Curl of Function: Constant Magnitude or Way Off?

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Discussion Overview

The discussion revolves around the properties of the curl of a function, specifically whether the gradient of the curl being zero implies that the magnitude of the curl is constant. Participants explore the definitions and relationships between curl, gradient, and divergence in the context of vector calculus.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions if the gradient of the curl being zero means the magnitude of the curl is constant, expressing uncertainty about their understanding.
  • Another participant corrects the misconception, stating that the gradient of the curl is not always zero and clarifies the distinction between divergence and gradient.
  • A participant highlights two properties related to curl: that the divergence of the curl is always zero and that a function written as the gradient of another has a curl of zero.
  • Further clarification is provided that the curl of a gradient is zero, contrasting with the original claim about the gradient of a curl.
  • One participant discusses the trivial nature of the fact that the curl of the gradient is zero, linking it to fundamental theorems in calculus, while suggesting that there may be interesting cases of forms with zero curl that are not gradients.
  • A later reply points out that the term "grad of curl" is not defined, indicating confusion in terminology between gradient and divergence.

Areas of Agreement / Disagreement

Participants express disagreement regarding the initial claim about the gradient of the curl, with some clarifying and correcting misconceptions. The discussion remains unresolved as participants explore different properties without reaching a consensus.

Contextual Notes

There are limitations in the original post's terminology, particularly the confusion between gradient and divergence, which may affect the clarity of the discussion.

nolanp2
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if the grad of the curl of a function is always zero does this mean the magnitude of the curl is constant? or am i way off here?
 
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edit: misread the question- sorry!
 
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Firstly note that it is NOT true that the grad of the curl of a function is always zero.

Actually nolan I think you are mixing up two separate properties of "curl" there to come to this misconception. The properties that you are thinking of are probably these two :

1. The div of the curl of a function is always zero. (div is not the same thing as grad ok).

2. If a given function can be written as the grad of another function then that given function has a curl of zero.

Note that neither of thse two things (taken separately or together) imply what you have written.
 
Ok looking at your question again I can see that it's probably the second property that I wrote that is the one you have gotten mixed up. It's the curl of a grad that is zero (a zero vector) not the way you put it, grad of a curl is not neccessarily zero
 
that the curl of the grad is zero is essentially trivial. it follows from the trivial fact that the integral of GRAD is zero around a closed curve. this is because that integral is evaluated by subtracting the values of some function at the two endpoints, which are equal.

that in turn is true by the FTC, which holds by the trivial fact that the derivative of the area function is the height functon, which holds trivially because the area of a rectangle is the base times the height.

on the other hand, sticking all these trivialities together, maybe we have a pathologically amazing theorem!

more interesting is the investigation of forms with zero curl which are not gradients, like dtheta.
 
One other thing I should have noticed about the original post. "Grad of Curl" isn't even defined. Grad is normally applied to a scaler field, giving a vector result. So "div of curl" makes sense but "grad of curl" doesn't. So my guess is that the OP is getting confused between grad and div.
 

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