how to convert surface to vector field to use it in line integral

Question

example for what i want to say how to convert the scalar form of line integral :- $\int g(x,y)\ ds = \int p\ dx + \int q\ dy$

real world example:-

$\int xy \ ds \ s: unit \ circle \ centered at origin$

to

$ \int p\ dx + \int q\ dy $

Answer 1

The unit circle centered at the origin is given by parametric equations $x= \cos(t), y= \sin(t)$ with $t$ going from 0 to $2\pi$ so $dx= -\sin(t)dt$ and $dy= \cos(t)dt$. $ds^2= (dx)^2+ (dy)^2$ so $ds= \sqrt{(dx)^2+ (dy)^2}= \sqrt{(-\sin(t)dt)^2+ (\cos(t)dt)^2}= \sqrt{\sin^2(t)+ \cos^2(t)}dt= dt$. (More simply, the angle t, measured in radians, measures the circumference of the circle. With "t" in radians, $ds= dt$).

$\int xy ds= \int_0^{2\pi} cos(t)dt$.

There is no $\int dx+ dy$.