Essential Mathematical Tools for IB Physics | IB物理常用重要数学结论

📚 Essential Mathematical Tools for IB Physics | IB物理常用重要数学结论

Physics is, at its core, a quantitative science. Throughout the IB Physics syllabus, you will encounter mathematical relationships that describe everything from the motion of particles to the decay of radioactive nuclei. Mastering a small set of key mathematical conclusions can greatly improve both your problem-solving speed and your conceptual understanding.

物理学本质上是一门定量科学。在整个IB物理课程中，你会遇到描述从粒子运动到放射性核衰变等各种现象的数学关系。掌握一小部分关键的数学结论，可以显著提高你的解题速度和概念理解能力。

1. Proportionality and Linearization | 比例关系与线性化

Direct proportion means that if one variable increases, the other increases at a constant rate: y ∝ x implies y = kx, where k is a constant. Inverse proportion, y ∝ 1/x, implies y = k/x, and appears often in gravity and electromagnetism, such as F ∝ 1/r².

正比例意味着一个变量增大时，另一个变量以恒定速率增大：y ∝ x 意味着 y = kx，其中 k 为常数。反比例 y ∝ 1/x 意味着 y = k/x，常见于引力和电磁学，例如 F ∝ 1/r²。

To test whether data fits a power law y = axⁿ, plot ln y against ln x. If the relationship is linear, the slope gives the exponent n and the intercept gives ln a.

为了检验数据是否符合幂律关系 y = axⁿ，可以绘制 ln y 对 ln x 的图。如果图形为直线，斜率给出指数 n，截距给出 ln a。

Linearization is a powerful tool: by choosing appropriate axes, you can turn curved relationships into straight lines, making graphical analysis much easier.

线性化是一个强大的工具：通过选择合适的坐标轴，可以把曲线关系转化为直线，从而大大简化图形分析。

2. Exponentials and Logarithms | 指数函数与对数

An exponential function has the form y = A eᵏˣ. It grows or decays at a rate proportional to its current value. In physics, this describes capacitor discharge, radioactive decay, and the absorption of radiation.

指数函数的形式为 y = A eᵏˣ。它的增长或衰减速率与当前值成正比。在物理中，它描述电容器放电、放射性衰变以及辐射吸收等现象。

The natural logarithm is the inverse of the exponential: if y = eˣ, then x = ln y. Key rules include ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, and ln(aⁿ) = n ln a.

自然对数是指数函数的逆运算：如果 y = eˣ，则 x = ln y。主要法则包括 ln(ab) = ln a + ln b、ln(a/b) = ln a − ln b 以及 ln(aⁿ) = n ln a。

For radioactive decay N = N₀ e^(−λt), taking natural logs gives ln N = ln N₀ − λt, so a graph of ln N against t is a straight line with slope −λ.

对于放射性衰变 N = N₀ e^(−λt)，取自然对数可得 ln N = ln N₀ − λt，因此 ln N 对 t 的图形是一条直线，斜率为 −λ。

3. Trigonometry | 三角函数

Right-angled triangle definitions are essential: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. These are used in resolving forces and calculating components of vectors.

直角三角形的三角函数定义至关重要：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。它们用于力的分解和向量分量的计算。

Small-angle approximations are extremely useful for oscillators and optics: when θ is small and measured in radians, sin θ ≈ θ, tan θ ≈ θ, and cos θ ≈ 1.

小角度近似对于振动和光学非常有用：当 θ 很小且以弧度为单位时，sin θ ≈ θ，tan θ ≈ θ，cos θ ≈ 1。

For non-right-angle triangles, the sine rule a/sin A = b/sin B = c/sin C and the cosine rule a² = b² + c² − 2bc cos A allow you to solve any triangle geometry.

对于非直角三角形，正弦定理 a/sin A = b/sin B = c/sin C 和余弦定理 a² = b² + c² − 2bc cos A 可以用于求解任意三角形几何问题。

4. Vectors | 向量

A vector has both magnitude and direction, while a scalar has only magnitude. Displacement, velocity, force, and momentum are vectors; speed, mass, energy, and temperature are scalars.

向量既有大小又有方向，而标量只有大小。位移、速度、力和动量是向量；速率、质量、能量和温度是标量。

To add perpendicular vectors A and B, the resultant magnitude is R = √(A² + B²), and the direction is given by θ = tan⁻¹(B/A), measured from A.

要合成互相垂直的向量 A 和 B，合向量大小为 R = √(A² + B²)，方向由 θ = tan⁻¹(B/A) 给出，从 A 的方向开始测量。

Any vector can be resolved into components: Fₓ = F cos θ and Fᵧ = F sin θ. These components can then be added separately.

任何向量都可以分解为分量：Fₓ = F cos θ，Fᵧ = F sin θ。然后可以分别对这些分量求和。

5. Differentiation | 微分

Differentiation measures the rate of change of one variable with respect to another. In mechanics, velocity is the time derivative of displacement: v = dx/dt, and acceleration is a = dv/dt.

微分用于衡量一个变量相对于另一个变量的变化率。在力学中，速度是位移对时间的导数：v = dx/dt，加速度是 a = dv/dt。

Standard derivatives: d/dx(xⁿ) = n xⁿ⁻¹, d/dx(sin ax) = a cos ax, d/dx(cos ax) = −a sin ax, d/dx(eᵃˣ) = a eᵃˣ, and d/dx(ln x) = 1/x.

常用导数：d/dx(xⁿ) = n xⁿ⁻¹、d/dx(sin ax) = a cos ax、d/dx(cos ax) = −a sin ax、d/dx(eᵃˣ) = a eᵃˣ 以及 d/dx(ln x) = 1/x。

The gradient of a displacement-time graph at any point gives instantaneous velocity; the gradient of a velocity-time graph gives instantaneous acceleration.

位移-时间图像上任意一点的切线斜率给出瞬时速度；速度-时间图像上的切线斜率给出瞬时加速度。

6. Integration | 积分

Integration is the inverse of differentiation and can be interpreted as the area under a curve. It is used to find total quantities from rates of change.

积分是微分的逆运算，可以理解为曲线下的面积。它用于从变化率求出总量。

Standard integrals: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, ∫ sin ax dx = −(1/a) cos ax + C, ∫ cos ax dx = (1/a) sin ax + C, ∫ eᵃˣ dx = (1/a) eᵃˣ + C.

常用积分：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C、∫ sin ax dx = −(1/a) cos ax + C、∫ cos ax dx = (1/a) sin ax + C、∫ eᵃˣ dx = (1/a) eᵃˣ + C。

In physics, the area under a force-extension graph equals elastic potential energy; the area under a pressure-volume graph equals work done by a gas.

在物理中，力-伸长图像下的面积等于弹性势能；压强-体积图像下的面积等于气体所做的功。

7. Uncertainty and Error Propagation | 不确定度与误差传播

Every measurement has an uncertainty. Absolute uncertainty has the same unit as the quantity, while fractional uncertainty is absolute uncertainty divided by the measured value, often expressed as a percentage.

每次测量都有不确定度。绝对不确定度与物理量具有相同单位，而相对不确定度是绝对不确定度除以测量值，通常用百分比表示。

When adding or subtracting quantities, add absolute uncertainties. For example, (a ± Δa) + (b ± Δb) gives total absolute uncertainty Δa + Δb.

加减运算时，绝对不确定度相加。例如，(a ± Δa) + (b ± Δb) 的总绝对不确定度为 Δa + Δb。

When multiplying or dividing quantities, add fractional or percentage uncertainties. If c = a × b, then Δc/c = Δa/a + Δb/b.

乘除运算时，相对或百分比不确定度相加。若 c = a × b，则 Δc/c = Δa/a + Δb/b。

When raising a quantity to a power, multiply the fractional uncertainty by the power: if c = aⁿ, then Δc/c = n × Δa/a.

当一个量取幂次时，相对不确定度乘以指数：若 c = aⁿ，则 Δc/c = n × Δa/a。

8. Slopes and Areas under Graphs | 图像斜率与曲线下面积

Many physical quantities can be obtained from the slope of a graph. The slope of a displacement-time graph is velocity, and the slope of a velocity-time graph is acceleration.

许多物理量可以通过图像斜率获得。位移-时间图像的斜率是速度，速度-时间图像的斜率是加速度。

The area under a velocity-time graph gives displacement, while the area under a force-time graph gives impulse. These interpretations are central to kinematics and momentum questions.

速度-时间图像下的面积给出位移，力-时间图像下的面积给出冲量。这些解释是运动学和动量问题的核心。

For curved graphs, the instantaneous slope is the gradient of the tangent at that point, and the area can be found by counting squares or using integration.

对于曲线图像，瞬时斜率是某点切线的斜率，而面积可以通过数方格或积分来求得。

9. Geometry, Units and Approximations | 几何、单位与近似

Certain geometrical results appear repeatedly in IB Physics: the circumference of a circle is 2πr, area is πr²; the volume of a sphere is (4/3)πr³, and its surface area is 4πr².

某些几何结论在IB物理中反复出现：圆的周长为 2πr，面积为 πr²；球的体积为 (4/3)πr³，表面积为 4πr²。

Pythagoras’ theorem, a² + b² = c², is essential for finding vector magnitudes, periodic time in pendulums, and path differences in wave interference.

勾股定理 a² + b² = c² 对于求向量大小、单摆周期以及波干涉中的光程差至关重要。

For small x, the binomial approximation (1 + x)ⁿ ≈ 1 + nx is useful in relativity and gravitational potential energy derivations. The quadratic formula x = [−b ± √(b² − 4ac)] / (2a) remains a standard tool in projectile and circuit problems.

对于小 x，二项式近似 (1 + x)ⁿ ≈ 1 + nx 在相对论和引力势能推导中很有用。二次方程求根公式 x = [−b ± √(b² − 4ac)] / (2a) 是抛体运动和电路问题中的标准工具。

Finally, always check units and use radians for angle calculations in physics unless degrees are explicitly required.

最后，始终检查单位，并在物理计算中使用弧度制，除非题目明确要求角度制。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Essential Mathematical Tools for IB Physics | IB物理常用重要数学结论