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For what smooth functions $f:\mathbb R^3\to \mathbb R$ is there a smooth vector field $W:\mathbb R^3\to \mathbb R^3$ with $\operatorname{curl}W=V$, where $$V(x,y,z)=(y,x,f(x,y,z)).$$ For $f$ in this class, find such a $W$. Is it unique?

I wrote out $\operatorname{curl}W$ explicitly as $$\operatorname{curl}W=(D_2w_3-D_3w_2,D_3w_1-D_1w_3,D_1w_2-D_2w_1)$$ and the condition in question is $$(D_2w_3-D_3w_2,D_3w_1-D_1w_3,D_1w_2-D_2w_1)=(y,x,f(x,y,z))$$ I guess I have to suppose that everything takes place in the same $\mathbb R^3$ with the same coordinates $x,y,z$. Then $w_i=w_i(x,y,z)$ and $$D_2w_3-D_3w_2=y\\D_3w_1-D_1w_3=x\\D_1w_2-D_2w_1=f(x,y,z)$$

For the first two equations I can choose $w_3=y^2/2-x^2/2$ and $w_1,w_2$ arbitrary. But what to do with the third condition? Should I write $w1,w_2$ as one-variable integrals of $f$ or something? The above was for the "find $W$ part". Also how to find the $f$'s we need?

user557
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    You also have that $\operatorname{div}(V)=\operatorname{div}(\operatorname{curl}(W))=0$ – Botond Aug 11 '18 at 21:18
  • Once you use @Botond’s key hint and have chosen an $f$, you can use the method in this answer to find $W$, but there are some particularly simple choices for $f$ that don’t require all of that machinery. – amd Aug 11 '18 at 22:02
  • $w_1$ and $w_2$ are not arbitrary. Plugging your $w_3$ into, say, the second equation gives you the condition $D_3w_1=0$ on $w_1$. – amd Aug 11 '18 at 22:07
  • @amd Why do I need to choose $f$? The problem asks to find all such $f$'s. Botond's hint gives that $f$ is independent of $z$, which gives a necessary condition. But is it sufficient? – user557 Aug 11 '18 at 22:14
  • I read the problem as asking for a specific example. If that’s not what’s being asked of you, then the best you can do is to express $W$ as an integral. – amd Aug 11 '18 at 22:49
  • You need some conditions on the domain to ensure that $\operatorname{div}V=0 \implies V=\operatorname{curl}W$ for some $W$, but those are satisfied here. – amd Aug 11 '18 at 22:50
  • Related: https://math.stackexchange.com/questions/2379196/when-does-this-partial-differential-equation-system-have-solutions – user557 Aug 19 '18 at 15:27

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