A force field is described by <F_x,F_y,F_z> = < xy , 2z-y +x, 2y -z > . Is this force field conservative?

1 Answer
Aug 10, 2016

Non-conservative

Explanation:

The definitive evidence of a conservative field vec F(x,y,z) is the existence of potential function f(x,y,z) which defines the potential at every point and makes the transitions between points in the field path-independent.

And because vec F = \nabla f and thus nabla times vec F = nabla times nabla f = 0 by definition as curl grad is zero....

.....then a necessary but insufficient condition is that the curl of the field be zero. (IOW the curl could be zero but the field still be non -conservative, but if the field is conservative the curl has to be zero)

We test that here:

nabla times vec F = det((hat x, hat y, hat z),(del_x, del_y, del_z),(xy, 2z-y+x, 2y-z))

= hat x (2-2) - hat y (0-0) + hat z (1-x) ne 0

Non-conservative