Question About Exact Differential Form

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Discussion Overview

The discussion revolves around the concept of exact differential forms and the conditions under which they are considered exact, particularly focusing on the compatibility condition involving mixed partial derivatives. Participants explore the implications of the theorem regarding the equality of continuous mixed partial derivatives in the context of functions defined by differential forms.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the compatibility condition $$\frac{\partial}{\partial y}M(x,y)=\frac{\partial}{\partial x}N(x,y)$$ and its relation to the equality of continuous mixed partial derivatives.
  • Another participant provides a link to a Wikipedia article on the symmetry of second derivatives, suggesting it may be relevant.
  • A participant clarifies that the theorem states if a function has two continuous partial derivatives, then $$\frac{\partial^2 F}{\partial x \partial y} = \frac{\partial^2 F}{\partial y \partial x}$$, indicating a necessary condition for the functions M and N.
  • One participant confirms the understanding that the equality of mixed partial derivatives holds when the derivatives are continuous, but notes that there exist functions where this does not hold if the derivatives are not continuous.
  • A question is raised about the applicability of the theorem in ranges where the second order mixed partials are continuous.
  • A later reply affirms that the theorem is applicable in neighborhoods where the conditions of continuity are met, emphasizing the local nature of differentiability and continuity.

Areas of Agreement / Disagreement

Participants generally agree on the conditions under which the theorem applies, particularly regarding the necessity of continuity for the equality of mixed partial derivatives. However, there is some uncertainty about the implications when continuity is not present, leading to a nuanced discussion.

Contextual Notes

The discussion highlights the importance of continuity in the context of mixed partial derivatives and the potential existence of functions where the second order partials are not continuous, which complicates the application of the theorem.

Drakkith
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My book is going through a proof on exact differential forms and the test to see if they're exact, and I'm lost on one part of it.

It says:

If $$M(x,y)dx + N(x,y)dy = \frac{\partial F}{\partial x}dx + \frac{\partial F}{\partial y}dy$$ then the calculus theorem concerning the equality of continuous mixed partial derivatives $$\frac{\partial }{\partial y}\frac{\partial F}{\partial x}=\frac{\partial }{\partial x}\frac{\partial F}{\partial y}$$ would dictate a "compatibility condition" on the functions ##M## and ##N##: $$\frac{\partial}{\partial y}M(x,y)=\frac{\partial}{\partial x}N(x,y)$$

What does this mean? What is the "calculus theorem concerning the equality of continuous mixed partial derivatives" it talks about?
 
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the calculus theorem says that if F has 2 continuous partial derivatives then ∂^2F/∂x∂y = ∂^2F/∂y∂x. Thus a necessary condition for M to equal ∂F/∂x and for N to equal ∂F/∂y, is that we must have ∂M/∂y = ∂^2F/∂x∂y = d^2F/∂y∂x = ∂N/∂x.
 
So, this is basically saying that given a function ##F(x,y)##, the 2nd order mixed partial derivatives of that function are equal even if you flip the order in which you take the derivatives?

If so, then the compatibility condition that ##\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}## makes sense.
It helps when you understand what a 2nd order mixed partial derivative is. :rolleyes:
 
the theorem does not say that the second order partials must always be equal in both orders, but it does hold if they are continuous. i.e. there do exist functions whose second order partials exist but are not continuous, and then the mixed order partials do not need to be equal.
 
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If the 2nd order mixed partials are not continuous, I assume the theorem is still true over the range in which they are continuous?
 
Drakkith said:
If the 2nd order mixed partials are not continuous, I assume the theorem is still true over the range in which they are continuous?
Yes, because differentiability as well as continuity are local properties. If you narrow down the domain to an open neighborhood where the conditions are met, the theorem is applicable.
 
Got it. Thanks all!
 

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