Calculus III - Conservative Vector Fields

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Section 16.6 : Conservative Vector Fields

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1. Determine if the following vector field is conservative.

$\vec F = \left( {{x^3} - 4x{y^2} + 2} \right)\vec i + \left( {6x - 7y + {x^3}{y^3}} \right)\vec j$ 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,

$\begin{align*}P & = {x^3} - 4x{y^2} + 2 & \hspace{0.5in}{P_y} & = - 8xy\\ Q & = 6x - 7y + {x^3}{y^3} & \hspace{0.5in}{Q_x} & = 6 + 3{x^2}{y^3}\end{align*}$

Okay, we can clearly see that ${P_y} \ne {Q_x}$ and so the vector field is not conservative.