Curl of Gradient of a Scalar Field

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Discussion Overview

The discussion revolves around the mathematical property of the curl of the gradient of a scalar field, specifically questioning why it is considered a null vector when mixed partial derivatives may not always be equal. The scope includes theoretical exploration and mathematical reasoning regarding the conditions under which this property holds.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions how the curl of a gradient can be a null vector if mixed partial derivatives are not always equal.
  • Another participant suggests that if a function is twice continuously differentiable, then the second derivatives should be equal, leading to the conclusion that the curl of the gradient is zero.
  • A participant emphasizes that the condition of continuous second derivatives must be satisfied for the curl of the gradient to be a null vector.
  • There is mention of the existence of "nice" functions in physics, but also acknowledgment of potentially pathological cases that may violate the equality of mixed partials.
  • One participant expresses difficulty in finding a counter-example of a function where second partial derivatives differ based on the order of differentiation.
  • A later reply raises the question of whether it is safe to assume that the scalar function has continuous second-order mixed partials.

Areas of Agreement / Disagreement

Participants express differing views on the conditions under which the curl of the gradient is a null vector, with some asserting that continuous second derivatives are necessary while others question the universality of this condition. The discussion remains unresolved regarding the existence of functions that do not satisfy this condition.

Contextual Notes

Participants reference Clairaut's theorem on the equality of mixed partials, indicating that the discussion is dependent on the assumptions of differentiability and continuity of the scalar functions involved.

Nishant Garg
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Hello, new to this website, but one question that's been killing me is how can curl of a gradient of a scalar field be null vector when mixed partial derivatives are not always equal??

consider Φ(x,y,z) a scalar function
consider the determinant [(i,j,k),(∂/∂x,∂/∂y,∂/∂z),(∂Φ/∂x, ∂Φ/∂y, ∂Φ/∂z)] (this is from ∇×(∇Φ))
When you expand this you will get
[(∂^2Φ/∂y∂z)-(∂^2Φ/∂z∂y)]i-[(∂^2Φ/∂x∂z)-(∂^2Φ/∂z∂x)]j+[(∂^2Φ/∂x∂y)-(∂^2Φ/∂y∂x)]k
Now this can only be null vector when individual components are 0, and that's only when mixed partials are equal, but they are not always equal now are they?
 
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Hello Nishant, welcome to PF :smile: !

If I (like you probably also did) simply google
proof that curl of a gradient is zero
then I find (e.g. here , but in many other places) that cross derivatives are equal.
You place a question mark there, though. Perhaps the definition of derivative -- writing out the lot in limit terms -- can help you see they really are equal. Or do you know a counter-example ?:wink:

Wiki on this subject.
 
The last line of that post, like I said

If f is twice continuously differentiable, then its second derivatives are independent of the order in which the derivatives are applied. All the terms cancel in the expression for curl∇f, and we conclude that curl∇f=0.

So it must satisfy that condition above for curl of gradient to be null vector? It's not always null vector right, it must satisfy that condition?

Counter example, I try hard but can't think of a function whose second partial derivatives are different depending on respect to which you took derivative first
 
Point is that in physics, functions are always "nice". But (see Wolfram ) seemingly nice functions can be pathological in this respect.

[edit] more goodies: Clairaut[/PLAIN] theorem and this thread with our colleagues
 
Last edited by a moderator:
So is it safe to assume that our scalar function has continuous second order mixed partials?
 
Ok, thank you. How do I close this thread?
 
If we stop posting, that's closing the thread. It stays on the forum for the benefit of all !
 

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