Question about conditions for conservative field

Click For Summary
SUMMARY

The discussion centers on the conditions required for a conservative vector field, specifically the necessity of the region being simply connected and open. It highlights that common textbooks often lack detailed explanations or proofs regarding these assumptions. The inquiry seeks to determine whether these conditions are merely for computational convenience or if they are fundamentally required for the existence of a conservative field. The consensus is that both conditions are essential for the mathematical properties of conservative fields to hold true.

PREREQUISITES
  • Understanding of vector calculus and conservative fields
  • Familiarity with the concepts of simply connected and open regions
  • Knowledge of fundamental theorems in vector analysis
  • Basic proficiency in mathematical proofs and reasoning
NEXT STEPS
  • Research the implications of simply connected regions in vector calculus
  • Study the role of open sets in the context of conservative fields
  • Explore the Fundamental Theorem of Line Integrals
  • Examine counterexamples of conservative fields in non-simply connected regions
USEFUL FOR

Mathematicians, physics students, and educators seeking a deeper understanding of conservative vector fields and their foundational requirements.

kelvin490
Gold Member
Messages
227
Reaction score
3
Question about conditions for conservative field
In common textbooks' discussions about conservative vector field. There is always two assumptions about the region concerned, namely the region is simply connected and open.

Usually in textbooks there is not much explanations on why these assumptions are necessary, no proof is given on why conservative field is not possible if the region is not simply connected or not open.

I wonder whether these two assumptions are just for computational convenience or it is really logically not possible to have a conservative field in region that is not simply connected or is not open?
 
Physics news on Phys.org

Similar threads

Undergrad Equivalence of alternative definitions of conservative vector fields and line integrals in different metric spaces
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Vector field and Helmholtz Theorem
  • · Replies 20 ·
Replies
20
Views
4K
Graduate An inconsistent conservative vector field
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Sufficient condition for a vector field to be conservative
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad What would a calculus author have to say on ##\int r^2dm##?
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad Conservative vector field conditions
  • · Replies 2 ·
Replies
2
Views
4K
MHB Question about the differential in Calculus
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Is any constant vector field conservative?
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Mechanism of Energy Conservation in Zero-Amplitude Sum of EM Waveforms
  • · Replies 21 ·
Replies
21
Views
3K
Graduate Determining if a vector field is conservative
  • · Replies 4 ·
Replies
4
Views
3K