Path integral/Stokes's and Green's theorem

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SUMMARY

The discussion centers on the application of Stokes' theorem and Green's theorem to evaluate the path integral of a vector function F over a closed path in Euclidean space where z = 0. The user initially questioned the validity of using both theorems sequentially, ultimately concluding that applying Stokes' theorem followed by Green's theorem is permissible, leading to the result that div(curl(F)) equals zero. This confirms the relationship between the two theorems in the context of vector calculus.

PREREQUISITES
  • Understanding of vector calculus concepts, specifically Stokes' theorem and Green's theorem.
  • Familiarity with path integrals and their applications in Euclidean space.
  • Knowledge of vector fields and their properties, including curl and divergence.
  • Basic proficiency in mathematical notation and problem-solving techniques in calculus.
NEXT STEPS
  • Study the proofs and applications of Stokes' theorem in vector calculus.
  • Explore Green's theorem and its implications for line integrals in the plane.
  • Investigate the relationship between divergence and curl in vector fields.
  • Practice solving path integrals using both Stokes' and Green's theorems with various vector functions.
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are working with vector calculus, particularly those interested in applying Stokes' and Green's theorems to solve complex problems involving path integrals.

PhysicsGente
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I meant Line integral.

Homework Statement



I want to find the path integral of a vector function F over a closed path in Euclidean space with z = 0.

Homework Equations


The Attempt at a Solution



I was wondering if it is allowed to first use Stoke's theorem and then Green's theorem. I would end up getting div(curl(F)) which I believe equals to zero. But I am not sure if I'm allowed to use Green's theorem in this case.
 
Last edited:
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Can you post the original problem and your work so far?
 
I have figured it out. Thank you!
 

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