Difference between a conservative and nonconservative vector fields

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Discussion Overview

The discussion revolves around the distinction between conservative and non-conservative vector fields, particularly in the context of a calculus test problem involving work and line integrals. Participants explore the implications of these concepts in both mathematical and physical terms.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant describes their understanding of conservative vector fields, noting that they can be expressed as the gradient of a function and that the work done around a closed loop should be zero.
  • The same participant recounts a personal experience where they calculated a non-zero work value for a closed loop, leading them to question whether the force field was non-conservative.
  • Another participant hints at the curl of a magnetic field as a potential example of a non-conservative field.
  • There is a suggestion that the participant should verify whether the field in question is conservative, as this would affect the work done along a closed path.
  • The original poster expresses uncertainty about real-life examples of non-conservative fields and questions whether they are purely mathematical constructs.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the field in the original problem is conservative or non-conservative, and there is an acknowledgment of uncertainty regarding the implications of this distinction.

Contextual Notes

The discussion highlights the importance of verifying the nature of the vector field when calculating work, but does not resolve the specific mathematical or physical details regarding the field in question.

mjordan2nd
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From a mathematical standpoint I have no trouble understanding the difference between a conservative vector field and a non-conservative vector field. It's rather simple. The conservative field can be reduced to some functions gradient vector, doesn't care what path you decide to take, and always returns 0 on a closed loop. However, on my calculus test today on a problem about work I got an unexpected answer. A particle started at the origin, went around the plane z=y/2 ranging from {(x, y), -1<=x<=1, -2<=y<=2} and returned to the origin. The fact that it traversed a closed loop made bells go off in my head. I though, "what a stupid question. No-brainer." But for some reason I didn't put down 0 and worked out the line integral. I got 3, no matter how many times I checked myself. I'm absolutely convinced I did the math correctly on that problem. Our physics professor had always told us that if the displacement vector is 0 then no work was done. At the time of my test I was so convinced to what my physics professors had told me, I convinced myself I was wrong and put down 0 anyway. I was baffled by that problem, so I thought about it on my way to the computer lab once I got out of my test. And it hit me. What if the force field wasn't conservative? It would make perfect sense. That's why the answer, was in fact, 3. However I don't quite understand. What would be a real life example of a non-conservative force-field? Or any real non-conservative vector field for that matter. What are the physical differences between them? Or are non-conservative fields purely a mathematical construct? I'm still grappling with this. I can't get my head quite around it, though I'm glad I figured out why my answer was always 3.

Sorry if this should be in calculus instead...
 
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Hint: What is the curl of a magnetic field?

Zz.
 
Heh, I have no idea.

I'll look it up. Thanks.
 
If the field was "conservative", then there would be no work done in moving around a closed path. Did you check to make sure the field was conservative?
 
No, unfortunately I did not even think of the possibility that the field could have been non-conservative at the time. I have been so used to seeing work being force * displacement, I didn't check. I checked on plenty of other problems, but it eluded me on this one. Heh, guess it happens. Now that I think back on it, however, I'm almost certain that it was not a conservative field.
 

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