Finding Vector Fields that Satisfy Certain Curls

In summary, the conversation discusses the possibility of finding a vector field with a curl of yi and xi, and the attempts made to find such a vector field. It is mentioned that a vector potential can be found for a vector field with a zero divergence, but since the divergence of xi is 1, it is not possible to find a vector field with a curl of xi.
  • #1
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Homework Statement


Is it possible to find a vector field whose curl is yi? xi?

Homework Equations





The Attempt at a Solution


I found that if F = <0,0,y^2/2>, its curl will be yi. However, I cannot figure out a vector field whose curl will be xi. I tried using exponentials and just different combination of x's, y's and z's as components of F, and just building my function as I go through the 'curl determinant', but haven't been able to find anything. If someone could please help me out that would be greatly appreciated. Thank you
 
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  • #2
Take the divergence of the vector field you're interested. If it's zero, then you can find the "vector potential" for the vector field you're interested in (for more info, see http://en.wikipedia.org/wiki/Vector_potential).

Since the divergence of xi is 1, you cannot find a vector field whose curl is xi.

Hope that helps!
 
  • #3
Thanks for the help!
 

1. What is a vector field?

A vector field is a mathematical concept that assigns a vector (a quantity with magnitude and direction) to every point in a given space.

2. What does it mean for a vector field to have a certain curl?

The curl of a vector field is a measure of its rotation at a given point. A vector field with a nonzero curl means that the vectors are rotating around that point.

3. How do you find a vector field that satisfies a certain curl?

To find a vector field with a specific curl, you need to use mathematical techniques such as partial derivatives and the curl operator to manipulate the components of the vector field and make sure it meets the given criteria.

4. Why is it important to find vector fields with specific curls?

Vector fields with certain curls have many applications in physics, engineering, and other fields. For example, they can be used to model fluid flow or electromagnetic fields.

5. What are some common techniques for finding vector fields with specific curls?

Some common techniques for finding vector fields with specific curls include using the Helmholtz decomposition theorem, using the gradient, and using Green's theorem.

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