For theorem 4 above, what is an open connected region? The physical interpretati
ID: 2880248 • Letter: F
Question
For theorem 4 above, what is an open connected region?
The physical interpretation is that the work done by a conservative force field as it moves an object around a closed path is 0. The following theorem says that the only vector fields that are independent of path are conservative. It is stated and proved for plane curves, but there is a similar version for space curves. Theorem Suppose F is a vector field that is continuous on an open connected region D. If integral_c F middot dr is independent of path in D. then F is a conservative vector field on D: that is. There exists a function f such that nabla f = F.Explanation / Answer
I can try to explain this through a simple example.
Consider the annulus {(x,y)|1<x2+y2<2}2. This region is open and connected, but not simply connected.
On the other hand, the closed disk {(x,y)|x2+y21}2 is simply connected, but not open.
In other words, a simply connected domain is a path-connected domain where one can continuously shrink any simple closed curve into a point while remaining in the domain.
For two-dimensional regions, a simply connected domain is one without holes in it and an open connected region will have holes as in the example above where a open connected region is a ring whereas a simple connected domain is the filled circle with no emply space / hole in the centre.