Simple question about circulation integral

[nonequilibrium]

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

$$\int_{C} x dy$$ 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)

 

[elibj123]

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.

 

[nonequilibrium]

Thank you!