9665-FM03 Further Mathematics: Core Topics Explained | 9665-FM03 进阶数学核心专题精讲

📚 9665-FM03 Further Mathematics: Core Topics Explained | 9665-FM03 进阶数学核心专题精讲

Unit FM03 of the International A-Level Further Mathematics specification (9665) gathers the advanced pure-mathematics techniques that separate a competent Further Maths candidate from an outstanding one. The topics below follow the style of the 2016 mark scheme and cover hyperbolic functions, reduction formulae, arc length, matrices, eigenvalues and eigenvectors, vectors in three dimensions and polar coordinates. Mastery of these topics requires fluency with identities, differentiation, integration techniques and geometric interpretation.

国际 A-Level 进阶数学（9665）的 FM03 单元汇集了区分合格与优秀进阶数学考生的高等纯数技巧。以下专题参照 2016 年评分方案的命题风格整理，涵盖双曲函数、递推（归约）公式、弧长、矩阵、特征值与特征向量、三维向量以及极坐标。掌握这些专题需要熟练运用恒等式、微分、积分技巧及其几何解释。

Each section pairs a short theory reminder with the examinable result, then flags the mistakes examiners report most often. Revision should be active rather than passive: derive every result once by hand, then practise recognising when it applies.

每个小节先给出简明的理论回顾与可考查结论，再指出考官最常提及的高频错误。复习应当主动而非被动：每个结论至少手推一次，再练习判断它在何时适用。

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1. Hyperbolic Functions: Definitions and Identities | 双曲函数：定义与恒等式

nctions are built from the exponential function, so every property can be traced back to eˣ. The two base functions are sinh x = (eˣ − e⁻ˣ)/2 and cosh x = (eˣ + e⁻ˣ sinh x / cosh x, coth x = cosh x / sinh x, sech x = 1 / cosh x and cosech x = 1 / sinh x.

双曲函数由指数函数构造，因此每个性质都可追溯到 eˣ。两个基本函数定义为 sinh x = (eˣ − e⁻ˣ)/2 与 cosh x = (eˣ + e⁻ˣ)/2。其余四个由商或倒数定义：tanh x = sinh x / cosh x，coth x = cosh x / sinh x，sech x = 1 / cosh x，cosech x = 1 / sinh x。

cosh² x − sinh² x = 1, 1 − tanh² x = sech² x, coth² x − 1 = cosech² x

h is even; the graph of y = cosh x lies above the x-axis with minimum value 1 at x = 0, tracing a catenary. The graphs of sinh and tanh pass through the origin and tanh x → ±1 as x → ±∞.

第一条恒等式是 sin² x + cos² x = 1 的双曲版本，展开指数式一行即可证明。注意 sinh 是奇函数、cosh 是偶函数；y = cosh x 的图像位于 x 轴上方，在 x = 0 处取最小值 1，形成悬链线。sinh 与 tanh 的图像都过原点，且当 x → ±∞ 时 tanh x → ±1。

x cosh y + sinh x sinh y. Hence cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x and sinh 2x = 2 sinh x cosh x. The practical rule is Osborn' rigonometric identity and change the sign of every term containing a product of two sines.

加法与倍角公式与三角版本相似，只是符号有别：sinh(x + y) = sinh x cosh y + cosh x sinh y，cosh(x + y) = cosh x cosh y + sinh x sinh y。于是 cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 1 + 2 sinh² x，sinh 2x = 2 sinh x cosh x。实用的口诀是奥斯本法则：把三角恒等式照搬，凡含两个 sin 之乘积的项改变符号。

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2. Inverse Hyperbolic Functions and Logarithmic Forms | 反双曲函数及其对数形式

Because the hyperbolic functions are defined by exponentials, their inverses can be written as natural logarithms. These logarithmic forms are essential for integration and for solving equations where the answer is expected as an exact value.

由于双曲函数由指数定义，它们的反函数可以写成自然对数形式。这些对数形式在积分和求精确解的方程中至关重要。

arsinh x = ln(x + √x² + 1), x ∈ ℝ

arcosh x = ln(x + √x² − 1), x ≥ 1

artanh x = ½ ln((1 + x)/(1 − x)), |x| < 1

< 1 for artanh. The proof for arsinh is standard: set y = arsinh x so that x = sinh y = (eʸ − e⁻ʸ)/2, multiply by 2eʸ and solve the resulting quadratic e²ʸ − 2x eʸ − 1 = 0. Taking the positive root gives eʸ = x + √x² + 1, and the negative root is discarded because eʸ > 0.

定义域分别为：arsinh 为全体实数，arcosh 为 x ≥ 1，artanh 为 −1 < x < 1。arsinh 的证明是标准套路：令 y = arsinh x，则 x = sinh y = (eʸ − e⁻ʸ)/2，两边乘以 2eʸ 得二次方程 e²ʸ − 2x eʸ − 1 = 0，取正根得 eʸ = x + √x² + 1，负根因 eʸ > 0 而舍去。

Also useful are the reciprocal-type results such as arcosech x = ln(1/x + √1/x² + 1) for x ≠ 0 and arcoth x = ½ ln((x + 1)/(x − 1)) for |x| e converted back into an inverse hyperbolic one, a conversion examiners frequently reward.

此外还有倒数型结论，例如 x ≠ 0 时 arcosech x = ln(1/x + √1/x² + 1)，|x| > 1 时 arcoth x = ½ ln((x + 1)/(x − 1))。识别这些形式可以把对数答案还原为反双曲形式，考官常常对这一步给予分数。

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3. Differentiation of Hyperbolic and Inverse Hyperbolic Functions | 双曲函数与反双曲函数的微分

The derivatives of the six hyperbolic functions follow a clean pattern, with the 'co-' functions carrying a minus sign. These results must be recalled instantly because they underpin most later integration.

六个双曲函数的导数规律清晰，凡带 co- 的函数都带一个负号。这些结果必须做到条件反射式反应，因为后面大多数积分都依赖它们。

Function	Derivative	中文
sinh x	cosh x	双曲正弦
cosh x	sinh x	双曲余弦
tanh x	sech² x	双曲正切
coth x	−cosech² x	双曲余切
sech x	−sech x tanh x	双曲正割
cosech x	−cosech x coth x	双曲余割

The inverse functions differentiate to algebraic expressions rather than hyperbolic ones:

反函数的导数则是代数表达式，而不是双曲函数：

d/dx arsinh x = 1 / √x² + 1

d/dx arcosh x = 1 / √x² − 1, x > 1

d/dx artanh x = 1 / (1 − x²), |x| < 1

nd d/dx ln(cosh x) = tanh x. For a function of the form arsinh(x/a), the derivative is 1 / √x² + a², which is exactly the integrand met in the next section. Always state the condition x > 1 or |x| < 1 where it is needed, since domain statements are printed on the mark scheme.

实际题目中它们通过链式法则出现：d/dx sinh(3x) = 3 cosh 3x，d/dx cosh(x²) = 2x sinh(x²)，d/dx ln(cosh x) = tanh x。对于 arsinh(x/a) 形式的函数，导数为 1 / √x² + a²，这正好是下一节出现的被积函数。凡是需要条件的地方都要写明 x > 1 或 |x| < 1，因为定义域说明在评分方案中有对应分数。

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4. Integration with Hyperbolic Functions | 双曲函数的积分

Integration reverses the differentiation table and adds the standard results obtained from the inverse hyperbolic derivatives. The three results in the second block are the ones that turn awkward surds and quadratics into logarithms, and they should be memorised in both forms.

积分是微分表的逆运算，再加上由反双曲函数导数得到的三条标准结论。下面第二组的三条结论能把难处理的根式与二次式化为对数，且两种形式都应记住。

∫ sinh x dx = cosh x + c, ∫ cosh x dx = sinh x + c

∫ tanh x dx = ln(cosh x) + c, ∫ sech² x dx = tanh x + c

∫ 1/√x² + a² dx = arsinh(x/a) + c = ln(x + √x² + a²) + c

∫ 1/√x² − a² dx = arcosh(x/a) + c = ln(x + √x² − a²) + c, x > a

∫ 1/(a² − x²) dx = (1/a) artanh(x/a) + c = (1/(2a)) ln|(a + x)/(a − x)| + c, |x| < a

plicated integrands use a hyperbolic substitution. With x = a sinh u, dx = a cosh u du and √x² + a² becomes a² sech² u. A useful identity for odd powers is cosh 2u = 2 cosh² u − 1, which converts squares into double angles.

遇到更复杂的被积函数就使用双曲代换。令 x = a sinh u，则 dx = a cosh u du，且 √x² + a² = a cosh u，因为 cosh² u − sinh² u = 1 直接消去了根号。令 x = a cosh u 时对应根式为 a sinh u；令 x = a tanh u 时 a² − x² 变为 a² sech² u。处理奇次幂的常用恒等式是 cosh 2u = 2 cosh² u − 1，可把平方项化为倍角。

Worked pattern: to find ∫ √x² + 4 sh u. Answering in terms of x, with the logarithm form where required, is what the mark scheme expects.

典型套路：求 ∫ √x² + 4 dx 时令 x = 2 sinh u，积分化为 4∫ cosh² u du = 2∫ (1 + cosh 2u) du = 2u + sinh 2u + c，再用 u = arsinh(x/2) 与 sinh 2u = 2 sinh u cosh u 换回。用 x 表示答案，并在需要时写成对数形式，正是评分方案所期望的。

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5. Reduction Formulae | 递推（归约）公式

ndard example is Iₙ = ∫₀^(π/2) sinⁿ x dx, for which integration by parts applied to sinⁿ x

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