Determine whether the vector field is conservative.
A) conservative
B) not conservative

Question

Determine whether the vector field is conservative. 
A) conservative 
B) not conservative

Quizplus · Accepted Answer

A vector field is conservative if and only if it is the gradient of a scalar function, so that $	extbf{F} = 
abla f$ for some scalar function $f$. To test whether a vector field is conservative, we can check if its curl is zero ($
abla 	imes 	extbf{F} = 	extbf{0}$). 
Using the curl formula, we find that 
$$
abla 	imes 	extbf{F} = \left(\frac{\partial Q}{\partial y} - \frac{\partial P}{\partial z} ight) 	extbf{i} + \left(\frac{\partial R}{\partial z} - \frac{\partial P}{\partial x} ight) 	extbf{j} + \left(\frac{\partial P}{\partial y} - \frac{\partial Q}{\partial x} ight) 	extbf{k}.$$
Substituting the given components of $	extbf{F}$, we get
$$
abla 	imes 	extbf{F} = 2y 	extbf{i} + 2x 	extbf{j} + (-2z) 	extbf{k}.$$
This is nonzero, so the vector field is not conservative.

Determine Whether the Vector Field Is Conservative