Cauchy,s theorem on real plane?(R^2)

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In summary: This means that the area enclosed by C_1 and C_2 is the same as the area enclosed by the union of C_1 and C_2. Therefore:\oint_C \omega = \oint_{C_1} \int_C_1 \omega^2+\oint_{C_2} \int_C_2 \omega^2 You can use the divergence theorem to express the charge of monopoles as the result of a singularity of the form a/r^{2}.
  • #1
eljose
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could actually the Cauchy theorem be applied to [tex] R^{2} [/tex] of course with some restrictions..namely

-The differential form must be invariant under rotation in the plane
-The differential form must come from a gradient [tex] \nabla{S}=f/(|r|-a) [/tex]

with this using the same techniques by Cuachy we could se the equality:

[tex] \Int_{C}dq.F(x,y)//(|r|-a)=2\piF_{r}(a) [/tex]

this means somehow that the integral over a curve on RxR is equal to the normal component evaluated at the point r=a where r is the modulus of the vector (x,y,z)
 
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  • #2
Of course i meant the formula:

[tex] \int_{C}dq.F(x,y)/(|r|-a)=2\pi{F_{r}(a)} [/tex]

where the integral along a curve of a 2-differential form with a singularity at the point whose distance from the origin is equal to "a", is set to be the normal component (invariant under rotations in RxR) of the differential form F..the derivation would be the same..we set a circle of radius [tex] \epsilon [/tex] and make this epsilon tend to zero...
 
  • #3
By the way, you can use that button "Edit" to edit your posts to fix the LaTeX. (You will have to reload the page to get the updated image)


Anyways, you want to do differential geometry! The key theorem you want to use here is the generalized Stoke's theorem. A special case is Green's theorem:

If D is a region of R², and C is the boundary of D, and P and Q are scalar fields with continuous partial derivatives, then:

[tex]
\oint_C P \, dx + Q \, dy = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA
[/tex]

Notice that if the vector field (P, Q) is the gradient of some scalar field, then the right hand side is zero.


The proof idea for what you want to show is similar to the proof of the similar equation in complex analysis, but with Green's theorem being the theorem that you use to justify changing the curve.



Now, I suspect you haven't stated your problem right. Did you really intend for your integrand to be singular along the entire circle of radius a centered on the origin?
 
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  • #4
-No Hurkyl..i perform the integration about two curves in the form:

[tex] \oint_{C}F.dq-\oint_{\epsilon}{Pdx+Qdy} [/tex]

the first curve encloses the area marked by the curve except a 2-D "ball" of leghnt [tex] \epsilon{2\pi} [/tex] in which the point (x0,y0) is and note that:

[tex] a^{2}=(x_{0})^{2}+(y_{0})^{2} [/tex] of course on the contrary to Cauchy,s theorem in which "a" was a complex number the corresponding value of a must be somehow an scalar, expressed as a distance from the origin of the point (x0,y0) and we must impose that at least the "radial" part of the differential Form F must be invariant under rotations on x-y plane..by the way generalizing to Surface integrals and using divergence theorem (Gauss) we could express the "charge of monopoles" as the result of a singularity of the form [tex] a/r^{2} [/tex] for the Magnetic field B (vector)

-a is some kind of distance (note that there are no 2-D real number)
 
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  • #5
If R is a ring in 2-space, then the boundary of R consists (which I'll call C) of the two curves: C_1, the outer boundary, and C_2, the inner boundary. By definition:

[tex]
\oint_C \omega = \oint_{C_1} \omega - \oint_{C_2} \omega
[/tex]
 

1. What is Cauchy's theorem on the real plane (R^2)?

Cauchy's theorem on the real plane is a fundamental theorem in complex analysis that states that if a function is holomorphic (complex differentiable) on a simply connected domain in the complex plane, then its integral over any closed curve in that domain is equal to zero.

2. How is Cauchy's theorem used in mathematics?

Cauchy's theorem has numerous applications in mathematics, particularly in complex analysis and geometry. It is used to prove other key theorems, such as Cauchy's integral formula and the Cauchy-Riemann equations, and it plays a crucial role in the development of the theory of analytic functions.

3. What is the difference between Cauchy's theorem on the real plane and Cauchy's theorem on the complex plane?

The main difference between these two theorems is the dimension of the space on which they are defined. Cauchy's theorem on the real plane is concerned with functions defined on the two-dimensional real plane (R^2), while Cauchy's theorem on the complex plane deals with functions defined on the two-dimensional complex plane (C^2).

4. Can Cauchy's theorem be applied to non-simply connected domains?

No, Cauchy's theorem only holds for simply connected domains. A domain is considered simply connected if any closed curve within it can be continuously deformed into a single point without leaving the domain. If the domain is not simply connected, then the theorem does not apply.

5. What are some real-world applications of Cauchy's theorem?

Cauchy's theorem has practical applications in fields such as physics and engineering. It is used to solve problems involving electric and magnetic fields, fluid dynamics, and heat transfer. It also has applications in image processing and computer graphics.

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