Determine if is conservative, and if so, find a potential function of the vector field.
A) is conservative with
B) is conservative with
C) is conservative with
D) is not conservative

Question

Determine if  is conservative, and if so, find a potential function  of the vector field.
A) is conservative with 
B) is conservative with 
C) is conservative with 
D) is not conservative

Quizplus · Accepted Answer

A vector field is conservative if and only if it is the gradient of some scalar potential function. The given vector field has components that do not explicitly depend on any of the variables, so it is possible that it is conservative. To determine whether it is conservative, we can check whether its curl is zero. Computing the curl and simplifying using the given vector notation yields:

curl(F) = (-4x^2y - 4xy^2)i + (4x^3 - 3y^2 - 8xy)j

The curl is not zero, so the vector field is not conservative. Therefore, none of the potential functions listed can be correct.

Determine If Is Conservative, and If So, Find a Potential