Vector field curvature in the complex plane

In summary, the conversation is about determining the curvature of a vector field described by a complex potential function. The individual is looking for a formula or method to express the curvature at every point, but has been unable to find one in their books or through online research. They are specifically interested in the curvature of the potential and streamlines, and are seeking examples or relevant reading material to help them understand the concept better. They have been advised to try working out simple examples and parameterizing curves by arclength.
  • #1
meldraft
281
2
Hey all,

I have a vector field described by a complex potential function (so I have potential lines and streamlines). I am looking for a way to express its curvature at every point, but I can't find such a formula in my books. I have searched in wikipedia and I read that the way to define it in cartesian coordinates is through a parametric curve, but I'm not sure how I should go about it in the complex plane.

If anyone can give me a pointer on how to derive the curvature of the field or to relevant reading material, I would be grateful :biggrin:
 
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  • #2
meldraft said:
Hey all,

I have a vector field described by a complex potential function (so I have potential lines and streamlines). I am looking for a way to express its curvature at every point, but I can't find such a formula in my books. I have searched in wikipedia and I read that the way to define it in cartesian coordinates is through a parametric curve, but I'm not sure how I should go about it in the complex plane.

If anyone can give me a pointer on how to derive the curvature of the field or to relevant reading material, I would be grateful :biggrin:

I am not sure what you mean by the curvature of a vector field.

Do you mean the curvature of the potential and stream lines or do you mean the curvature of the conformal metrics determined by the potential?
 
  • #3
You are right, I wasn't clear, I mean the curvature of the potential and stream lines!
 
  • #4
Shameless bump :biggrin:
 
  • #5
meldraft said:
You are right, I wasn't clear, I mean the curvature of the potential and stream lines!

I haven't had a chance to figure out if there is a general description but I would try working out some simple examples first - e.g. with algebraic functions in the plane.

You need to parameterize the curves u = constant, and v = constant by arclength then differentiate the unit length tangent vector.
 

1. What is a vector field in the complex plane?

A vector field in the complex plane is a mathematical function that assigns a vector to each point in the complex plane. The vector at each point represents the direction and magnitude of the field at that specific point.

2. How is curvature defined in a vector field in the complex plane?

Curvature in a vector field in the complex plane is defined as the rate of change of the direction of the vectors as we move along a curve in the field. It is a measure of how much the field is curving or changing direction at a specific point.

3. What does it mean for a vector field in the complex plane to have zero curvature?

If a vector field in the complex plane has zero curvature, it means that the direction of the vectors does not change along any curve in the field. This can also be interpreted as the field being flat or not curving.

4. How is the curvature of a vector field in the complex plane visualized?

The curvature of a vector field in the complex plane can be visualized using a color mapping, where different colors represent different levels of curvature. Another way is to use streamlines, which are curves that are tangential to the direction of the vectors at each point and can reveal the pattern of curvature in the field.

5. What are some real-life applications of studying vector field curvature in the complex plane?

Vector field curvature in the complex plane has various applications in physics, engineering, and fluid dynamics. It can be used to describe the flow of electromagnetic fields, the motion of particles in a magnetic field, and the movement of fluids in a vortex or turbulent flow. It is also used in computer graphics and animation to create realistic simulations of fluid dynamics.

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