How can I explain that this vector field is not conservative?

In summary, the conversation discusses the concept of conservative vector fields and how to determine if a given field is conservative. It is mentioned that for a field to be conservative, the partial derivatives of its components must be equal. The speaker attempted to find the components of the given vector field, but was unsuccessful. The conversation then moves on to discussing how to determine if a field is not conservative, using the example of a closed curve that follows the direction of the field. It is noted that this may not be enough to prove non-conservatism. The conversation concludes by discussing the idea of work done in a circular motion within the field, with the speaker suggesting that the amount of work would be the same in both clockwise and counterclockwise directions
  • #1
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Homework Statement



upload_2014-11-21_0-12-36.png
I have to explain why this vector field is not conservative.

Homework Equations


Maybe it is: if ##\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}## then F(x, y) = p(x, y)i + Q(x, y)j is a conservative field. I tried to figure out what P and Q is, but that

The Attempt at a Solution


I tried to figure out what P and Q is, but that was unsuccessfu. By book says that when I can draw a closed, continuous curve that always goes with where the vector field goes, that field is not conservative. But, I don't think it's enough.
 

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  • #2
Suppose those arrows represent force of a water current. Do you think you would do the same amount of work swimming in a circle clockwise about the origin as swimming counterclockwise? Zero in both cases?
 
Last edited:
  • #3
I think the amount would be same, but will have negative value. But is that an enough explanation?
 
  • #4
Surely you can word it more completely and carefully than that.
 

1. How can I determine if a vector field is conservative or not?

To determine if a vector field is conservative, you can use the curl test or the path independence test. The curl test involves taking the curl of the vector field and if it is equal to zero, then the field is conservative. The path independence test involves evaluating the line integral of the vector field along two different paths and if they are equal, then the field is conservative.

2. What does it mean for a vector field to be conservative?

A conservative vector field is one in which the line integral along any closed path is equal to zero. This means that the work done by the field on any particle moving along a closed path is zero, and the path taken does not affect the final outcome of the work.

3. Why is it important to know if a vector field is conservative or not?

Knowing if a vector field is conservative or not is important in many areas of science, especially in physics and engineering. Conservative vector fields often represent physical forces such as gravity or electromagnetism, and understanding their properties can help in solving problems and making predictions.

4. Can a vector field be partially conservative?

Yes, a vector field can be partially conservative, meaning that it may be conservative in some regions but not in others. This can occur when there are certain constraints or boundaries in the field that prevent it from being completely conservative.

5. How can I visually determine if a vector field is conservative?

One way to visually determine if a vector field is conservative is by looking at its potential function. If a potential function exists for the vector field, then it is conservative. Additionally, you can graph the field and see if it forms closed loops, which is another indication of a conservative field.

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