****************
a) because along lines parallel to the y-axis all of the vectors are contant and all pointing in the positive x-direction. Each dot product on upper half of the circle will have the negative of dot product on the other half of the circle because the x-direction of dr is pointing the opposite direction.
b) all of the arrows of the vector field are pointing perpendicular to the circle, hence dot product is zero at each point.
c) this vector field is the gradient of x^2-y^2 (check it out). but the path integral along a closed path of any gradient is zero .
