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What are the necessary and sufficient conditions on functions $h,k:{\Bbb R}^2\to{\Bbb R}$ such that given any smooth $F:{\Bbb R^3}\to{\Bbb R}^3$ of the form $F=(F_1(y,z),F_2(x,z),0)$ and whose divergence is zero, there is a smooth $G:{\Bbb R}^3\to{\Bbb R}^3$ of the form $G=(G_1,G_2,0)$ such that $\nabla\times G=F$ in ${\Bbb R}^3$ and $G=(h,k,0)$ on $z=0$?

Can anyone explain what $G$ means in this problem and how the assumption $\nabla \cdot F=0$ might be used?

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if the divergence isn't zero, F can't be written as the curl of G. You can see this by calculating the divergence of the curl of G and using schwartz.

G is the vectorpotential of F.

camel
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  • What do you mean by "using Schwartz"? –  Dec 25 '13 at 20:52
  • type, supposed to be schwarz. However, I can't find any information on it online, only in my calculus notes. It went something like "If f is from C2, then diff(diff(f(x,y),x),y)==diff(diff(f(x,y),y),x)" – camel Dec 25 '13 at 20:55
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$G$ is called a vector potential for $F$, and $G_3 = 0$ is a gauge condition. The basic existence result for vector potentials is that if $F$ is a smooth vector field on a domain $D$ that has no "holes" (I'll leave out the technicalities of that) and $\nabla \cdot F = 0$ everywhere in $D$, then a vector potential for $F$ always exists. Moreover, you can impose some gauge conditions ($G_3 = 0$ and a little more...).

Robert Israel
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  • Thank you for your answer. I've never seen "gauge condition" before in a multivariable calculus book. (And I )Is it something related to physics? Would you please point me to some mathematical references about this result? (Is http://en.wikipedia.org/wiki/Vector_potential the result you refer to?) –  Dec 25 '13 at 20:47
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    The result in your link is a special case of the more general result. See e.g. Adams & Essex, "Calculus: A Complete Course" 7th edition, sec. 16.2, or Davis & Snyder, "Introduction to Vector Analysis", sec. 4.6, for the result on star-shaped domains. – Robert Israel Dec 25 '13 at 22:41
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    If you want to see the phrase "gauge condition" in a calculus book, there's my "Calculus the Maple Way", 2nd edition, p. 230. http://books.google.com/books?id=SFPuAAAACAAJ – Robert Israel Dec 25 '13 at 22:44