Given Curl V, can you find V?

  • Thread starter Jerbearrrrrr
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In summary, there is a way to find a vector field V given its curl, as long as certain conditions are met. This can be done through a transformation within the kernel of the curl operator. However, there may be some ambiguity as V can only be determined up to an arbitrary gradient field. This concept is discussed in multivariable calculus textbooks.
  • #1
Jerbearrrrrr
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Say V is a vector field.
Is there a way (or rather a reasonable algorithm) to find V, given Curl V?

Thanks
 
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  • #2
If I told you curlV=0, can you give me a bunch of different functions that satisfy this requirement?
 
  • #3
Okay, let's impose V satisfies boundary conditions on a (hyper)surface or something. (Though I don't remember the conditions for uniqueness and regularity and so on of PDE solutions)

I guess I asked the wrong question.
Is there a way to find V within a transformation in what we might call the Kernel of the curl operator?

(Like when someone asks what differentiates to f, it's understood that we can add a constant...)
 
  • #4
Yes, if the domain is good enough. The conventional way to phrase the problem is: given a vector field T with div T = 0, find V so that curl V = T. This should be in all multivariable calculus textbooks. But perhaps only after the discussion of vector integration.
 
  • #5
You can determine V out of its curl, up to an arbitrary gradient field.
 

1. What is curl?

Curl is a mathematical operation that describes the rotation or spinning behavior of a vector field.

2. How do you calculate curl?

Curl can be calculated using the cross product of the gradient operator and the vector field. In other words, it is the change in direction of a vector field per unit area.

3. What is the relationship between curl and vorticity?

Vorticity is the measure of the local rotation of a fluid, while curl is the mathematical representation of rotation. In physics and fluid dynamics, vorticity is often described using curl.

4. Can you find V if only the curl is given?

Yes, given the curl of a vector field, it is possible to find the original vector field using mathematical calculations.

5. Is curl a vector or a scalar quantity?

Curl is a vector quantity, as it has both magnitude and direction. It is often represented using a vector symbol with three components.

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