Question

Find out if the force is conservative or non-conservative if the work done is
W = x^2.y - x.z^3 - z.

Answer 1

If the work done is a function of x y and z, then the force is conservative.

Answer 2

That is the whole point of a potential field. It does not matter you you get from point x1,y1,z1 to point x2,y2,z2
The work done is the same by any path.
It may not seem that way if you are taking different trails, steep ones that go up and down and easy ones with a constant slope, from the base camp to the summit, but it is :)

Answer 3

Thanks Damon.
but i need to show W=0 then force is conservative or W not=0 for non conservative force.

My book says if Work done along a closed path is zero then force is conservative.
That is W = line integration of (F.dr)=0

I don't know how to prove it.

Answer 4

If W1 = f(x1,y1,z1)
and W2 = f(x2,y2,z2)
then the work from point 1 to point two
is W2-W1
the work going back is W1 - W2
the net work for the round trip is zero.
That is the point.
If you can define potential as a function of position, the field is conservative.
It is really a trick question.

Answer 5

Thanks Damon.
I converted x,y,z into spherical polar coordinates and proved it when theta =0 and 2pi.
but your method is easy, all things cancel out and W=0 proved.
Thanks