Answer
a) It has been proved that a constant force field does zero work on a particle that moves once uniformly around the circle $x^2
+y^2=1$
b) Yes
Work Step by Step
a) Let us consider $F(x,y)=A(x,y) i+B(x,y) j$
Work done, $W=\int_C F\cdot ds=\int_C a dx +\int_C b dy$
or, $W=\int_0^{2 \pi} a [-\sin t dt]+\int_0^{2 \pi} b [\cos t dt]=a(0)+b(0)=0$
Hence, it has been shown that a constant force field does zero work on a particle that moves once uniformly around the circle $x^2
+y^2=1$
b) Let us consider $F(x,y)=A(x,y) i+B(x,y) j$
Work done, $W=\int_C F\cdot ds=\int_C A dx +\int_C B dy=\int_0^{2 \pi} [k\cos t dt][-\sin t] dt+\int_0^{2 \pi} k \sin t$
and $W=k \int_0^{2 \pi} -\cos t \sin t +\sin t \cos t dt=0$
Yes, for a force field $F(x)=kx$