Continuous 2nd Partials a Substantial Requirement for Conservative Field?

  • Thread starter breez
  • Start date
  • #1
65
0

Main Question or Discussion Point

It can be shown that if F has continuous 2nd partials, then div curl F = 0. According to Stoke's Theorem, the work done around a closed path C is equal to the flux integral of curl F on a surface sigma that has C as its boundary in positive orientation. However, this integral is equal to the volume triple integral of div curl F. But if F has continuous 2nd partials, then div curl F = 0, and hence the work around C must be 0. Doesn't this show that F is conservative if F has continuous 2nd partials?
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,794
923
It isn't "F" that is "conservative"!

A vector field [itex]\vec{f}[/itex] is said to be "conservative" (that's physics terminology; I prefer "is an exact derivative") if there exist a scalar function F, having continuous partials, such that [itex]\vec{f}= \nabla F[/itex]. IF there exist such an F, then yes, [itex]\vec{f}[/itex] is "conservative"!
 
  • #3
nicksauce
Science Advisor
Homework Helper
1,272
5
So you're saying that because

[tex] \int_{\partial S}\vec{F}\cdot d \vec{l} = \int_{S} \nabla \times \vec{F} \cdot d \vec{S}[/tex]

and because

[tex] \int_{\partial V}\vec{F} \cdot d \vec{S} = \int_{V} \nabla \cdot \vec{F} dV[/tex]

it follows that

[tex] \int_{\partial V}\vec{F}\cdot d \vec{l} = \int_{V} \nabla \cdot \nabla \times \vec{F} dV = 0 [/tex]

assuming F is smooth enough? The problem is that you can't go from the first theorem to the second. In the first S is an open surface (which has a boundary curve) and in the second S = delV is a closed surface (which does not have a boundary curve)
 
  • #4
nicksauce
Science Advisor
Homework Helper
1,272
5
The first integral in the third equation should be around delS not delV.
 

Related Threads for: Continuous 2nd Partials a Substantial Requirement for Conservative Field?

  • Last Post
Replies
6
Views
811
Replies
1
Views
1K
  • Last Post
Replies
2
Views
36K
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
3
Views
1K
Replies
1
Views
2K
Replies
4
Views
4K
  • Last Post
Replies
5
Views
5K
Top