Applications of Green's theorem to physics

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SUMMARY

Green's theorem has significant applications in physics, particularly in calculating line and surface integrals involving vector fields. It allows for the determination of potential energy by integrating force over a closed path, eliminating the need for parametrizations. Additionally, Green's theorem is a special case of Stokes' theorem, which is fundamental in electromagnetism, especially in transitioning between differential and integral forms of Maxwell's equations. The theorem also finds relevance in fluid mechanics, showcasing its versatility in various physical contexts.

PREREQUISITES
  • Understanding of vector fields and line integrals
  • Familiarity with Stokes' theorem
  • Knowledge of Maxwell's equations in electromagnetism
  • Basic principles of fluid mechanics
NEXT STEPS
  • Explore applications of Stokes' theorem in electromagnetism
  • Study the relationship between Green's theorem and fluid dynamics
  • Investigate potential energy calculations using line integrals
  • Review examples of Green's theorem in various physical scenarios
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Students and professionals in physics, particularly those focused on electromagnetism, fluid mechanics, and vector calculus applications.

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I am reading Etgen's Calculus: One and Several Variables section on Green's theorem. I was wondering if there is any direct application of this concept to physics or is it only used to calculate areas?
 
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Well, since Green's theorem may facilitate the calculation of path (line) integrals, the answer is that there are tons of direct applications to physics. Line or surface integrals appear whenever you have a vector function (vector fields) in the integrand. Potential energies are obtained wen you integrate a force over a path. If that path is closed, then you can find the potential energy by integrating over the region bound by that path (do away with parametrizations). Moreover, Green's theorem is a special case of Stokes theorem, which appears everywhere in electromagnetism (think about how you get from the differential form to integral form of the last two Maxwell equations). I'm sure a lot people here will come up with tons of specific examples (which can also come from fluid mechanics, for example).
 

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