📚 Mathematical tools for physics | 物理中的数学工具
Physics relies heavily on mathematics to describe natural phenomena quantitatively. From the motion of planets to the behaviour of subatomic particles, mathematical tools provide the language and framework for modelling, predicting, and understanding the physical world. This article surveys essential mathematical techniques that every IB physics student should master, including vectors, calculus, trigonometry, complex numbers, exponentials, differential equations, error analysis, dimensional analysis, and series approximations.
物理学高度依赖数学来定量描述自然现象。从行星运动到亚原子粒子的行为，数学工具为建模、预测和理解物理世界提供了语言和框架。本文概述了每位IB物理学生都应掌握的基本数学技巧，包括向量、微积分、三角函数、复数、指数函数、微分方程、误差分析、量纲分析和级数近似。
1. Vectors and Scalars | 向量与标量
Vectors have both magnitude and direction, while scalars have only magnitude. In physics, displacement, velocity, acceleration, force, and momentum are vectors; distance, speed, mass, energy, and time are scalars. Vector addition follows the triangle or parallelogram law. Components resolve a vector into perpendicular directions, typically using cosine and sine: Fx = F cosθ, Fy = F sinθ. The dot product gives the scalar projection, A·B = |A||B| ed to calculate work (W = F·d). The cross product yields a vector perpendicular to both, A×B = |A||B|sinθ n, essential for torque (τ = r×F) and magnetic force (F = qv×B).
向量既有大小又有方向，而标量只有大小。物理中，位移、速度、加速度、力和动量是向量；距离、速率、质量、能量和时间是标量。向量加法遵循三角形或平行四边形法则。分量法将一个向量分解为相互垂直的方向，通常使用余弦和正弦：Fx = F cosθ，Fy = F sinθ。点积给出标量投影，A·B = |A||B|cosθ，用于计算功（W = F·d）。叉积产生垂直于两者的向量，A×B = |A||B|sinθ n，对于扭矩（τ = r×F）和磁力（F = qv×B）至关重要。
2. Differentiation – Rates of Change | 微分——变化率
ity: a = dv/dt = d²s/dt². If s(t) = ut + ½at², then v = u + at and a = constant. In Newton' second law, F = dp/dt expresses force as the rate of change of momentum. For a spring, F = –kx relates force to the derivative of potential energy: F = –dU/dx. The table below summarises common differentiation rules used in physics.
微分用于求函数的瞬时变化率。在运动学中，速度v是位移s对时间的导数：v = ds/dt。加速度a是速度的导数：a = dv/dt = d²s/dt²。若s(t) = ut + ½at²，则v = u + at，a为常数。在牛顿第二定律中，F = dp/dt将力表示为动量的变化率。对于弹簧，F = –kx将力与势能的导数联系起来：F = –dU/dx。下表总结了物理学中常用的导数规则。
| Function f(x) | Derivative f'(x) | 物理应用示例 |
| xⁿ | n xⁿ⁻¹ | s = kt² ⇒ v = 2kt |
| sin(kx) | k cos(kx) | x = A sin(ωt) ⇒ v = ωA cos(ωt) |
| cos(kx) | –k sin(kx) | v = ωA cos(ωt) ⇒ a = –ω²A sin(ωt) |
| eᵏˣ | k eᵏˣ | I = I₀ e–t/RC ⇒ dI/dt = –(1/RC)I₀ e–t/RC |
| ln(x) | 1/x | 用于半衰期线性拟合 |
3. Integration – Area under a Curve | 积分——曲线下面积
f differentiation and calculates the accumulation of a quantity. The area under a velocity–time graph gives displacement: s = ∫ v dt. The area under an acceleration– th limits provide specific values, while indefinite integrals recover original functions. Numerical integration may be used when analytical forms are unavailable.
积分是微分的逆运算，用于计算量的累积。速度–时间图下的面积给出位移：s = ∫ v dt。加速度–时间图下的面积给出速度变化量。变力做的功为W = ∫ F(x) dx。力在时间上的冲量为J = ∫ F dt = Δp。在电学中，电容器储存的电荷为Q = ∫ I dt，电感器中储存的能量与∫ P dt有关。带有上下限的定积分给出具体数值，而不定积分则恢复原函数。当解析形式不可用时，可使用数值积分。
4. Trigonometric Functions and Harmonic Motion | 三角函数与简谐运动
π/T. Velocity and acceleration follow sinusoidal patterns: v = dx/dt = ωA cos(ωt + φ), a = – θ = sin 2θ, simplify expressions. Small-angle approximations (sinθ ≈ θ, tanθ ≈ θ, cosθ ≈ 1 – pendulum motion. Phase difference between waves can be expressed using sine/cosine shifts.
正弦和余弦函数描述振动和波动。简谐运动（SHM）的位移为x = A sin(ωt + φ) 或 x = A cos(ωt + φ)。角频率ω = 2πf = 2π/T。速度和加速度遵循正弦模式：v = dx/dt = ωA cos(ωt + φ)，a = –ω²x。三角恒等式，如sin²θ + cos²θ = 1 和 2 sinθ cosθ = sin 2θ，可简化表达式。小角度近似（sinθ ≈ θ, tanθ ≈ θ, cosθ ≈ 1 – θ²/2）对于单摆运动非常有用。波之间的相位差可以用正弦/余弦的平移来表示。
5. Complex Numbers for AC Circuits and Waves | 复数在交流电路与波中的应用
Complex numbers combine a real and an imaginary part, z = a + ib, where i² = – ideal for alternating current (AC) analysis. Voltage and current are expressed as V = V₀ eiωt and I = I₀ e^(i(ωt – or Z = 1/(iωC). Phasor diagrams use the complex plane to visualise rotating vectors. Euler's formula, eiθ trigonometric functions, greatly simplifying wave superposition and interference calculations.
复数由实部和虚部组成，z = a + ib，其中i² = –1（物理学中常用j）。它们表示同时具有振幅和相位的量，非常适合交流电分析。电压和电流表示为V = V₀ eiωt 和 I = I₀ e^(i(ωt – φ))。阻抗Z = V/I是复数，结合了电阻R和电抗X：Z = R + iX。电阻Z = R，电感Z = iωL，电容Z = 1/(iωC)。相量图使用复平面来可视化旋转向量。欧拉公式eiθ = cosθ + i sinθ将指数函数与三角函数联系起来，极大地简化了波的叠加和干涉计算。
6. Exponentials and Logarithms in Decay and Growth | 指数与对数在衰变与增长中的应用
here the rate of change is proportional to the current amount. Radioactive decay: N = N₀ e–λt, where λ is the decay constant. Capacitor discharge: Q = Q₀ e–t/RC and I = I₀ e–t/RC. Newton's law of cooling: T – Tₐ = (T₀ – Tₐ) e–kt hmic transforms linearise such data: plotting ln N vs. t yields a straight line with slope –λ. The half-life is t₁/₂ = ln 2 / λ. In log– es with slope n. Exponential growth also appears in population models and chain reactions.
指数函数用于建模变化率与当前数量成正比的过程。放射性衰变：N = N₀ e–λt，其中λ是衰变常数。电容器放电：Q = Q₀ e–t/RC 和 I = I₀ e–t/RC。牛顿冷却定律：T – Tₐ = (T₀ – Tₐ) e–kt。对数变换可将此类数据线性化：绘制ln N 对 t 的图得到斜率为 –λ 的直线。半衰期 t₁/₂ = ln 2 / λ。在双对数图中，幂律关系 y = kxⁿ 变成斜率为 n 的直线。指数增长也出现在人口模型和链式反应中。
7. Differential Equations in Physics | 物理中的微分方程
Many physical laws are expressed as differential equations (DEs). Newton' x = 0 has solutions x = A cos(ωt + φ). First-order DEs describe exponential decay: dN/dt = –λN ⇒ N = N₀ e–λt. Maxwell's equations link electric and magnetic fields using partial derivatives. Solving DEs often requires integration, separation of variables, or recognising standard forms. Understanding the physical meaning of initial conditions and boundary conditions is crucial for obtaining specific solutions that match real observations.
许多物理定律都以微分方程（DE）的形式表达。牛顿第二定律：m d²x/dt² = ΣF(x, v, t)，引出运动方程。简谐运动方程：d²x/dt² + ω²x = 0，其解为 x = A cos(ωt + φ)。一阶微分方程描述指数衰减：dN/dt = –λN ⇒ N = N₀ e–λt。麦克斯韦方程组使用偏导数将电场和磁场联系起来。求解微分方程通常需要积分、分离变量或识别标准形式。理解初始条件和边界条件的物理意义对于获得与真实观测相符的特定解至关重要。
8. Uncertainty and Error Analysis | 不确定度与误差分析
/z = n Δx/x. Standard deviation and confidence intervals quantify random errors. In IB Physics, uncertainties in gradients and intercepts from graphs are determined using min– maintain precision. Chi-squared tests can evaluate goodness of fit for theoretical models.
所有测量都具有不确定度，表示为绝对（±Δx）或相对（Δx/x）。当测量结果合并时，误差会传播。加减运算：Δz = Δx + Δy（绝对误差相加）。乘除运算：Δz/z = Δx/x + Δy/y（相对误差相加）。对于幂运算：若 z = xⁿ，则 Δz/z = n Δx/x。标准差和置信区间量化随机误差。在IB物理中，图中斜率和截距的不确定度通过使用最小–最大值线或电子表格中的LINEST函数来确定。有效数字和科学记数法保持精度。卡方检验可以评估理论模型的拟合优度。
9. Dimensional Analysis | 量纲分析
], force [M L T⁻²]. Any valid equation must have consistent dimensions on both sides. Dimensional analysis can suggest the form of an unknown relation: e.g., the period of a pendulum T might be assumed to depend on length l, mass m, and g: T = k lᵃ mᵇ gᶜ
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图