how are you?
I have this little exercise. It seems to be so easy but I don't know what to do. It's of the topic of vectorial integrals (gradient, divergence and curl).
Calculate the integral $\int_{} r \cdot \, \vec{dr}$ in a ring of radio R that's centered in the origin. Was the result predictable? Explain.
As the statement said, we have to integrate in a ring, that could be seen as an arc. That's a closed curve, so if we now the integral theorem of the gradient, we know that:
$\int_{c} \vec{F} \cdot \, \vec{dr} = \int_{c} \nabla \phi \cdot \, \vec{dr} = \int_{} d\phi = \phi_2 - \phi_1$
That's correct if $\vec{F} = \nabla \phi$. But the statements it's not as explicit as I'd like it to be, because I don't know if they want me to do this:
$\int_{c} r \cdot \, \vec{dr} = \int_{0}^R r \cdot \, \vec{dr}$
Or if they want me to change the coordenates to the cartesian coordenate system. I don't know what they want me to demostrate.
Thanks for your help. :)