科学与数学web.williams.edu编辑精选
希望我在开始教微分方程前就明白的事（1997）
Lessons I wish I had learned before I started teaching differential equations [pdf] (1997)
2017 — 2022，它一次次回到首页
黑点是上首页（≥ 50 分）的那次投稿，空心点是没冲上去的投稿，点越大分数越高。
导读
这是数学家 Gian-Carlo Rota 对微分方程教学的反思，列出他希望在开始教这门课之前就明白的经验教训。他讨论了'函数'与'密度函数'两种概念在拉普拉斯变换和测度论中的混淆，主张学生应远离微分符号（differentials），并质疑是否值得花时间教解的存在唯一性等理论细节。
为什么经久不衰：Rota 以数学家的犀利眼光审视微分方程教学中的概念混乱与取舍问题，文中关于密度函数记号与微分符号的批评，对今天仍在学数学和教数学的人依然切中要害。
HN 评论精选
不同意忽略存在唯一性的建议，指出 Lipschitz 条件下存在唯一性通常不难证明。
常微分方程最有趣的应用之一是控制理论，它让自主系统、机器人或车辆能自我导航（Brian Douglas 有很棒的入门视频）。我不完全同意第5节（忘记解的存在唯一性）：对 xdot = f(t,x) 型的 ODE，只要 f 对 t 连续、对 x Lipschitz 连续，解在某个区间上就存在且唯一，这通常不太难证明。若无法证明唯一性，就可能要把系统当作微分包含来处理。
ubj查看英文原文
One of the most interesting applications of ODEs (in my opinion) is the field of control theory, which empowers autonomous systems/robots/vehicles to guide themselves. Brian Douglas has some fantastic introductory videos on the subject [1]. I don't think I agree completely with section 5 (Forget About Existence and Uniqueness of Solutions). For an ODE in the form xdot = f(t,x), existence and uniqueness on an interval follows if f is continuous in t and Lipschitz continuous in x [2]. This often isn't too difficult to demonstrate for a given system. In addition, if you can't prove uniqueness of solutions then you likely have to treat the system as a differential inclusion [3]. I personally find differential inclusions fascinating, but the math can be much more dense. [1]: https://engineeringmedia.com/videos [2]: https://en.m.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem [3]: https://en.m.wikipedia.org/wiki/Differential_inclusion
数学课与物理课对微分符号的处理不一致，导致学习困难，呼应'远离微分'的建议。
数学课上每个对象都有定义、定理都证明，我们学了微分方程和微分形式；但物理课上 dy/dx = f(x) 可以变成 dy = f(x)dx 再两边积分，'dx' 突然成了无穷小量，而它之前要么是线性形式的记号、要么是导数的记号。物理课用的数学没被形式化定义，似乎遵循要自己在期末考试中猜出来的不成文规则，这让我很困扰。
yodsanklai查看英文原文
> 7. stay away from differentials I always had issues with this. My undergraduate math classes were very formal and well thought, every object was defined, almost all theorems were proved in class. In particular, we were taught differential equations theories, some general theories like systems of differential equations with constant coefficients, Cauchy's theorem, and a bunch of recipes. In a separate class, we were also taught differential forms (e.g. a.dx + b.dy being a linear combination of base elements dx and dy - if I remember well). But then came the physics class, why we could go from dy/dx = f(x) to dy = f(x).dx and integrate both sides remain a mystery. "dx" suddenly became an infinitely small things, when it used to be a notation for a linear form OR a notation for f'(x) === df/dx. Generally speaking, I had a lot of issues in physics because the maths used in that context were not formally defined and seemed to obey to some unwritten set of rules that you were supposed to find out by yourself during your final exam...
赞同 Rota 对函数/密度函数记号混淆的批评，认为把测度当作密度来写更自然。
Rota 指出拉普拉斯变换中'函数'的两种用法被危险地混淆：一种是有图像的普通函数，另一种是作为密度（如质量或概率密度）的函数，数学家通过 Stieltjes 积分、测度等方式回避面对后者，但物理学和工程学中密度函数的记号可证明更优。我也曾建议把测度论中的 ∫f(x)dμ(x) 改写成 ∫f(x)·μ(x)dx，把测度当作密度函数，但常被数学家问'这有什么意义'。Radon-Nikodym 导数写成 μ/λ 也比 dμ/dλ 更自然——它看起来就像除法。
ogogmad查看英文原文
I find the following remark by Rota interesting: Insofar as the Laplace transform goes, two radically different uses of the word “function” are dangerously confused with each other. The first is the ordinary notion of function as a something that has a graph. The second is the radically different notion of function as density, whether mass density or probability density. For the sake of the argument, let us agree to call this second kind of function “density function.” Professional mathematicians have avoided facing up to density functions by a variety of escapes, such as Stieltjes integrals, measures, etc. But the fact is that the current notation for density functions in physics and engineering is provably superior, and we had better face up to it squarely I've suggested to mathematicians that the notation ⌠ | f(x) dμ(x) ⌡ in measure theory where μ is a measure should be replaced with ⌠ | f(x)⋅μ(x) dx ⌡ where the measure μ is treated as a density function. What I've found is a mixture of either not understanding the logic, or the response "What's the point?" So it's interesting seeing Gian-Carlo Rota suggest the same idea AFAICT. As a bonus, the Radon-Nikodym "derivative" would be written as μ/λ instead of dμ/dλ. The RN derivative then doesn't appear like a derivative - it looks simply like division.
课程内容严重过载，学生机械记住 Jordan 标准型算法却无法回答基本的可逆性问题。
我做过两学期 ODE 课程助教。主要问题是单学期塞了太多内容：学生事先几乎不懂线性代数，却要在六周内消化整个线性代数课再加微分方程。为解常系数线性 ODE 系统要算 Jordan 标准型，学生靠死记步骤在考试中做出来了，但同一份考卷上问一个特征值只有 2 和 -2 的 5x5 矩阵是否可逆，没人能合理回答。这有什么意义？
woopwoop查看英文原文
I TA'd for an undergrad ODE course for two semesters. I'm a grad student at a public university with very good undergrad engineering students. Nonetheless, with the course I taught, at least, the main problem was that WAY too much was packed into a single semester. I suspect that this is the same everywhere. For example, about a month and a half of the course was devoted to systems of linear equations. These were students who, a priori, knew basically no linear algebra, being asked to understand an entire linear algebra course, with some extra stuff thrown in (you know, the differential equations), in six weeks. In order to solve a general system of constant coefficient linear odes, you have to take a matrix and compute its Jordan canonical form. The students I taught were performing this calculation by the end of the unit, but it was of course a joke. On their exam, they were asked to do this calculation (in differential equations language), and most of them were able to do it because they were good students and had memorized the procedure. Then, in another question, they were presented a 5x5 matrix, told that its only eigenvalues were 2 and -2, and asked if it was invertable. I don't think anyone gave a reasonable answer. What is the point of that? Edit: Everyone should be aware of this amazing Gian-Carlo Rota quote, the entirety of a book review on contemporary philosophers: "When pygmies cast such long shadows, it must be very late in the day."
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