Expert: Socrates - 10/1/2009

Question
determine wheter the force F is conservative by finding the value of the line integral F.dx for
F=xy^2i = x^2yj
along the path r(t)=ti+(1/t)j ,   1<=t<=2

im having a brain freexe, not sure how to approach this problem, please help

Answer
The notation I use might be different from yours , but here it is

dx,dy stand for the derivatives of the path components

Substitute t for x , 1/t for y , 1 for dx and -t^-2 for dy  in

S xy^2 dx + x^2y dy

Then evaluate the integral from t = 1 to t = 2

S (t)(t^-2)(1) + (t^2)(t^-1)(-t^-2) dt =


S t^-1 - t^-1 dt  =

S 0 dt = 0

(nothing to evaluate , the integrand is zero)

The force is conservative along this path because the integral is 0