指出 (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)，这是求复数的幂和根的有力工具。
The complex conjugate is z* = x − iy. Note that z z* = |z| )* = z. The roots of polynomial equations with real coefficients occur in conjugate pairs.
复共轭为 z* = x − iy。注意 z z* = |z|² 且 (z*)* = z。实系数多项式方程的根总是成共轭对出现。
2. Matrices and Transformations | 矩阵与变换
Matrices are used to represent linear transformations, solve systems of linear equations, and model geometric operations. Key properties include determinant and inverse calculations.
矩阵用于表示线性变换、求解线性方程组以及建模几何操作。关键性质包括行列式和逆矩阵的计算。
For a 2×2 matrix M =
, det(M) = ad − bca b c d
The inverse M⁻¹ exists only if det(M) ≠ 0 and is given by (1/det(M))
| d | −b |
| −c | a |
仅当 det(M) ≠ 0 时存在逆矩阵 M⁻¹，其表达式为 (1/det(M))
| d | −b |
| −c | a |
satisfies Mv = λv. Diagonalisation is possible when eigenvectors are linearly independent.
特征值 λ 通过解 det(M − λI) = 0 求得。对于每个 λ，对应的特征向量 v 满足 Mv = λv。当特征向量线性无关时，矩阵可以对角化。
3. Vectors in 3D | 三维向量
Vectors in three-dimensional space describe quantities with both magnitude and direction. Operations include addition, scalar multiplication, dot product, and cross product.
三维空间中的向量描述了具有大小和方向的量。运算包括加法、数乘、点积和叉积。
a = a₁i + a₂j + a₃k, |a| = √a₁² + a₂² + a₃²
The dot product a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| s the angle between the vectors. It is used for projections and determining orthogonality.
点积 a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ，其中 θ 是两向量间的夹角。它用于投影和判断正交性。
The cross product a × b yields a vector perpendicular to both a and b, with magnitude |a||b| sin θ. In component form:
叉积 a × b 得到一个垂直于 a 和 b 的向量，其大小为 |a||b| sin θ。分量形式为：
a × b = (a₂b₃ − a₃b₂)i + (a₃b₁ − a₁b₃)j + (a₁b₂ − a₂b₁)k
s the direction vector. The plane is given by r · n = a · n, where n is the normal vector.
直线的向量方程为 r = a + t d，其中 a 是直线上一点，d 是方向向量。平面方程为 r · n = a · n，其中 n 是法向量。
4. Hyperbolic Functions | 双曲函数
Hyperbolic functions are analogues of trigonometric functions, defined using exponential functions. They arise in the study of hanging cables, relativity, and differential equations.
双曲函数是三角函数的类似物，通过指数函数定义。它们在悬链线、相对论和微分方程的研究中出现。
sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2
rigonometric identity. Their derivatives are d(sinh x)/dx = cosh x, d(cosh x)/dx = sinh x.
基本恒等式为 cosh²x − sinh²x = 1，类似于三角恒等式。它们的导数为 d(sinh x)/dx = cosh x，d(cosh x)/dx = sinh x。
Inverse hyperbolic functions can be expressed in logarithmic forms, for example:
反双曲函数可以用对数形式表达，例如：
arsinh x = ln(x + √x² + 1), arcosh x = ln(x + √x² − 1) (x ≥ 1)
These functions are essential for integrating certain forms like 1/√x² ± a² and 1/√a² − x² after appropriate substitutions.
这些函数对于积分诸如 1/√x² ± a² 和 1/√a² − x² 等特定形式至关重要，在适当代换后使用。
5. Polar Coordinates | 极坐标
Polar coordinates (r, θ) provide an alternative to Cartesian coordinates, using radial distance from the origin and angle from the positive x-axis. They are ideal for curves with circular symmetry.
极坐标 (r, θ) 提供了笛卡尔坐标的替代方案，使用到原点的径向距离以及与正 x 轴的夹角。它们非常适合具有圆对称性的曲线。
x = r cos θ, y = r sin θ, r = √x² + y²
The area bounded by a polar curve r = f(θ) between θ = α and θ = β is given by:
极坐标曲线 r = f(θ) 在 θ = α 和 θ = β 之间所围成的面积由下式给出：
A = ½ ∫αβ r² dθ
ioids (r = a(1 + cos θ)), limacons, and roses (r = a cos(nθ)). Arc length in polar form is s = ∫ √(r² + (dr/dθ)²) dθ.
常见的极坐标曲线包括心形线 (r = a(1 + cos θ))、蚌线和玫瑰线 (r = a cos(nθ))。极坐标形式的弧长为 s = ∫ √(r² + (dr/dθ)²) dθ。
6. Series and Summation | 级数与求和
Series expansions are used to approximate functions, sum sequences, and analyse convergence. Standard series include arithmetic, geometric, and power series.
级数展开用于近似函数、求和序列以及分析收敛性。标准级数包括算术级数、几何级数和幂级数。
Arithmetic: Sₙ = n/2 [2a + (n−1)d]; Geometric: Sₙ = a(1 − rⁿ)/(1 − r), S∞ = a/(1 − r) for |r| < 1
f(r) − f(r+1)] into f(1) − f(n+1). It is frequently used to prove results by cancellation.
差分法将形如 ∑ [f(r) − f(r+1)] 的和化简为 f(1) − f(n+1)。它常被用于通过消去法证明结果。
The binomial expansion for (1+x)ⁿ, where n is rational, is valid for |x| < 1:
对于有理数 n，(1+x)ⁿ 的二项展开式在 |x| < 1 时有效：
(1+x)ⁿ = 1 + nx + n(n−1)x²/2! + n(n−1)(n−2)x³/3! + …
7. Differential Equations | 微分方程
Ordinary differential equations (ODEs) model dynamic systems. First-order linear ODEs can be solved using an integrating factor.
常微分方程 (ODE) 用于建模动态系统。一阶线性 ODE 可使用积分因子求解。
For dy/dx + P(x) y = Q(x), the integrating factor is μ = e∫P dx
Second-order linear ODEs with constant coefficients are solved via the auxiliary equation.
乘以 μ 后，左侧变为全导数：d(μ y)/dx = μ Q(x)。常系数二阶线性 ODE 通过辅助方程求解。
For a d²y/dx² + b dy/dx + c y = 0, auxiliary: am² + bm + c = 0
- Distinct real roots m₁, m₂: y = A em₁x + B em₂x
- Repeated root m: y = (A + Bx) emx
- Complex roots α ± iβ: y = eαx(A cos βx + B sin βx)
For the non-homogeneous case, the particular integral is found using the form of the RHS (polynomial, exponential, trigonometric) and added to the complementary function.
对于非齐次情况，特解通过 RHS 的形式（多项式、指数、三角函数）求得，并加到补函数上。
- Different real roots m₁, m₂: y = A em₁x + B em₂x
- 等根 m: y = (A + Bx) emx
- 复根 α ± iβ: y = eαx(A cos βx + B sin βx)
8. Maclaurin Series | 麦克劳林级数
expresses a function as an infinite sum of terms calculated from its derivatives at zero.
麦克劳林级数是函数在 x = 0 处的泰勒展开。它将函数表示为根据其在零点的导数计算的无穷项之和。
f(x) = f(0) + f'(0) x + f''(0) x²/2! + f'''(0) x³/3! + …
Key standard expansions include eˣ, sin x, cos x, ln(1+x), and (1+x)ⁿ. Their general terms are often tested.
关键的标准化展开包括 eˣ、sin x、cos x、ln(1+x) 和 (1+x)ⁿ。它们的一般项常被考查。
| Function | Maclaurin Series | Validity |
|---|---|---|
| eˣ | ∑ xⁿ/n! | all x |
| sin x | ∑ (−1)ⁿ x²ⁿ⁺¹/(2n+1)! | all x |
| cos x | ∑ (−1)ⁿ x²ⁿ/(2n)! | all x |
| ln(1+x) | ∑ (−1)ⁿ⁻¹ xⁿ/n | −1 < x ≤ 1 |
These expansions can be composited, e.g., eˣ sin x can be approximated by multiplying respective series up to required powers.
这些展开可以复合，例如，eˣ sin x 可以通过将各自级数乘到所需次幂来近似。
9. Proof by Induction | 归纳法证明
Mathematical induction is a rigorous method for proving statements that are true for all positive integers. It consists of a base case, an inductive hypothesis, and an inductive step.
数学归纳法是一种严格的证明方法，用于证明对所有正整数成立的命题。它包含基础情形、归纳假设和归纳步骤。
Step 1: Prove statement true for n = 1 (base case).
Step 2: Assume true for n = k (inductive hypothesis).
Step 3: Prove true for n = k+1 using the hypothesis.
Common induction proofs involve summation formulas, divisibility, matrix powers, and inequalities. The key is to manipulate the (k+1) expression to reveal the assumed form for k.
常见的归纳证明包括求和公式、整除性、矩阵幂和不等式。关键是将 (k+1) 表达式变形，以揭示假设的 k 形式。
步骤一：证明 n = 1 时命题成立（基础情形）。
步骤二：假设 n = k 时命题成立（归纳假设）。
步骤三：利用假设证明 n = k+1 时命题成立。
visor. For matrix powers Mᵏ, one often shows Mᵏ⁺¹ = M·Mᵏ and substitutes the assumed form.
对于整除性，将 f(k+1) 表示为 f(k) 加上除数的倍数。对于矩阵幂 Mᵏ，通常证明 Mᵏ⁺¹ = M·Mᵏ 并代入假设的形式。
10. Further Calculus Techniques | 进阶微积分技巧
Advanced integration techniques include integration by parts, substitution, and the use of partial fractions. These methods extend the scope of standard integrals.
高级积分技巧包括分部积分法、换元法和部分分式的使用。这些方法扩展了标准积分的范围。
Integration by parts: ∫ u dv/dx dx = uv − ∫ v du/dx dx
The substitution rule states ∫ f(g(x)) g' dx = ∫ f(u) du, where u = g(x). Trigonometric substitutions are useful for integrands with √a² − x², √a² + x² or √x² − a².
换元法指出 ∫ f(g(x)) g'(x) dx = ∫ f(u) du，其中 u = g(x)。对于包含 √a² − x²、√a² + x² 或 √x² − a² 的被积函数，三角换元非常有用。
Partial fractions decompose rational functions into simpler fractions that are easier to integrate. For distinct linear factors:
部分分式将有理函数分解为更简单的分式，使积分更容易。对于不同的线性因子：
1/[(x−a)(x−b)] ≡ A/(x−a) + B/(x−b)
integration by parts. For example, ∫ sinⁿ x dx can be expressed in terms of ∫ sinⁿ⁻² x dx.
约化公式用于积分三角函数的幂，通常通过分部积分推导。例如，∫ sinⁿ x dx 可以用 ∫ sinⁿ⁻² x dx 表示。
then dy/dx = (dy/dt)/(dx/dt). The second derivative is d²y/dx² = (d/dt)(dy/dx) / (dx/dt).
参数微分：若 x = f(t)，y = g(t)，则 dy/dx = (dy/dt)/(dx/dt)。二阶导数为 d²y/dx² = (d/dt)(dy/dx) / (dx/dt)。
11. Inequalities and Modulus | 不等式与绝对值
Solving inequalities involving algebraic fractions or modulus functions requires careful handling of sign changes and critical values. The set of solutions is often expressed as unions of intervals.
解涉及代数分式或绝对值函数的不等式时，需要仔细处理符号变化和临界值。解集常表示为区间的并集。
|x − a| < b ⇔ a − b < x < a + b
|x − a| > b ⇔ x < a − b or x > a + b
0, construct a sign table considering zeros of numerator and denominator. Never multiply across by an expression whose sign is unknown.
对于 (x−a)/(x−b) ≥ 0 这类不等式，需构建考虑分子和分母零点的符号表。切勿乘以符号未知的表达式。
The triangle inequality states |a + b| ≤ |a| + |b|. It is frequently used in complex number and vector proofs.
三角不等式指出 |a + b| ≤ |a| + |b|。它在复数与向量的证明中经常使用。
12. Curve Sketching and Rational Functions | 曲线草图与有理函数
Sketching graphs of rational functions involves identifying vertical, horizontal, and oblique asymptotes, intercepts, and stationary points. A systematic approach helps reveal key features.
绘制有理函数的图形涉及识别垂直渐近线、水平渐近线和斜渐近线、截距以及驻点。系统性的方法有助于揭示关键特征。
Vertical asymptote: denominator = 0 (after simplification)
Horizontal asymptote: y = (leading coefficient ratio) as x → ±∞
Oblique asymptote: if degree of numerator exceeds denominator by 1
To find oblique asymptotes, perform polynomial division; the quotient (excluding remainder) gives the asymptote's equation.
要找到斜渐近线，进行多项式除法；商式（不含余数）给出了渐近线的方程。
Use differentiation to find stationary points and determine their nature via the first or second derivative test. Consider domain restrictions, especially when square roots or logarithms are present.
利用微分找到驻点，并通过一阶或二阶导数检验确定其性质。考虑定义域的限制，尤其是当存在平方根或对数时。
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