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Figuring Out if A Force Field is Conservative or Not

  1. Oct 9, 2016 #1
    1. The problem statement, all variables and given/known data
    There is a collection of different force fields, for example:
    $$F_{x}=ln z$$
    $$F_{y}=-ze^{-y}$$
    $$F_{z}=e^{-y}+\frac{x}{z}$$
    We are supposed to indicate whether they are conservative and find the potential energy function.

    2. Relevant equations
    See Above

    3. The attempt at a solution

    Is it a conservative force if it is the gradient of a scalar field?

    So if $$\vec{F}=(\frac{\delta u}{\delta x},\frac{\delta u}{\delta y},\frac{\delta u}{\delta z})$$

    You also have to check that $$
    \Delta\times\vec{F}=\vec{0}$$

    Which is true.

    So for this particular case the answer would be yes, it is conservative, because $$u(x,y,z) = ze^{-y}+xlnz$$ fulfills this requirement.

    So the actual potential energy would just be $$-u(x,y,z)$$

    Is this the whole process I can do for any three dimensional force field? Am I missing any subtle details here?

    Thank you!
     
  2. jcsd
  3. Oct 9, 2016 #2

    kuruman

    User Avatar
    Homework Helper
    Gold Member

    If the curl of the force is zero, the force is conservative.
    If the force can be written as the gradient of a scalar field, it is conservative.
     
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