数学物理教学 | 学科联结、公式标准与惯例理解

注：本文为 “数学物理公式学习” 相关合辑。
英文引文，机翻未校。
如有内容异常，请看原文。

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Formulas in Physics Have a “Standard” Form

物理学公式的“标准”形式

Matthew J. Moelter, California Polytechnic State University, San Luis Obispo, CA
Martin Jackson, University of Puget Sound, Tacoma, WA
马修·J·莫尔特 （加利福尼亚州立理工大学，圣路易斯奥比斯波，加利福尼亚州）
马丁·杰克逊 （普吉特海湾大学，塔科马，华盛顿州）

We discuss the importance of the ordering of symbols in physics formulas and identify implicit conventions that govern the “standard” form for how formulas are written and interpreted. An important part of writing and reading this form is understanding distinctions among constants, parameters, and variables. We delineate these conventions and encourage instructors to make them explicit for students.
我们探讨了物理学公式中符号排序的重要性，并明确了支配公式书写与解读的“标准”形式所遵循的隐含惯例。掌握这种形式的读写核心在于理解常量、参数和变量之间的区别。本文详细阐述了这些惯例，并建议教师向学生明确传授这些内容。

In most cases the natural phenomena described by physical theories are represented in the language of mathematics. To succeed, students must understand this language both in terms of the underlying mathematics and how it is used in physics. Physics curricula typically require students to have mathematical (pre)co-requisites along with their physics courses. Mathematics, as used by physicists, is a language complete with a grammar and notational conventions. In particular, formulas used by physicists have particular conventions that help to expose different aspects of the physical content. Romer¹ has an excellent discussion of the process of “reading the equations” of physics. More recently, Hewitt² encourages the conceptual understanding of the equations as a guide to problem solving. However, this type of understanding is only possible if one can read the language of the formulas. We suggest that there are implicit grammatical rules concerning the writing of formulas in physics, and that these conventions should be explicitly identified and taught to enhance student understanding.
在大多数情况下，物理理论所描述的自然现象都是通过数学语言来呈现的。学生要想学好物理，必须同时理解其背后的数学原理以及数学在物理学中的应用方式。物理课程通常要求学生在修读物理的同时，具备相应的（先修）数学基础。物理学家所使用的数学是一种完整的语言，包含语法和符号惯例。特别是，物理学家使用的公式具有特定惯例，这些惯例有助于揭示物理内容的不同方面。罗默（Romer）¹ 对“解读物理学方程”的过程进行了精彩论述。最近，休伊特（Hewitt）² 倡导将方程的概念理解作为解题的指导。然而，只有当人们能够读懂公式这一语言时，这种理解才有可能实现。我们认为，物理学公式的书写存在隐含的语法规则，应当明确识别并传授这些惯例，以提升学生的理解能力。

Example of a student’s formula

学生公式示例

A student’s improper use of mathematical notation can readily lead to confusion for both the student and instructor. Many of us are familiar with the following difficulties: mixing upper and lower case ( $M$ versus $m$ ); failing to properly indicate the vector nature of a quantity ( $(F=ma)$ ); or careless use of sub/superscripts, $m2$ versus $m^2$ versus $m_2$ . These are examples of incorrect symbol usage and students are typically instructed on their proper use. There is, however, a more subtle issue regarding the way symbols are arranged in a formula, even when all the symbols are present and correct.
学生对数学符号的不当使用很容易给自身和教师带来困惑。我们很多人都遇到过以下问题：大小写混淆（ $M$ 与 $m$ ）；未正确表示物理量的矢量性质（ $(F=ma)$ ）；或粗心使用下标/上标，如 $m2$ 、 $m^2$ 与 $m_2$ 混用。这些都是符号使用错误的典型例子，学生通常会被教导如何正确使用这些符号。然而，即使所有符号都齐全且正确，公式中符号的排列方式也存在一个更微妙的问题。

Consider the following exam problem posed to students in an introductory calculus-based course:
(a) Use Gauss’s law to find the electric field outside of an object with a spherically symmetric charge distribution and total charge $2Q$ .
(b) Sketch the magnitude of the electric field outside the sphere as a function of the distance from the center of the sphere, $E_r$ versus $r$ .
考虑以下给微积分基础物理入门课程学生出的考试题目：
(a) 利用高斯定理求具有球对称电荷分布、总电荷量为 $2Q$ 的物体外部的电场强度。
(b) 绘制球外电场强度的大小随到球心距离变化的图像，即 $E_r$ 对 $r$ 的图像。

A student solved the problem correctly using the appropriate reasoning (concentric spherical Gaussian surface, constant electric field magnitude on surface, area of sphere, charge enclosed, etc.). Here is the student’s answer for part (a):
一名学生通过恰当的推理（同心球形高斯面、面上电场强度大小恒定、球面面积、包围电荷等）正确解出了该问题。以下是该学生对第 (a) 问的答案：
$$|\mathbf{E}|=\frac{2Q}{4\pi r^2{\epsilon }_0}.\text{\hspace{1em}(1)}$$

To the practiced eye this appears a bit odd. Why? The formula is not written in the “standard” form (see below). But does this really matter? For this student, yes.
对于有经验的人来说，这个公式看起来有些奇怪。为什么？因为它没有按照“标准”形式书写（见下文）。但这真的重要吗？对这名学生来说，答案是肯定的。

Part (b) of the problem asked the student to make a sketch based on the result from the first part. The student’s sketch was a horizontal line on properly labeled $E_r$ -versus- $r$ axes. During a face-to-face discussion after the exam, it became clear that the student did not “see” the factor of $r^2$ buried in the denominator when attempting to recognize and sketch the functional form of the spatial dependence of the field.
问题的第 (b) 问要求学生根据第一问的结果绘制图像。该学生在标注正确的 $E_r$ -对- $r$ 坐标轴上画了一条水平线。考试后的面对面讨论显示，该学生在试图识别并绘制电场空间依赖的函数形式时，未能“发现”隐藏在分母中的 $r^2$ 因子。

The student’s answer, Eq. (1), contains all of the correct symbols in a mathematically acceptable position (products, numerator, denominator, etc.). From a mathematical standpoint the answer is exactly “right,” by which we mean that if you substituted numbers for each of the symbols you would obtain the correct numerical result. However, colleagues with whom we shared examples such as this described the formulas as “awkward,” or “difficult to interpret,” or “confusing.” What is it that makes them seem odd?
学生的答案（式 (1)）包含了所有正确的符号，且符号在数学上的位置合理（乘积、分子、分母等）。从数学角度来看，这个答案完全“正确”——也就是说，如果将每个符号替换为具体数值，将会得到正确的数值结果。然而，我们向同事展示此类例子时，他们认为这些公式“别扭”“难以解读”或“令人困惑”。是什么让它们显得奇怪呢？

The “standard” form of physics formulas

物理学公式的“标准”形式

Examining the displayed formulas in almost any physics text reveals a standard convention used when physicists write formulas. Let’s look at two typical formulas:
查阅几乎所有物理学教材中的公式都会发现，物理学家书写公式时遵循一种标准惯例。我们来看两个典型公式：

$$x=x_0+v_{0x}t+\frac{1}{2}{a}_x{t}^2\text{and}\mathbf{B}=\frac{{\mu }_0}{4\pi }\frac{Id\mathbf{I}\times \hat{r}}{r^2}..\text{\hspace{1em}(2)}$$

The placement of each of the symbols in these formulas appears in an arrangement that follows canonical (but unwritten) rules. Almost all expressions used in physics texts conform to a “standard” order in which each of the terms is written in the form
这些公式中每个符号的位置都遵循着规范（但未成文）的规则。物理学教材中几乎所有表达式都符合一种“标准”顺序，即每个项都按照以下形式书写：
$$\begin{array}{r}(\text{constants})(\text{parameters})(\text{variables})\end{array}\text{\hspace{1em}(3)}$$

If the term is a fraction, the ordering can be applied to the numerator and denominator separately or factored into a product. The ordering is also applied within the arguments of transcendental functions.
如果该项是分数，该顺序可分别应用于分子和分母，或分解为乘积形式。超越函数的自变量内部也遵循这一顺序。

A first step in helping students learn to read and write formulas using this standard order is to help them classify the various quantities at hand into the categories of constants, parameters, and variables.
帮助学生学会按照这种标准顺序读写公式的第一步，是帮助他们将手头的各种物理量分类为常量、参数和变量。

Constants

常量

These are mathematical or physical quantities that never change such as:
常量是指永远不变的数学或物理量，例如：

• numbers in fractional or decimal form: $\frac{1}{2}$ , $4$ , $2.43$ , …
  分数或小数形式的数字： $\frac{1}{2}$ 、 $4$ 、 $2.43$ 等；
• named numbers: $e$ , $\pi$ , …
  有名称的数： $e$ （自然常数）、 $\pi$ （圆周率）等；
• physical constants: $G$ , ${\epsilon }_0$ , $c$ , $k_B$ , …

物理常量: $G$ (万有引力常量)､ ${\epsilon }_0$ (真空介电常数)､ $c$ (真空中的光速)､ $k_B$ (玻尔兹曼常量)等｡

If more than one “constant” appears, then they are usually ordered in the sequence given above, e.g., in $4\pi {\epsilon }_0$ , the order is number/named number/physical constant.
如果出现多个“常量”，它们通常按上述顺序排列，例如在 $4\pi {\epsilon }_0$ 中，顺序为数字/有名称的数/物理常量。

Parameters

参数

These “quantities are constant for a particular experimental run but can change from run to run.”⁵ Parameters are quite important and many times crucial to our physical understanding. Let’s take the example of a ball dropped from rest near the surface of a planet, where the position is given by:

参数是“在特定实验中保持不变，但在不同实验中可以变化的物理量”⁵。参数非常重要，很多时候对我们的物理理解至关重要。以在行星表面附近从静止释放的小球为例，其位置由下式给出：
$$y(t)=h_0-\frac{1}{2}g{t}^2$$

To appreciate the significance of the parameters $h_0$ and $g$ , think in terms of making measurements. For each experimental run, the quantities $h_0$ and $g$ have a fixed value, while $t$ and $y$ change during the experiment. In another experimental run, we might change the initial height, so $h_0$ would be different, or we could go to a different planet and $g$ would change. Thus, $h_0$ and $g$ are not “constants” in the same way that $\pi$ or $c$ represent particular mathematical or physical constants. So, a parameter is a quantity such as $h_0$ or $g$ that is constant for a particular experimental run, but might change from run to run.
要理解参数 $h_0$ （初始高度）和 $g$ （重力加速度）的意义，可以从测量的角度思考。在每次实验中， $h_0$ 和 $g$ 具有固定值，而 $t$ （时间）和 $y$ （位置）在实验过程中变化。在另一次实验中，我们可能会改变初始高度，因此 $h_0$ 会不同；或者我们前往另一个行星， $g$ 会发生变化。因此， $h_0$ 和 $g$ 并不像 $\pi$ 或 $c$ 那样是代表特定数学或物理常量的“常量”。因此，参数是诸如 $h_0$ 或 $g$ 之类的物理量——在特定实验中保持不变，但在不同实验中可能变化。

Variables

变量

These are usually the quantities of most physical interest and correspond to time, position, electric field, or …. They are most like the inputs and outputs of the mathematical expression of functions.
变量通常是物理意义上最受关注的量，对应时间、位置、电场强度等。它们最类似于函数数学表达式中的输入量和输出量。

Of course one person’s parameter might be another person’s variable. The distinction generally depends on the context and which quantity is of mathematical or physical interest.
当然，一个人的参数可能是另一个人的变量。这种区分通常取决于具体情境以及哪个物理量是数学或物理意义上的关注焦点。

Be explicit with students: The “standard” ordering matters

向学生明确说明：“标准”顺序至关重要

When physicists discuss phenomena using mathematics, the “standard” ordering can be a crucial part of the discussion. Typically we want to understand the essential behavior of one physical quantity as a function of some other quantity.
当物理学家用数学语言讨论现象时，“标准”顺序可能是讨论的关键部分。通常，我们希望理解一个物理量作为另一个物理量的函数所表现出的基本行为。

For example, how does the electric field depend on distance or on charge? Specifically, consider the magnitude of the electric field for a long, thin rod with uniform linear charge density ${\lambda }_0$ . We show three versions of the formula that exhibit different emphases [as indicated in square brackets]:
例如，电场强度如何依赖于距离或电荷量？具体来说，考虑具有均匀线电荷密度 ${\lambda }_0$ 的细长直杆产生的电场强度大小。我们给出该公式的三个版本，分别突出不同重点[方括号内标注]：

$$\begin{array}{rl}|\mathbf{E}|=E& =\frac{{\lambda }_0}{2\pi {ϵ}_0}\frac{1}{r}[everything]\\ |\mathbf{E}|=E& =(\_)\frac{{\lambda }_0}{r}[parameter/variable]\\ |\mathbf{E}|=E& =(\_)\frac{1}{r}[variable]\end{array}\text{\hspace{1em}(4)(5)(6)}$$

From the second version the experienced reader can readily see that the field doubles upon doubling the charge density or halving the distance. This “standard” form is important when physicists interpret and discuss the physical meaning of a formula.
从第二个版本中，有经验的读者可以很容易地看出：电荷密度加倍或距离减半时，电场强度加倍。这种“标准”形式在物理学家解读和讨论公式的物理意义时非常重要。

While some students will write formulas properly by imitation or through repetition, many will not. Because there is such a strong adherence to the formula conventions, we can help students communicate more effectively by being explicit about these conventions. To help students better use and appreciate the power of the mathematical description of the world, we suggest that instructors (and textbook authors) explicitly discuss the conventions regarding the “standard” form and the ordering of quantities in formulas. This might be done by following a treatment similar to that presented here. This should include a discussion surrounding the distinctions among constants, parameters, and variables.
虽然有些学生通过模仿或反复练习能够正确书写公式，但很多学生做不到。由于公式惯例的遵循度很高，明确传授这些惯例可以帮助学生更有效地进行交流。为了帮助学生更好地运用并理解数学描述世界的力量，我们建议教师（和教材作者）明确讨论公式“标准”形式和物理量排序的相关惯例。可以按照本文呈现的方式进行讲解，其中应包括对常量、参数和变量之间区别的讨论。

As with most things, achieving facility requires repetition and practice with timely feedback. Therefore, instructors should provide explicit practice in working with the standard form and should enforce the standard form when evaluating student work. For example, instructors could ask students to rewrite Eq. (1) in “standard” form and then expect one of the following:

与大多数事情一样，熟练掌握需要反复练习和及时反馈。因此，教师应提供关于标准形式的明确练习，并在评价学生作业时强制执行标准形式。例如，教师可以要求学生将式 (1) 改写为“标准”形式，并期望得到以下结果之一：
$$|\mathbf{E}|=\frac{2Q}{4\pi {\epsilon }_0}\frac{1}{r^2}=\frac{1}{2\pi {\epsilon }_0}\frac{Q}{r^2}.\text{\hspace{1em}(7)}$$

In evaluating student work, we might invent a new copyediting symbol such as “NSF” for “not in standard form.”
在评价学生作业时，我们可以引入一个新的编辑符号，例如“NSF”（即“未采用标准形式”）。

Conclusion

结论

Physics formulas follow implicit conventions and are written in a “standard” form. We have identified the elements of this form consisting of constants, parameters, and variables, in that order. To assist students, instructors are encouraged to make this convention explicit and encourage its use.
物理学公式遵循隐含惯例，具有“标准”书写形式。本文明确了该形式的构成要素：常量、参数和变量，且按此顺序排列。为帮助学生学习，建议教师明确传授这一惯例并鼓励学生使用。

Acknowledgments

致谢

We thank T. Bensky, E. Demsetz, and J. Keller for helpful discussions. MJM thanks the Physics Education Research Group in the School of Physics at the Dublin Institute of Technology for their hospitality during a sabbatical leave.
感谢 T·本斯基（T. Bensky）、E·德姆塞茨（E. Demsetz）和 J·凯勒（J. Keller）提供的有益讨论。MJM 感谢都柏林理工学院物理学院物理教育研究小组在其休假期间提供的热情款待。

Physics teachers… get your students registered for the preliminary exam in the U.S. Physics Team selection process.
物理教师们……请让你们的学生报名参加美国物理奥林匹克队选拔的预赛。

All physics students are encouraged to participate in the American Association of Physics Teachers’ $F_{\text{net}}=ma$ Contest! The U.S. Physics Team Program provides a once-in-a-lifetime opportunity for students to enhance their physics knowledge as well as their creativity, leadership, and commitment to a goal.
鼓励所有物理专业学生参加美国物理教师协会举办的 $F_{\text{net}}=ma$ 竞赛！美国物理奥林匹克队项目为学生提供了千载难逢的机会，助力他们提升物理知识水平，同时培养创造力、领导力和对目标的执着追求精神。

References

参考文献

1. Robert H. Romer, “Reading the equations and confronting the phenomena–The delights and dilemmas of physics teaching," Am. J. Phys. 61, 128–142 (Feb. 1993).
   罗伯特·H·罗默(Robert H. Romer),《解读方程与直面现象——物理学教学的乐趣与困境》,《美国物理学杂志》(Am. J. Phys.),第 61 卷,第 128-142 页(1993 年 2 月)｡

2. Paul G. Hewitt, “Equations as guides to thinking and problem solving," Phys. Teach. 49, 264 (May 2011).
   保罗·G·休伊特(Paul G. Hewitt),《方程作为思考与解题的指南》,《物理教师》(Phys. Teach.),第 49 卷,第 264 页(2011 年 5 月)｡

3. Discussion and follow-up interview with student in calculus-based introductory course for science and engineering students.
   与理工科微积分基础物理入门课程学生的讨论及后续访谈｡

4. In advanced courses, especially those in high energy or particle physics, common practice is to use a system of units such that $\hslash =1$ or $c=1$ and the like. However, this is not common at the introductory or intermediate level, so we ignore it here.
   在高级课程中,特别是高能物理或粒子物理相关课程,常用自然单位制(如令 $\hslash =1$ 或 $c=1$ 等)｡但这种做法在入门或中级课程中并不常见,因此本文暂不考虑｡

5. Regarding parameters we closely follow the discussion in Andrew F. Rex and Martin Jackson, Integrated Physics and Calculus (Addison-Wesley, New York, 2000), p. 39.

关于参数的定义,我们主要参考安德鲁·F·雷克斯(Andrew F. Rex)和马丁·杰克逊(Martin Jackson)所著《整合物理与微积分》(Integrated Physics and Calculus,艾迪生-韦斯利出版社,纽约,2000 年)第 39 页的论述｡

Martin Jackson has been a professor of mathematics at the University of Puget Sound since 1990. In collaboration with colleagues from physics, he has team-taught a course that integrates the introductory physics and calculus sequences. The text Integrated Physics and Calculus, coauthored with Andrew Rex, is based on that course.
Department of Mathematics and Computer Science, University of Puget Sound, Tacoma, WA 98416; martinj@pugetsound.edu
马丁·杰克逊（Martin Jackson）自 1990 年起担任普吉特海湾大学数学教授。他与物理学科的同事合作，共同讲授一门整合了入门物理和微积分系列内容的课程。与安德鲁·雷克斯（Andrew Rex）合著的《整合物理与微积分》一书即基于该课程编写。
普吉特海湾大学数学与计算机科学系，塔科马，华盛顿州 98416；电子邮箱：martinj@pugetsound.edu

Matthew Moelter is a professor of physics at Cal Poly and teaches all manner of introductory physics. He has particular interests in physics education and condensed matter.
Department of Physics, California Polytechnic State University, San Luis Obispo, CA 93407; mmoelter@calpoly.edu
马修·莫尔特（Matthew Moelter）是加利福尼亚州立理工大学物理学教授，讲授各类入门物理课程。他的研究兴趣主要集中在物理教育和凝聚态物理领域。
加利福尼亚州立理工大学物理系，圣路易斯奥比斯波，加利福尼亚州 93407；电子邮箱：mmoelter@calpoly.edu

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The Role of Math in Physics: Why Mathematics Is the Language of the Universe

数学在物理学中的作用：为什么数学是宇宙的语言

Published on February 27, 2025
By EdQuill Academy

Physics seeks to explain how the universe works—from the motion of planets to the behavior of subatomic particles. But behind every physical law and scientific discovery is something essential: mathematics.
物理学试图解释宇宙是如何运作的——从行星的运动到亚原子粒子的行为。但在每一条物理定律和每一个科学发现的背后，都有一样不可或缺的东西：数学。

Math is not just a tool in physics; it is the very language that allows scientists to describe, predict, and understand natural phenomena. Without math, many of the breakthroughs in physics—from Newton’s laws of motion to Einstein’s theory of relativity—would never have been possible.
数学不仅仅是物理学中的一个工具；它是科学家用来描述、预测和理解自然现象的语言。如果没有数学，物理学中的许多突破——从牛顿的运动定律到爱因斯坦的相对论——都将无法实现。

At EdQuill Academy, we know that one of the biggest challenges for students learning physics is not the science itself, but understanding how math connects to the concepts being taught. In this guide, we’ll explore the crucial role math plays in physics and how students can build the skills they need to succeed in both.
在 EdQuill 学院，我们知道，学生在学习物理学时面临的最大挑战之一并不是科学本身，而是理解数学与所教授概念之间的联系。在本指南中，我们将探讨数学在物理学中发挥的关键作用，以及学生如何培养在两者中都取得成功所需的技能。

Why Math Is Essential in Physics

数学在物理学中为何必不可少

At its core, physics is about observing the natural world and asking questions like:
物理学的核心是观察自然世界并提出如下问题：

• How fast is an object moving?
  一个物体移动的速度有多快？
• How much energy does a system have?
  一个系统有多少能量？
• How does force affect motion?
  力如何影响运动？

To answer these questions clearly and accurately, we need more than words—we need numbers, relationships, and formulas. This is where math comes in. Math allows physicists to describe what they see, make predictions, and test theories with precision.
要清晰准确地回答这些问题，我们需要的不仅仅是文字——我们需要数字、关系和公式。这就是数学的用武之地。数学使物理学家能够精确地描述他们所看到的，进行预测，并验证理论。

Mathematics transforms abstract ideas into measurable quantities, providing a framework to express how things change and interact.
数学将抽象的概念转化为可测量的量，提供了一个表达事物如何变化和相互作用的框架。

The Connection Between Physical Laws and Mathematical Equations

物理定律与数学方程之间的联系

Every major concept in physics has a mathematical counterpart. Some examples include:
物理学中的每一个主要概念都有一个数学对应物。一些例子包括：

• Newton’s Second Law of Motion:
  牛顿第二运动定律：
  $F=ma$ (Force = mass × acceleration)
  力 = 质量 × 加速度

• Kinetic Energy:
  动能：
  $KE=\frac{1}{2}m{v}^2$ (Kinetic energy is half the mass times the square of the velocity)
  动能是质量的一半乘以速度的平方

• Ohm’s Law (Electricity):
  欧姆定律（电学）：
  $V=IR$ (Voltage = current × resistance)
  电压 = 电流 × 电阻

• Einstein’s Energy-Mass Equivalence:
  爱因斯坦的质能方程：
  $E=m{c}^2$ (Energy equals mass times the speed of light squared)
  能量等于质量乘以光速的平方

These equations are not just math exercises—they represent the underlying principles of how the universe operates.
这些方程不仅仅是数学练习——它们代表了宇宙运行的基本原理。

Why Physics Needs Algebra, Geometry, Trigonometry, and Calculus

物理学为何需要代数、几何、三角学和微积分

Different branches of mathematics play specific roles in physics:
数学的不同分支在物理学中扮演着特定的角色：

Algebra

代数

Algebra allows students to rearrange equations and solve for unknowns, which is critical when working with formulas in physics.
代数使学生能够重新排列方程并求解未知数，这在使用物理学中的公式时至关重要。

Geometry

几何

Geometry helps describe shapes, angles, and spatial relationships—important when studying vectors, forces, and motion paths.
几何有助于描述形状、角度和空间关系——在研究矢量、力和运动路径时非常重要。

Trigonometry

三角学

Trigonometry is essential for resolving vectors, calculating angles, and analyzing wave behavior. It’s especially useful in topics like projectile motion, optics, and electromagnetism.
三角学对于解析矢量、计算角度和分析波的行为至关重要。它在抛体运动、光学和电磁学等主题中特别有用。

Calculus

微积分

Calculus introduces concepts of change and accumulation, which are necessary for understanding rates of change (derivatives) and total quantities (integrals). Calculus makes it possible to analyze motion, growth, energy, and fields that vary over time or space.
微积分引入了变化和积累的概念，这对于理解变化率（导数）和总量（积分）是必要的。微积分使得分析随时间或空间变化的运动、增长、能量和场成为可能。

One List: How Math Helps Explain Key Concepts in Physics

一份清单：数学如何解释物理学中的关键概念

Here’s a clear breakdown of how math directly supports important areas of physics:
以下清晰地列出了数学如何直接支持物理学中的重要领域：

• Describing motion: Using equations of motion to calculate speed, velocity, and acceleration.
  描述运动： 使用运动方程计算速度、加速度和速度。
• Calculating forces: Applying Newton’s laws with algebraic equations to determine the effects of force on objects.
  计算力： 使用代数方程应用牛顿定律来确定力对物体的影响。
• Analyzing energy: Computing kinetic, potential, and thermal energy using mathematical formulas.
  分析能量： 使用数学公式计算动能、势能和热能。
• Understanding waves and oscillations: Using sine and cosine functions to model wave motion and vibration.
  理解波和振动： 使用正弦和余弦函数来模拟波的运动和振动。
• Exploring electricity and magnetism: Applying algebra and calculus to analyze circuits, charge, and electromagnetic fields.
  探索电学和磁学： 应用代数和微积分分析电路、电荷和电磁场。
• Studying fluids and pressure: Using ratios, proportions, and calculus to solve problems involving fluid dynamics.
  研究流体和压力： 使用比例、比率和微积分解决流体动力学问题。
• Modeling planetary motion: Employing geometry and calculus to predict orbits and gravitational effects.
  模拟行星运动： 使用几何和微积分预测轨道和引力效应。
• Explaining thermodynamics: Applying equations to calculate heat transfer, work, and energy efficiency.
  解释热力学： 应用方程计算热传递、功和能量效率。

Why Physics and Math Together Build Problem-Solving Skills

物理学和数学结合为何能培养解决问题的能力

Learning to apply math in physics does more than help students pass exams. It builds critical thinking and problem - solving skills that are valuable across disciplines. Students learn to:
学习在物理学中应用数学不仅仅是帮助学生通过考试。它还能培养在各个学科中都有价值的批判性思维和解决问题的能力。学生学会：

• Break complex problems into manageable parts.
  将复杂问题分解为可管理的部分。
• Identify the knowns and unknowns in a scenario.
  识别场景中的已知量和未知量。
• Choose the right formula or approach for the problem.
  选择适合问题的正确公式或方法。
• Check their solutions for accuracy and logic.
  检查他们的解决方案是否准确和合理。
• Understand why a solution works—not just how to get the answer.
  理解为什么一个解决方案有效——而不仅仅是如何得到答案。

This approach to thinking prepares students not only for advanced studies in science and engineering but also for real - life problem - solving.
这种思维方式不仅为学生在科学和工程领域的高级学习做好了准备，也为现实生活中的问题解决做好了准备。

The Real - World Impact of Math in Physics

数学在物理学中的现实世界影响

The role of math in physics extends far beyond the classroom. Some examples of real - world applications include:
数学在物理学中的作用远远超出了课堂。一些现实世界的例子包括：

• Engineering and architecture: Designing bridges, buildings, vehicles, and machines with precise calculations.
  工程和建筑： 使用精确计算设计桥梁、建筑、车辆和机器。
• Medicine and healthcare: Using physics - based imaging technologies like MRIs and CT scans.
  医学和保健： 使用基于物理学的成像技术，如磁共振成像（MRI）和计算机断层扫描（CT）。
• Environmental science: Modeling climate systems, predicting weather patterns, and analyzing pollution data.
  环境科学： 建模气候系统、预测天气模式和分析污染数据。
• Technology and electronics: Developing computers, smartphones, and communication systems based on principles of electricity and magnetism.
  技术和电子： 基于电学和磁学原理开发计算机、智能手机和通信系统。
• Space exploration: Calculating trajectories, fuel needs, and navigation for spacecraft.
  太空探索： 计算航天器的轨迹、燃料需求和导航。

These examples highlight that understanding the math behind physics opens doors to innovation and discovery across multiple fields.
这些例子突显了理解物理学背后的数学知识可以为多个领域的创新和发现打开大门。

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10 Formula Memorisation Hacks for Maths and Physics

数学和物理公式的 10 个记忆技巧

Discover ten practical ways to remember mathematics and physics formulas. Each method uses simple steps that help with steady practice, clear understanding, and long - term recall.
发现十种实用的方法来记忆数学和物理公式。每种方法都使用简单的步骤，有助于稳定练习、清晰理解以及长期记忆。

MC EDUCATION DESK
NOVEMBER 27, 2025 / 17:06 IST

1. Break Long Formulas into Smaller Parts

将长公式分解为较小的部分

When a formula looks long, it becomes easier to remember when you divide it into small sections. Look at each part and understand what it means in the problem. This method helps you recall the order of terms because you learn them piece by piece. It also reduces confusion during revision, as you focus on one section at a time instead of the entire formula at once.

当一个公式看起来很长时，将其分解为小部分后会更容易记忆。查看每个部分并理解它在问题中的含义。这种方法可以帮助你回忆各项的顺序，因为你是一点一点地学习它们的。在复习时，这种方法也可以减少混淆，因为你一次只关注一个部分，而不是一次看整个公式。

2. Understand the Meaning Behind Each Formula

理解每个公式背后的含义

Formulas become easier to remember when you know how they work. Try to understand why each symbol is used and what the formula helps you calculate. When you connect the formula to a real idea, it becomes much simpler to recall. This also helps during exams because you know when to use the formula and how to apply it correctly in different questions.

当你知道公式是如何运作的时候，公式会变得更容易记忆。尝试理解每个符号的用途以及公式帮助你计算的内容。当你将公式与一个实际的想法联系起来时，它会变得更容易回忆。这在考试中也很有帮助，因为你知道何时使用公式以及如何在不同问题中正确应用它。

3. Use Real Examples to Practise the Formula

使用实际例子来练习公式

Learning becomes easier when you use examples from real situations. Pick simple problems and apply the formula several times. This repetition helps the formula move into long - term memory without feeling forced. When you see how the formula works in different conditions, you also develop the ability to use it with confidence during tests and assignments.

当你使用实际情境中的例子时，学习会变得更容易。选择简单的问题并多次应用公式。这种重复有助于公式进入长期记忆，而不会让人觉得是被迫的。当你看到公式在不同条件下是如何运作的时候，你也会培养出在考试和作业中自信使用它的能力。

4. Create Your Own Patterns and Associations

创建你自己的模式和联想

Sometimes it helps to link a formula with a pattern or a personal connection. You can relate certain terms with objects, directions, or steps that make sense to you. These links act like small reminders. When your brain connects two things, it becomes easier to bring back the formula when you need it. This method works well for both short and long formulas.

有时，将公式与一个模式或个人联系联系起来会很有帮助。你可以将某些术语与对你有意义的物体、方向或步骤联系起来。这些联系就像小提示一样。当你的大脑将两件事联系起来时，当你需要时，回忆起公式会变得更容易。这种方法对于短公式和长公式都很有效。

5. Practise Spaced Repetition While Revising

复习时练习间隔重复

Instead of revising everything in one sitting, spread your practice over different days. Look at the formulas today, then again after a few hours, and once more the next day. This steady revision strengthens memory and prevents forgetting. It also gives you time to understand the formula better and use it with less effort during problem solving.

不要一次复习所有内容，而是将你的练习分散在不同的日子。今天看看公式，然后几个小时后再看一次，第二天再看一次。这种稳定的复习可以加强记忆并防止遗忘。它也给你时间更好地理解公式，并在解决问题时更轻松地使用它。

6. Write Formulas by Hand While Studying

学习时手写公式

Writing formulas helps you remember them better because your mind stays engaged while your hand moves. Copy the formulas on paper several times while saying each step clearly to yourself. This method builds muscle memory and mental recall at the same time. Keeping a small notebook of formulas also gives you a quick guide to revise before an exam or class test.

手写公式可以帮助你更好地记住它们，因为当你手在动的时候，你的大脑也会保持参与。在纸上多次抄写公式，同时清晰地对自己说出每一步。这种方法同时建立了肌肉记忆和心理回忆。保持一个公式的笔记本也可以在考试或课堂测试前为你提供一个快速复习的指南。

7. Use Visual Memory with Diagrams or Colour Coding

使用图表或颜色编码来利用视觉记忆

Many students remember formulas better when they see them linked to a drawing or layout. Create simple diagrams to show how the formula works. You can also use different colours to highlight terms that often confuse you. These visual cues make the formula stand out in your mind and help you recognise the correct pattern when you see it again.

许多学生在看到公式与一幅画或布局联系在一起时，会更好地记住公式。创建简单的图表来展示公式是如何运作的。你也可以使用不同的颜色来突出那些经常让你困惑的术语。这些视觉提示会使公式在你的脑海中脱颖而出，并帮助你在再次看到它时识别出正确的模式。

8. Group Similar Formulas Together

将相似的公式放在一起

In subjects like mathematics and physics, many formulas follow a shared structure. Place formulas with similar patterns together and compare them. This helps you notice how they change and why those changes matter. When you understand the relationships between formulas, you find it much easier to remember them during revision, especially when two formulas look almost the same.

在数学和物理等学科中，许多公式遵循相同的结构。将具有相似模式的公式放在一起并进行比较。这有助于你注意到它们是如何变化的以及这些变化为什么重要。当你理解了公式之间的关系时，在复习时你会更容易记住它们，特别是当两个公式看起来几乎相同时。

9. Learn Through Teaching Someone Else

通过教别人来学习

Explaining a formula to another person forces your brain to recall it clearly. When you teach someone, you tend to break the formula into steps and explain why each part is important. This strengthens your understanding and makes the formula easier to remember later. Even if you do not have someone to teach, you can speak out loud as if you are explaining it to a class.

向另一个人解释一个公式会迫使你的大脑清晰地回忆起它。当你教别人时，你往往会将公式分解为步骤，并解释每个部分为什么重要。这会加强你的理解，并使公式在以后更容易记住。即使你没有可以教的人，你也可以大声说出来，就好像你在给一个班级讲解一样。

10. Use Short Daily Revision Sessions

使用短暂的日常复习课程

A short daily practice session is one of the best ways to remember formulas for a long time. Spend ten to fifteen minutes reading, writing, and applying formulas. This habit keeps them fresh in your mind and stops you from forgetting important steps. Regular repetition also builds confidence, as you do not rush to learn everything at the last moment before exams.

短暂的日常练习课程是长时间记住公式最好的方法之一。花十到十五分钟来阅读、书写和应用公式。这种习惯会使它们在你的脑海中保持新鲜，并阻止你忘记重要的步骤。定期重复也会建立信心，因为你不会在考试前的最后一刻匆忙学习所有内容。

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via:

• Formulas in Physics Have a “Standard” Form | Semantic Scholar
  https://www.semanticscholar.org/paper/Formulas-in-Physics-Have-a-“Standard”-Form-Moelter-Jackson/881d92fe2ed5a330f8fb922e057d73d73a2caaaf
• The Role of Math in Physics: Why Mathematics Is the Language of the Universe | EdQuill Academy
  https://blog.edquillacademy.com/the-role-of-math-in-physics-why-mathematics-is-the-language-of-the-universe/
• 10 Formula Memorisation Hacks for Maths and Physics
  https://www.moneycontrol.com/education/10-formula-memorisation-hacks-for-maths-and-physics-photo-gallery-13699699.html
• 4 Ways to Memorize Math and Physics Formulas - wikiHow
  https://www.wikihow.com/Memorize-Math-and-Physics-Formulas