I have a task at hand where I am to solve the integral of a conservative vector field F along a path that is the unit circle centered at origin, oriented CCW.
$${\bf F}(x,y) = (-\frac y{x^2+y^2}){\bf i} + (\frac x{x^2+y^2}){\bf j}$$
I have attempted to integrate both of the terms here to find the potential function, that proved to be difficult.
I have however 2 questions regarding this integral.
Given this follows a path that is the unit circle, does that not mean that the integral would evaluate to 0 as the integral is path independent and both starts and terminates at the same point?
Evaluating this integral by parameterizing the path yields 2pi as the answer:
$$\int_K{\bf \vec F(\vec r(t))}{\cdot}r´(t)\cdot{\rm d}{\bf t}$$
$$ r(t) = cos(t){\bf i} + sin(t){\bf j}$$ $$ r´(t) = -sin(t){\bf i} + cos(t){\bf j}$$
Would the answer be 2pi, 0 or something else that I have missed?