Let $\mathbf{F}(x,y,z) = y \hat{i} + x \hat{j} + z^2 \hat{k}$ be a vector field. Determine if its conservative, and find a potential if it is.

Attempt at solution:

We have that $\frac{\partial F_1}{\partial y} = 1 = \frac{\partial F_2}{\partial x} $, $\frac{\partial F_1}{\partial z} = 0 = \frac{\partial F_3}{\partial x}$, $\frac{\partial F_2}{\partial z} = 0 = \frac{\partial F_3}{\partial y}$, so the potential might exist.

Now we need to find a function $f$ such that $\nabla f = \mathbf{F}$.

For the first component, this means that $\frac{\partial f(x,y,z)}{\partial x} = y $, or after integrating, $f(x,y,z) = yx + C(y,z)$. Now I don't know how to determine the constant of integration $C(y,z)$, and I don't understand if I should add another constant when I integrate the second component.

For the second component, we have that $f(x,y,z) = xy + D(x,z)$, and for the third $f(x,y,z) = \frac{z^3}{3} + E(x,y)$. What now?

Any help please? In my textbook this is explained really in a terrible way.

  • $\begingroup$ If $\nabla f=\mathbf F$ then $f(\mathbf b)-f(\mathbf a)=\int_{\mathbf a}^{\mathbf b} \mathbf F(\mathbf r)\cdot\mathrm d\mathbf r$ over any path from $\mathbf a$ to $\mathbf b$. Let $\mathbf a=(0,0,0)$, $\mathbf b=(x,y,z)$, and pick any path you like from one to the other. $\endgroup$ – Rahul Mar 9 '15 at 20:17

You don't have to find the integration constant immediately. Keep proceeding as follows.

After you determined that $f(x,y,z) = xy+g(y,z)$, differentiate with respect to $y$.

This gives $\frac{\partial f}{\partial y}=x+\frac{\partial g}{\partial y}=F_y=x$.

Thus, $\frac{\partial g}{\partial y}=0$, which implies that $g$ is a function of $z$ only. In turn, this means that $f(x,y,z)=xy+h(z)$.

Next, differentiate $f$ with respect to $z$.

This gives $\frac{\partial f}{\partial z}=h'(z)=F_z=z^2$.

Thus, $h(z)=\frac13z^3+C$.

Finally, $f(x,y,z)=xy+h(z)=xy+\frac13z^3+C$.

To check this, we have $$\vec F=\nabla f(x,y,z)$$

$$=\hat x\frac{\partial f}{\partial x}+\hat y\frac{\partial f}{\partial y}+\hat z\frac{\partial f}{\partial z}$$

$$=\hat xy+\hat yx+\hat zz^2$$which completes the task!

  • $\begingroup$ Why do I have to differentiate it with respect to $y$? I just integrated it with respect to $x$, so it looks like double work to me. $\endgroup$ – Kamil Mar 9 '15 at 19:45
  • $\begingroup$ Your result to date is $f(x,y,z)x+g(y,z)$. The partial derivatives with respect to $y$ and $z$ still need to be equal to the $y$ and $z$ components of $\vec F$, do they not? You will find a contradiction in this attempt, thereby implying that $\vec F$ is NOT conservative. Your application of the Curl had an error in it. See if you can find it. $\endgroup$ – Mark Viola Mar 9 '15 at 19:49
  • $\begingroup$ I really have no clue what this method is about. I would set the three partial derivatives equal to the three components. Then I would integrate them all, which gives me 3 constants. I don't know how to handle those. $\endgroup$ – Kamil Mar 9 '15 at 19:52
  • $\begingroup$ After you changed the vector $\vec F$, I augmented the answer. I hope this helps now. $\endgroup$ – Mark Viola Mar 9 '15 at 20:40
  • $\begingroup$ Thanks, I think I'm starting to understand the process. But two questions: how do you know that the constant of integration (the function $g$) depends on $z$? Couldn't it be that it is just really a number? And second question, how do you know that the last constant you've found, $C$, does not depend on any variables anymore? $\endgroup$ – Kamil Mar 9 '15 at 21:06

The function $\phi(x,y,z) = xy + \frac{z^3}{3}$ is a potential for $\mathbf{F}$ since $$\operatorname{grad} \phi = \phi_x \mathbf{i} + \phi_y \mathbf{j} + \phi_z \mathbf{k} = y\mathbf{i} + x \mathbf{j} + z^2\mathbf{k} = \mathbf{F}.$$

To actually derive $\phi$, we solve $\phi_x = F_1, \phi_y = F_2, \phi_z = F_3$. Since $\phi_x = F_1 = y$, by integration $\phi(x,y,z) = xy + u(y,z)$. Now $\phi_y = x + u_y$, so from $\phi_y = F_2 = x$, we have $x + u_y = x$, or $u_y = 0$. By integration, $u(y,z) = v(z)$. Thus $\phi(x,y,z) = xy + v(z)$. Then $\phi_z = v'(z)$, so from $\phi_z = F_3 = \frac{z^3}{3}$, we get $v'(z) = \frac{z^3}{3}$, which by integration yields $v(z) = z^2 + C$, where $C$ is a constant independent of $x,y,z$. This gives the general solution $\phi(x,y,z) = xy + \frac{z^3}{3} + C$. For convenience we set $C = 0$, giving the particular solution $\phi(x,y,z) = xy + \frac{z^3}{3}$.

  • $\begingroup$ I don't understand the part where you say: "By integration, $u(y,z) = v(z)$". Where did that step come from? We found that the constant of integration of the first component equals zero, so what's next? $\endgroup$ – Kamil Mar 9 '15 at 20:29
  • $\begingroup$ Since $u_y(y,z) = 0$, $u$ depends only on $z$. So I can write this in the form $u(y,z) = v(z)$. As for your second question, I've already given the full analysis. $\endgroup$ – kobe Mar 9 '15 at 20:31

Take the derivative of $f(x,y,z)$ by y; this must be equal to $-x$. Then solve for $C(y,z)$ by Integration. You will obtain another Integration constant $C(z)$ that you can obtain by derivative by variable $z$ (must be equal to $z^2$).

  • $\begingroup$ But this will lead to a contradiction, which implies that the vector is not conservative. $\endgroup$ – Mark Viola Mar 9 '15 at 19:45

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