Grad f is proportional to grad g

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Discussion Overview

The discussion revolves around the relationship between the gradients of two functions, f and g, in Rn, specifically whether proportional gradients imply the existence of a differentiable function F such that F(f(x), g(x)) = 0 everywhere. The scope includes theoretical exploration and implications in fluid dynamics.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants propose that if the gradients of f and g are proportional everywhere, it might imply the existence of a differentiable function F such that F(f(x), g(x)) = 0 everywhere.
  • Others argue that while the converse is true by the chain rule, the original question remains uncertain and requires further exploration.
  • A participant suggests that taking F(x,y) to be identically zero is a trivial solution, but they seek a more interesting functional dependence between f and g.
  • One participant references barotropic fluid flows and questions whether the conditions involving gradients and pressure-density relationships are equivalent or if one is more general than the other.
  • Another participant asserts that if the gradients are everywhere proportional, the functions' level surfaces are identical, suggesting a local functional dependence between the two functions.
  • A follow-up question is raised about the existence of a global version of the relationship, indicating potential limitations in extending local results globally.

Areas of Agreement / Disagreement

Participants express differing views on the implications of proportional gradients, with some suggesting local relationships while others question the global applicability. The discussion remains unresolved regarding the existence of a global version of the proposed relationship.

Contextual Notes

Participants note that the relationship between f and g may not be one-to-one, and there are concerns about the conditions under which the gradients do not vanish, which could affect the validity of the proposed implications.

techmologist
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If the gradients of the two functions f and g are proportional everywhere in Rn, does that mean there is some differentiable function F of two variables such that F(f(x),g(x)) = 0 everywhere?

The converse is obviously true by the chain rule, so I was just wondering if this was true, too.
 
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techmologist said:
If the gradients of the two functions f and g are proportional everywhere in Rn, does that mean there is some differentiable function F of two variables such that F(f(x),g(x)) = 0 everywhere?

The converse is obviously true by the chain rule, so I was just wondering if this was true, too.
Yes, take F(x,y) identically zero. But that's not what you wanted, right? :smile:
 
Erland said:
Yes, take F(x,y) identically zero. But that's not what you wanted, right? :smile:


Heh, I didn't think about that. Yeah, I want there to be some more interesting functional dependence between f and g. But I think stating it as g(x) = F(f(x)) is too strong, because the relation between them might be one-to-many. I was reading about barotropic fluid flows, in which isobars are also surfaces of constant density, and I was just wondering if the conditions

[tex]\nabla p \times \nabla \rho = 0[/tex]

and

[tex]p = p(\rho)[/tex]

are equivalent, or if the first condition is more general.
 
The answer to your question is basically yes. One way to see this is to note that if the functions' gradients are everywhere proportional, then the functions' level surfaces are identical. So (at least in local patches) each must be a function of the other.
 
wpress said:
The answer to your question is basically yes. One way to see this is to note that if the functions' gradients are everywhere proportional, then the functions' level surfaces are identical. So (at least in local patches) each must be a function of the other.

Okay, thank you. Is there a global version of this, or does it generally fail to hold globally? I think I see that in regions where the gradients do not vanish, you can derive an explicit formula for the value of f as a function of the value of g, or vice-versa.
 

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