Vector field curvature in the complex plane

[meldraft]

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

 

[lavinia]

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

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?

 

[meldraft]

You are right, I wasn't clear, I mean the curvature of the potential and stream lines!

 

[meldraft]

Shameless bump

 

[lavinia]

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.