Analyzing Vector Fields: Determining Conservativeness

[Miike012]

Question: Which vector field is conservative?

I added two pictures of the vector fields in the paint document.

So far the only things I know about conservative vector fields are..

1. Net change is 0 for a closed path
2.there is some function f such that F = ∇f (F being the conserv. vector field)
3. Net change is independent of the path
4. Energy is always conserved

I don't know how to apply the top four facts to a picture of a vector field. So how can I determine if they fields are conservative?

 

[LCKurtz]

If you could find closed paths where the work going around wasn't zero that would show the field wasn't conservative. Or if you can convince yourself that you can't do that, it might be conservative.

 

[Miike012]

If you could find closed paths where the work going around wasn't zero that would show the field wasn't conservative. Or if you can convince yourself that you can't do that, it might be conservative.

Lets say I chose the closed path ... I created some randome one in the paint doc... how would you know if the work around the path is zero?

 

[LCKurtz]

What if that field represented wind velocity and you were riding a bike. Could you could find a path where the "wind is at your back" the whole way?

 

[haruspex]

Lets say I chose the closed path ... I created some randome one in the paint doc... how would you know if the work around the path is zero?

For the path you've chosen, it's not clear whether it could be zero. Try a path around the origin instead.

 

[Miike012]

For the path you've chosen, it's not clear whether it could be zero. Try a path around the origin instead.

Even if I picked a path around the origin I don't know what I am looking at. What are you focusing your attention on when evaluating the field?

 

[LCKurtz]

Even if I picked a path around the origin I don't know what I am looking at. What are you focusing your attention on when evaluating the field?

Did you read my post #4? Would it be the same going around the circle either way?

 

[haruspex]

Even if I picked a path around the origin I don't know what I am looking at. What are you focusing your attention on when evaluating the field?

Imagine going in a circle anticlockwise around the origin in the left-hand picture. At each point, the field E is making some angle to your direction of travel. As you advance a distance element vector dS, the energy transfer will be E.dS. If that scalar always has the same sign its integral cannot be zero.