# Thread: Vector fields and line integrals

1. ## Vector fields and line integrals

Hello!

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...

2. 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.

3. Thank you, but I think I need to find an expression for a specific closed curve. How do I do this?