I have to calculate the complex line integral of vector field $F(x,y)=(-y,x)$ on a segment from $(1,1)$ to $(3,2)$.
And this is what I've done, but I don't know if is it the proper way to do this problem:
$F(x,y)=(P(x,y),Q(x,y))=(-y,x)$
$\int_\gamma -y\;dx + x\;dy$ ;
$\gamma(t)=(x(t),y(t))=(1-t)(1,1)+t(3,2)=(1-t,1-t)+(3t,2t)=(1+2t,1+t)$
$\int_\gamma -y\;dx + x\;dy=\int_0^1(-1-t)2+(1+2t)\;dt=\int_0^1-1dt=-1$