### Finding a potential function for three-dimensional conservative vector fields

In this page, we give an example of finding a potential function of a three-dimensional conservative vector field. This procedure is an extension of the procedure of finding the potential function of a two-dimensional field .

The vector field we'll analyze is $$\dlvf(x,y,z) = (2xyz^3 + ye^{xy}, x^2z^3+xe^{xy}, 3x^2yz^2+\cos z).$$ We first check if it is conservative by calculating its curl, which in terms of the components of $\dlvf$, is $$\curl \dlvf = \left(\pdiff{\dlvfc_3}{y}-\pdiff{\dlvfc_2}{z}, \pdiff{\dlvfc_1}{z} - \pdiff{\dlvfc_3}{x}, \pdiff{\dlvfc_2}{x} - \pdiff{\dlvfc_1}{y} \right).$$

Since \begin{align*} \pdiff{\dlvfc_3}{y} &= \pdiff{\dlvfc_2}{z} = 3x^2z^2\\ \pdiff{\dlvfc_1}{z} &= \pdiff{\dlvfc_3}{x} = 6xyz^2\\ \pdiff{\dlvfc_2}{x} &= \pdiff{\dlvfc_1}{y} = 2xz^3+(1+xy)e^{xy}, \end{align*} the curl of $\dlvf$ is zero. The vector field is defined in all $\R^3$, which is simply connected, so $\dlvf$ is conservative.

We need to find a potential function $f(x,y,z)$ that satisfies $\nabla f = \dlvf$, i.e., the three conditions \begin{align*} \pdiff{f}{x}(x,y,z) &= 2xyz^3 + ye^{xy}\\ \pdiff{f}{y}(x,y,z) &= x^2z^3+xe^{xy}\\ \pdiff{f}{z}(x,y,z) &= 3x^2yz^2+\cos z. \end{align*}

We start with the first condition involving $\pdiff{f}{x}$. We integrate with respect to $x$, viewing $y$ and $z$ as constants, to obtain that $f(x,y,z)$ must satisfy \begin{align*} f(x,y,z) = x^2yz^3 + e^{xy} + g(y,z). \end{align*} Since we viewed $y$ and $z$ as constants, the constant of integration $g(y,z)$ can be an arbitrary function of $y$ and $z$. You can verify that $f(x,y,z)$ does satisfy the first condition.

Now, we simply need to determine what $g(y,z)$ must be for $f$ to satisfy the remaining two conditions involving derivatives with respect to $y$ and $z$. Let's differentiate our new expression for $f$ with respect to $y$, obtaining \begin{align*} \pdiff{f}{y}(x,y,z) = x^2z^3 + xe^{xy} + \pdiff{g}{y}(y,z) \end{align*} We need to $f$ to satisfy the second condition, above, involving $\pdiff{f}{y}$. For this to be true, we require \begin{align*} \pdiff{g}{y}(y,z) = 0. \end{align*}

Since $\dlvf$ is conservative, this equation for $\pdiff{g}{y}$ must be a function of $y$ and $z$ alone (and not involve $x$). If $x$ appeared, we would know we had made a mistake somewhere. Since $x$ is absent, we can keep going.

In this case, since we need $\pdiff{g}{y}(y,z)=0$, we conclude that $g(y,z)$ cannot depend on $y$. It must be a function of $z$ alone, which we'll call $h(z)$. Our expression for $f(x,y,z)$ simplifies to \begin{align*} f(x,y,z) = x^2yz^3 + e^{xy} + h(z) \end{align*}

We have one more condition to satisfy, the one involving $\pdiff{f}{z}$. We differentiate our new expression for $f$ with respect to $z$: \begin{align*} \pdiff{f}{z} = 3x^2yz^2 + \diff{h}{z}(z). \end{align*} For $f(x,y,z)$ to satisfy the third condition for $f$, the function $h(z)$ must satisfy \begin{align*} \diff{h}{z}(z) = \cos z. \end{align*} This is good news, as $\diff{h}{z}$ does not depend on $x$ or $y$. If it had, we would know something went wrong

We can easily integrate to obtain an expression for $h$, \begin{align*} h(z) = \sin z + k \end{align*} for an arbitrary constant $k$. A potential function can only be determined up to an arbitrary constant, since we only have conditions on its derivatives. But, line integrals of $\dlvf$ depend only on differences among the values of $f(x,y,z)$. The constant $k$ will always cancel out, so we can just set $k=0$.

Therefore, our potential function for $$\dlvf(x,y,z) = (2xyz^3 + ye^{xy}, x^2z^3+xe^{xy}, 3x^2yz^2+\cos z).$$ is $$f(x,y,z) = x^2yz^3 + e^{xy} + \sin z.$$

For any curve $\dlc$ from point $\vc{p}$ to point $\vc{q}$, the integral of $\dlvf$ is $$\dlint = f(\vc{q}) - f(\vc{p})$$ independent of the path taken by $\dlc$. Although we had to do a lot of work to calculate $f$, the last step of computing the integral is simple.

If $\dlc$ is the arc of a helix parametrized by $\dllp(t) = (\cos t, \sin t, t)$ for $0 \le t \le \pi/2$, the line integral of $\dlvf$ is simply \begin{align*} \dlint &= f(\dllp(\pi/2)) - f(\dllp(0))\\ &= f(0,1,\pi/2) - f(1,0,0)\\ &= 0 + e^0 + \sin \frac{\pi}{2} - 0 - e^0 - \sin 0 = 1 + 1 - 1 = 1. \end{align*}

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