I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 16.6 : Conservative Vector Fields
3. Determine if the following vector field is conservative.
\[\vec F = \left( {6 - 2xy + {y^3}} \right)\vec i + \left( {{x^2} - 8y + 3x{y^2}} \right)\vec j\] Show SolutionThere really isn’t all that much to do with this problem. All we need to do is identify \(P\) and \(Q\) then run through the test.
So,
\[\begin{align*}P & = 6 - 2xy + {y^3} & \hspace{0.5in}{P_y} & = - 2x + 3{y^2}\\ Q & = {x^2} - 8y + 3x{y^2} & \hspace{0.5in}{Q_x} & = 2x + 3{y^2}\end{align*}\]Okay, we can clearly see that \({P_y} \ne {Q_x}\) and so the vector field is not conservative.
Be careful with these problems. It is easy to get into a hurry and miss a very subtle difference between the two derivatives. In this case, the only difference between the two derivatives is the sign on the first term. That’s it. That is also enough for this vector field to not be conservative.