Mastering Core Mathematical Techniques from Jordan & Smith | 掌握 Jordan & Smith 核心数学技术

📚 Mastering Core Mathematical Techniques from Jordan & Smith | 掌握 Jordan & Smith 核心数学技术

Dominic Jordan and Peter Smith’s Mathematical Techniques is a widely used text that builds a bridge from school mathematics to the analytical tools needed in university physics, engineering and applied mathematics. This guide summarises the core methods IB students should master: differentiation and integration, complex numbers, vectors, matrices, ordinary and partial differential equations, Laplace transforms and numerical approaches.

Dominic Jordan 与 Peter Smith 的《数学技术》是一本广泛应用教材，在中学数学与大学物理、工程和应用数学所需的解析工具之间架起桥梁。本指南总结了 IB 学生应掌握的核心方法：微分与积分、复数、向量、矩阵、常微分方程与偏微分方程、拉普拉斯变换以及数值方法。

1. Differentiation and Curve Sketching | 微分与曲线草图

Differentiation gives the instantaneous rate of change of a function. For IB purposes, you should be confident with the product rule, quotient rule and chain rule. The second derivative d²y/dx² tells us about concavity: a positive value means the curve is concave up, a negative value concave down.

微分给出函数的瞬时变化率。就 IB 而言，你应熟练掌握乘法法则、除法法则和链式法则。二阶导数 d²y/dx² 反映凹凸性：正值表示曲线向上凹，负值表示向下凹。

d²y/dx² > 0 ⇒ local minimum; d²y/dx² < 0 ⇒ local maximum

Stationary points occur where dy/dx = 0. Classify them using the second derivative test or a sign table: if d²y/dx² > 0 the point is a local minimum; if d²y/dx² < 0 it is a local maximum; if d²y/dx² = 0 the test is inconclusive and an inflection point may occur.

驻点出现在 dy/dx = 0 处。使用二阶导数检验或符号表进行分类：若 d²y/dx² > 0，该点为局部极小值；若 d²y/dx² < 0，为局部极大值；若 d²y/dx² = 0，检验失效，可能出现拐点。

Curve sketching then combines intercepts, asymptotes, stationary points and concavity. Always check behaviour as x → ±∞ and near vertical asymptotes.

曲线草图结合截距、渐近线、驻点和凹凸性。务必检查 x → ±∞ 以及垂直渐近线附近的行为。

2. Integration Techniques | 积分技巧

Integration reverses differentiation and computes areas, volumes and accumulated change. Core methods include substitution, integration by parts, partial fractions and trigonometric integrals.

积分是微分的逆运算，用于计算面积、体积和累积变化。核心方法包括换元法、分部积分法、部分分式法和三角函数积分。

∫ u dv = uv − ∫ v du

A standard formula is ∫ u dv = uv − ∫ v du. For rational functions, split denominators into linear or quadratic factors before integrating. Trigonometric integrals often require identities such as sin²θ + cos²θ = 1 or double-angle formulas.

标准公式为 ∫ u dv = uv − ∫ v du。对于有理函数，先分解分母为一次或二次因子再积分。三角积分常需使用 sin²θ + cos²θ = 1 或倍角公式等恒等式。

Definite integrals give exact areas when limits are substituted. Improper integrals with infinite limits or discontinuities are evaluated via limits.

定积分在代入上下限后给出精确面积。无穷限或被积函数不连续的广义积分需借助极限计算。

3. Complex Numbers | 复数

A complex number z = x + iy has real part x and imaginary part y, where i² = −1. Its modulus is |z| = √(x² + y²) and its argument is θ = arctan(y/x).

复数 z = x + iy 的实部为 x，虚部为 y，其中 i² = −1。其模为 |z| = √(x² + y²)，辐角为 θ = arctan(y/x)。

e^(iθ) = cos θ + i sin

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