Figuring Out if A Force Field is Conservative or Not

[Summer95]

Homework Statement

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.

Homework Equations

See Above

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!

 

[kuruman]

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.