Question

Is the integral A.dr path independent if A =(2x-y)i+(x+y)j? Evaluate integral A.dr around a unit circle centered at the origin.what is the value of the integral if we take a clockwise direction?

Answer 1

Step-by-step explanation:

The condition for path independence of the line integral

Int(over C)[M(x,y)dx +N(x,y)dy]

is that the above integrand is an exact differential, which is the case if and only if

del(M)/del(y) = del(N)/del(x)

holds at each point of a region containing the curves C. In this case

del(M(x,y))/del(y) = del(2x-y)/del(y) = (-1),

and del(N(x,y))/del(x) = del(x+y)/del(x) = 1, and so the line integral is not path independent.