Residue calculus and gauss bonnet surfaces

In summary: In a way the vector field proof of the Gauss Bonnet formula is a winding number proof. If one normalizes a vector field with isolated zeros on an orientated surface to have length one away from its zeros, the the connection 1-form integrated over the image of a circle near a zero approximates the linking number of this image around the fiber circle above the zero. As one shrinks the circle towards the singularity this approximation improves and the linking number converges to a winding number.in summary,The two formulations of curvature and residue calculus are related in that on the Gauss Bonnet side you normalize the curv
  • #1
zwoodrow
34
0
I am not a mathematician but I have noticed how strangley similar the treatments of curvature and residues are when you compare the residues of residue calculus and the curviture of the gauss bonet forumlation of surfaces. Is there some generalization of things that contains both of these formulations as a subset?
 
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  • #2
You'll have to explain what you mean. On the surface, they don't have any relation at all.
 
  • #3
In both cases you are doing integration along a path that tells you something intrisic about the function/manifold inside of the curve as long as a certain point is inside the curve and not outside the curve. That simmalarity made me think that maybe the residue calculus was a subset of gauss bonnet.
 
  • #4
Ah, you're talking about Gauss-Bonnet for curves - then yes, the two are intimately related. On the Gauss Bonnet side, you're normalizing the curvature by integrating, and you're left with the winding number. On the Cauchy integral side, the integral of dz/z depends only on the winding number of the curve.
 
  • #5
zhentil said:
Ah, you're talking about Gauss-Bonnet for curves - then yes, the two are intimately related. On the Gauss Bonnet side, you're normalizing the curvature by integrating, and you're left with the winding number. On the Cauchy integral side, the integral of dz/z depends only on the winding number of the curve.

when you say gauss bonnet for curves what do you mean?
 
  • #6
In a way the vector field proof of the Gauss Bonnet formula is a winding number proof.
If one normalizes a vector field with isolated zeros on an orientated surface to have length one away from its zeros, the the connection 1-form integrated over the image of a circle near a zero approximates the linking number of this image around the fiber circle above the zero. As one shrinks the circle towards the singularity this approximation improves and the linking number converges to a winding number.

i can elaborate this picture if you like.
 

Related to Residue calculus and gauss bonnet surfaces

1. What is residue calculus?

Residue calculus is a branch of mathematics that deals with the computation of complex integrals using the theory of residues. It is based on the Cauchy residue theorem, which states that the value of a complex integral around a closed curve is equal to the sum of the residues of the singularities enclosed by the curve.

2. What are Gauss-Bonnet surfaces?

Gauss-Bonnet surfaces are a class of surfaces in differential geometry that satisfy the Gauss-Bonnet theorem. This theorem relates the curvature of a surface to its topological properties, such as the genus (number of holes) and the total angle defect. Examples of Gauss-Bonnet surfaces include the sphere, torus, and hyperboloid.

3. How are residue calculus and Gauss-Bonnet surfaces related?

Residue calculus can be used to calculate the total angle defect of a Gauss-Bonnet surface, which is a topological invariant that is related to the curvature of the surface. This relationship is given by the Gauss-Bonnet formula, which states that the total angle defect is equal to the integral of the Gaussian curvature over the entire surface.

4. What are some applications of residue calculus and Gauss-Bonnet surfaces?

Residue calculus and Gauss-Bonnet surfaces have various applications in mathematics, physics, and engineering. They are used to solve problems in fields such as complex analysis, differential geometry, and differential equations. They are also used in the study of surfaces in 3D graphics, computer vision, and robotics.

5. Are there any limitations to using residue calculus and Gauss-Bonnet surfaces?

While residue calculus and Gauss-Bonnet surfaces are powerful tools for solving certain problems, they do have limitations. For example, the Cauchy residue theorem only applies to functions that are analytic (smooth and infinitely differentiable) within a closed curve. Additionally, the Gauss-Bonnet theorem only applies to surfaces that are orientable (have a consistent direction of normal vectors) and have a constant curvature.

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