Q&A: Integrating dW, Conservative & Non-Conservative Vector Fields

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Discussion Overview

The discussion revolves around the integration of work done by vector fields, specifically focusing on the differences between conservative and non-conservative vector fields. Participants explore how the choice of path affects the limits of integration and the outcomes of the integrals in various scenarios.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the necessity of path selection for integration, suggesting that if a particle travels along a specific path, there should be only one set of limits.
  • Another participant clarifies that for conservative fields, the integral result is independent of the path taken, allowing for more straightforward calculations.
  • In contrast, for non-conservative fields, the path must be parametrized, which can complicate the integration process.
  • There is a discussion about integrating along different axes (x, y, z) and whether this approach yields consistent results, with some participants noting that it works in certain cases but not in others.
  • A participant seeks clarification on the condition for a field to be conservative, specifically asking if a zero curl is sufficient.
  • Another participant confirms that a zero curl is both a necessary and sufficient condition for a vector field to be conservative, listing several equivalent statements related to conservative fields.

Areas of Agreement / Disagreement

Participants express differing views on the implications of path selection for integration in conservative versus non-conservative fields. While some agree on the properties of conservative fields, the discussion remains unresolved regarding the integration methods and their applicability in various contexts.

Contextual Notes

Participants highlight the complexity of integrating in non-conservative fields and the potential for different outcomes based on the chosen path. There are also references to specific mathematical conditions that define conservative fields, but the implications of these conditions in practical integration scenarios remain under discussion.

Who May Find This Useful

This discussion may be useful for students and practitioners in physics and engineering who are grappling with the concepts of vector fields, work done by forces, and the implications of path dependence in integration.

Noone1982
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This thread is kind of an extension of my last, so pardon any overlap.

1) dW = F • dL

My teacher says "careful the path you integrate this on." But isn't there only one possible let of limits for something? I mean, if the particle is traveling on say y = x^2 from 0 to 5, what other path could there be? How do conservative and non-conservative vector fields play into these limits?

I noticed some problems you can just integrate dx with the x limits, dy with the y limits and dz with the z limits and get the right answer. However, some I notice you have to put everything in terms of say x and just integrate over x to get the right answer. Integrating over x,y,z limits gives me the right answers for some but not others. Why?

I would think the answer would be the same for an integral of dx integrated over x limits + dy integrated over y limits + dz integrated over z limits compared to an all x or all y or all z integral. Why does it matter?
 
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Noone1982 said:
This thread is kind of an extension of my last, so pardon any overlap.

1) dW = F • dL

My teacher says "careful the path you integrate this on." But isn't there only one possible let of limits for something? I mean, if the particle is traveling on say y = x^2 from 0 to 5, what other path could there be? How do conservative and non-conservative vector fields play into these limits?

For conservative fields, you can integrate along ANY path that has those limits and the result will be the same! This allows for easier computation. For non-conservative field, you MUST integrate along the path taken by the particle and thus you must parametrize the path, which is often times a lot of pain.

Noone1982 said:
I noticed some problems you can just integrate dx with the x limits, dy with the y limits and dz with the z limits and get the right answer.

It is not entirely clear to me what you mean, but if it's what I think it means the field is conservative and what you'Re doing when you integraate "dx with the x limit, then dy with the y limits and then dz with the z limits" is you integrate along a straight line on the x-axis (for which path y and z are constant), then along a straight line along the y-axis (for with x and z are constant), and finally along the z axis. This is the tactic I was referring to in paragraph one when talking about how with conservative field, you can integrate along any path, which eases computation.
 
Thank you, it is becoming clearer now. Now it is a conservative field if the curl is equal to zero?
 
It is a necessary and sufficient condition, yes.

Here's the answer to all your conservative fields needs and demands.

Theorem (Helmholtz):
The following statements are all logically equivalent (i.e. they are interlinked by a [itex]\Leftrightarrow[/itex] relation)

1) curl of F is 0
2) integral of F.dl is independent of path for any given end points
3) integral of F.dl = 0 for any closed loop.
4) F is the gradient of some scalar function V: [itex]\vec{F}(x,y,z) =-\nabla V(x,y,z)[/itex]
5) F is said to be a conservative vector field.
 
Last edited:

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