Since F=grad(U), F is conservative, so the work done by F along any closed path is 0.
That is, ∫Fdr = U(B)-U(A) = 0 since B=A for a closed path.
I have a vector field R^3 that is given by : F(x,y,z)=(yz,xz,xy)
Let y be the curve that is given by (x(t),y(t),z(t))=(cost, sint,t). t goes from 0 to pi/4.
-Calculate ∫Fdr by using the curve's parametrization. When I do this I get: pi/8
-Find a potential function for F:
I get: U(x,y,z)= yzx + D + F
Calculate the integral with the help of the potential function. So i get: U(B)- U(A)= pi/8 (same as before, which is good)
So far so good, but the last question I cannot solve.
For which closed curves C is the following fulfiled:
∫Fdr = 0
I know that Green's formula "gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C (wiki)"..
Can anyone please help me, I don't understand how to find the closed curves...