Can a Circulation Integral Be Calculated Using a Scalar Field?

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    Circulation Integral
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SUMMARY

The discussion centers on the calculation of a circulation integral, specifically the integral \(\int_{C} x \, dy\), using vector calculus principles. It is established that this integral can be computed using a line integral of a vector field, but the participants clarify that it cannot be directly calculated using a scalar field. Instead, a parametrization of the curve \(C\) is necessary, where \(ds = x(t) \, dy(t)\) can be applied. The conversation emphasizes the importance of understanding the theoretical limitations of scalar field integration in this context.

PREREQUISITES
  • Understanding of vector calculus concepts, particularly circulation integrals.
  • Familiarity with line integrals and vector fields.
  • Knowledge of parametrization techniques for curves.
  • Basic integration skills, particularly with respect to functions of multiple variables.
NEXT STEPS
  • Study the properties of circulation integrals in vector fields.
  • Learn about parametrization of curves in vector calculus.
  • Explore the differences between line integrals of vector fields and scalar fields.
  • Practice solving line integrals using specific examples and parametrizations.
USEFUL FOR

Students and professionals in mathematics, particularly those studying vector calculus, as well as educators looking to clarify the distinctions between scalar and vector field integrations.

nonequilibrium
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Hello. I'm new to vector calculus and I had a question about the following integral:

[tex]\int_{C} x dy[/tex] please note that this is a circulation (I didn't know the tex-code for the little circle sign on the integral)

They calculated this integral (for a specific curve) with the use of a line integral of the tangential component of F (i.e. line integral of a vector field).

But I was wondering, can this be calculated with a line integral of a scalar field? For example if C is the circle with center the origin and radius 1. I suppose for being able to do it with a scalar field, you'd then have to find a parametrization so that ds = x(t) dy(t) right? Is this doable?

(The reason I ask it is not for practical use, but to understand the theory more -- why this can't be done with a scalar field, while it looks so easy)
 
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No reason it can't be done by a scalar field integration.
A usual basic integration of x as a function of y will also suffice ( this will of course require you to break C into curves which are functions and not to forget the direction of integration)

Otherwise there is no reason not to parametrize the curve with x(t) and y(t) and do the integration in terms of t.
 
Thank you!
 

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