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Math Help - Vector fields and line integrals

  1. #1
    Junior Member
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    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...
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  2. #2
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    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.
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  3. #3
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    Thank you, but I think I need to find an expression for a specific closed curve. How do I do this?
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