Curl of Gradient of a Scalar Field

[Nishant Garg]

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?

 

[BvU]

Hello Nishant, welcome to PF !

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 ?

Wiki on this subject.

 

[Nishant Garg]

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

 

[BvU]

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

 

[Nishant Garg]

So is it safe to assume that our scalar function has continuous second order mixed partials?

 

[Nishant Garg]

Ok, thank you. How do I close this thread?

 

[BvU]

If we stop posting, that's closing the thread. It stays on the forum for the benefit of all !