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

  • Thread starter Noone1982
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In summary, statement 3 states that the integral of F.dl is 0 for any closed loop, statement 4 states that F is the gradient of a scalar function V, and statement 5 states that F is a conservative vector field.
  • #1
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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  • #2
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.
 
  • #3
Thank you, it is becoming clearer now. Now it is a conservative field if the curl is equal to zero?
 
  • #4
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 independant 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:

What is the difference between conservative and non-conservative vector fields?

A conservative vector field is a type of vector field where the line integral along any closed path is equal to zero. This means that the work done by the vector field is independent of the path taken. On the other hand, a non-conservative vector field is one where the work done depends on the path taken.

How can one tell if a given vector field is conservative or non-conservative?

A vector field is conservative if its curl is equal to zero. This means that the vector field is irrotational, or that it does not have any rotational component. On the other hand, a non-conservative vector field will have a non-zero curl, indicating that it does have a rotational component.

What is the relationship between conservative and non-conservative vector fields?

Conservative and non-conservative vector fields are closely related, as any vector field can be decomposed into a conservative and non-conservative component. The conservative component is the part that is irrotational, while the non-conservative component is the part that has a rotational component.

How can I integrate a conservative vector field?

Integrating a conservative vector field can be done using the fundamental theorem of calculus. This states that the line integral of a conservative vector field between two points is equal to the difference in potential between those two points. This allows for an easy integration of conservative vector fields by finding the potential function.

Can a vector field be both conservative and non-conservative?

No, a vector field cannot be both conservative and non-conservative. This is because a vector field is either irrotational (conservative) or has a rotational component (non-conservative). It cannot have both properties simultaneously.

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