Clairauts “equality of mixed partial derivatives” theorem

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Discussion Overview

The discussion revolves around Clairaut's theorem on the equality of mixed partial derivatives, focusing on its interpretation, visualization, and the conditions under which it holds. Participants explore the implications of the theorem in both mathematical and physical contexts.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested
  • Meta-discussion

Main Points Raised

  • One participant expresses a desire to understand the visualization of Clairaut's theorem and questions whether the theorem's assertion of symmetry in change applies only to second derivatives.
  • Another participant notes that Clairaut's theorem is valid when the first partial derivatives are continuous and emphasizes the importance of understanding the conditions for mathematical validity.
  • A physics student questions whether a prominent physicist, such as Feynman, would understand the conditions of the theorem, inviting speculation on the relationship between mathematics and physics knowledge.
  • A later reply suggests that in physical models, continuity of derivatives can often be justified on physical grounds, and highlights the practical balance between learning mathematics and physics for students in those fields.

Areas of Agreement / Disagreement

Participants generally agree on the importance of understanding the conditions under which Clairaut's theorem holds, but there is a divergence in perspectives regarding the necessity of this understanding for physics students compared to mathematics students.

Contextual Notes

Some assumptions about continuity and the applicability of the theorem in physical contexts remain unresolved, as well as the extent to which visualization aids understanding of the theorem.

Who May Find This Useful

Students and professionals in mathematics, physics, and engineering who are interested in the foundational aspects of calculus and the implications of mathematical theorems in physical applications.

davidbenari
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I know how to prove this via limits and I'm okay with that.

What I want to understand is the interpretation of the theorem and specifically a visualisation of why what the theorem states must be the case.

My guess is that this theorem is saying that change is symmetrical. But I don't know if this is only true for second derivatives.

If you don't know this theorem by its name the theorem basically says this:

∂/∂y(∂f/∂x)=∂/∂x(∂f/∂y)

Also, I would like to know if you consider my focus on visualisation to be not worthwhile and that I should instead just trust this theorem.

I thank you in advance.
 
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I am a physics student. I have a question about this "if you are a mathematics student then it is worthwhile" stuff. Would a great physicist, say like Feynman, know why this is so? Even if your answer is speculative, what do you think?
 
If the math is a model of some physical situation, you can often argue that derivatives are continuous, etc, on physical grounds. Usually, a math model only hits the "exceptional" conditions that mathematicians like to understand completely, if it's a poor model of the physics.

IMO it is always worthwhile (not to say essential) to know something about the conditions that make your math valid. But as a physicist or engineer you don't necessarily need to know the most general set of conditions that make it valid, or be able to prove why it is valid.

Of course you can never know "too much" math, but in real life, whether you learn more math or more physics is a time management problem.

(Full disclosure: I've seen both sides of this first hand - I have a math degree, and spent most of my life working on engineering problems).
 
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