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

1 Answer
Aug 1, 2016

Answer:

Not conservative

Explanation:

the conclusive test is the existence of a potential function #f# such that #vec F = nabla f#. you could try and reverse-engineer a potential function from the partials but...

.....a necessary (though insufficient) condition is that the curl of the force field is zero because, if #f# indeed exists, then #nabla times vec F = nabla times nabla f = 0# as curl ( grad ) =0.

here

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

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

So this is not conservative.