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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 Solution

There 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.


\[\begin{align*}P & = 6 - 2xy + {y^3} & \hspace{0.5in}{P_y} & = - 2x + 3{y^2}\\ Q & = {x^2} - 8y + 3x{y^2} & \hspace{0.5in}{Q_x} & = 2x + 3{y^2}\end{align*}\]

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.