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### Section 16.6 : Conservative Vector Fields

2. Determine if the following vector field is conservative.

$\vec F = \left( {2x\sin \left( {2y} \right) - 3{y^2}} \right)\vec i + \left( {2 - 6xy + 2{x^2}\cos \left( {2y} \right)} \right)\vec j$ 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.

So,

\begin{align*}P & = 2x\sin \left( {2y} \right) - 3{y^2} & \hspace{0.5in}{P_y} & = 4x\cos \left( {2y} \right) - 6y\\ Q & = 2 - 6xy + 2{x^2}\cos \left( {2y} \right)& \hspace{0.5in}{Q_x} & = - 6y + 4x\cos \left( {2y} \right)\end{align*}

Okay, we can clearly see that $${P_y} = {Q_x}$$ and so the vector field is conservative.