Determine whether the vector field is conservative. If it is, find a potential function for the vector field.
A) conservative with potential function
B) conservative with potential function
C) conservative with potential function
D) conservative with potential function
E) not conservative

Question

Determine whether the vector field is conservative. If it is, find a potential function for the vector field. 
A) conservative with potential function 
B) conservative with potential function 
C) conservative with potential function 
D) conservative with potential function 
E) not conservative

Quizplus · Accepted Answer

A vector field F is conservative if and only if it is the gradient of a scalar potential function f. That is, F = ∇f for some scalar function f. To determine whether a vector field F is conservative, we can check whether it satisfies the condition ∂F_i/∂x_j = ∂F_j/∂x_i for all i and j, where F = (F_1, F_2, F_3) are the components of the vector field in Cartesian coordinates.

Using this criterion, we can easily see that none of the given potential functions are the gradient of the vector field in question. Therefore, the vector field is not conservative.

Determine Whether the Vector Field Is Conservative