Section 16.6 : Conservative Vector Fields
6. Find the potential function for the following vector field.
\[\vec F = \left( {2{z^4} - 2y - {y^3}} \right)\vec i + \left( {z - 2x - 3x{y^2}} \right)\vec j + \left( {6 + y + 8x{z^3}} \right)\vec k\]Show All Steps Hide All Steps
Start SolutionNow, by assumption from how the problem was asked, we can assume that the vector field is conservative and because we don’t know how to verify this for a 3D vector field we will just need to trust that it is.
Let’s start off the problem by labeling each of the components to make the problem easier to deal with as follows.
\[\begin{align*}P & = 2{z^4} - 2y - {y^3}\\ Q & = z - 2x - 3x{y^2}\\ R & = 6 + y + 8x{z^3}\end{align*}\] Show Step 2To find the potential function for this vector field we know that we need to first either integrate \(P\) with respect to \(x\), integrate \(Q\) with respect to \(y\) or \(R\) with respect \(z\). It doesn’t matter which one we use chose to use and, in this case, it looks like none of them will be any harder than the others.
So, let’s go with the following integration for this problem.
\[\begin{align*}f\left( {x,y,z} \right) & = \int{{Q\,dy}}\\ & = \int{{z - 2x - 3x{y^2}\,dy}}\\ & = zy - 2xy - x{y^3} + h\left( {x,z} \right)\end{align*}\]Don’t forget that, in this case, because we were integrating with respect to \(y\) the “constant of integration” will be a function of \(x\) and/or \(z\)!
Show Step 3Next, we can differentiate the function from the previous step with respect to \(x\) and set equal to \(P\) or differentiate the function with respect to \(z\) and set equal to \(R\).
Again, neither looks any more difficult than the other so let’s differentiate with respect to \(z\).
\[{f_z} = y + {h_z}\left( {x,z} \right) = 6 + y + 8x{z^3} = R\hspace{0.25in} \Rightarrow \hspace{0.25in}{h_z}\left( {x,z} \right) = 6 + 8x{z^3}\]Now, recall that because we integrated with respect to \(y\) in Step 2 \(h\left( {x,z} \right)\), and hence \({h_z}\left( {x,z} \right)\) , should only be a function of \(x\)’s and \(z\)’s (as it is in this case). If there had been any \(y\)’s in \({h_z}\left( {x,z} \right)\) we’d know there was something wrong at this point. Either we’d made a mistake somewhere or the vector field was not conservative.
Also note that there is no reason to expect \({h_z}\left( {x,z} \right)\) to have both \(x\)’s and \(z\)’s in it. It is completely possible for one (or both) of the variables to not be present!
Show Step 4We can now integrate both sides of the formula for \({h_z}\left( {x,z} \right)\) with respect to \(z\) to get,
\[\,h\left( {x,z} \right) = 6z + 2x{z^4} + g\left( x \right)\]Now, because \(h\left( {x,z} \right)\) was a function of both \(x\) and \(z\) and we integrated with respect to \(z\) here the “constant of integration” in this case would need to be a function of \(x\), \(g\left( x \right)\) in this case.
The potential function is now,
\[f\left( {x,y,z} \right) = zy - 2xy - x{y^3} + 6z + 2x{z^4} + g\left( x \right)\] Show Step 5Next, we’ll need to differentiate the potential function from Step 4 with respect to \(x\) and set equal to \(P\). Doing this gives,
\[{f_x} = - 2y - {y^3} + 2{z^4} + g'\left( x \right) = 2{z^4} - 2y - {y^3} = P\hspace{0.25in} \Rightarrow \hspace{0.25in}g'\left( x \right) = 0\]Remember, that as in Step 3, we have to recall what variable we are differentiating with respect to here. In this case we are differentiating with respect to \(x\) and so \(g\left( x \right)\) should only be a function of \(x\). Had \(g'\left( x \right)\) contained either \(y\)’s or \(z\)’s we’d know that either we’d made a mistake or the vector field was not conservative.
Also, as shown in this problem, it is completely possible for there to be no \(x\)’s at all in \(g'\left( x \right)\).
Show Step 6Integrating both sides of the formula for \(g'\left( x \right)\) from Step 5 and we can see that we must have \(g\left( x \right) = c\).
Show Step 7Finally, putting everything together we get the following potential function for the vector field.
\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( {x,y,z} \right) = zy - 2xy - x{y^3} + 6z + 2x{z^4} + c}}\]