- #1

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**B**satisfying

**div B**= 0 everywhere

then there is a vector field

**A**such that

**B**=

**curl A**? If so, is it hard to prove? Of course, the converse is obviously true.

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- Thread starter techmologist
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- #1

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then there is a vector field

- #2

quasar987

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You could also have asked, given that curlA=0, does there exist a function f such that A=grad(f) ?

The general setting for answering questions such as these concerning the existence of a "primitive" in some sense or another is the subject of the de Rham theory of differential forms. In this context, to ask whether every smooth vector field of vanishing divergence defined on some open subset U of

Let B:R³\{0}-->R³ be the vector field

[tex]B(x,y,z)=\frac{x\hat{x}+y\hat{y}+z\hat{z}}{(x^2+y^2+z^2)^{3/2}}=\frac{r}{|r|^3}[/tex]

Suppose that B=curl(A) for some vector field A:R³\{0}-->R³. Then, by Stoke's theorem, we would have

[tex]\int_{S^2}B\cdot\hat{r}dA=\int_{S^2}\mbox{curl}(A)\cdot\hat{r}dA=\int_{\partial S^2}A\cdot dl = 0[/tex]

(because the sphere has no boundary: [itex]\partial S^2=\emptyset[/itex]). But, on the other hand, a direct calculation using spherical coordinates gives

[tex]\int_{S^2}B\cdot\hat{r}dA=\int_0^{2\pi}\int_0^{\pi}\sin(\phi)d\phi d\theta = 4\pi[/tex]

This is a contradiction that shows that B is not the curl of any vector field A.

- #3

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Just to make sure I understand right, it would be possible to find a vector field A of which B=r/|r|

- #4

quasar987

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It is a bit ambiguous to say "an open set that does not contain the origin", but your statement

*it would be possible to find a vector field A of which B=r/|r|3 is the curl if we restrict B's domain to some open subset U of R3 that does not contain the origin*

is correct if by "an open set that does not contain the origin" you mean an open set which is contained in a contractible set not containing the origin.

For instance, if you restrict the domain to the open "spherical shell" S define by 1<r<2 around the origin, then B still isn't the curl of any A. But if you restrict to the open "spherical shell" S around (3,0,0), then B is the curl of some A, because S is contained in the ball of radius 2 centered at (3,0,0) which is contractible.

is correct if by "an open set that does not contain the origin" you mean an open set which is contained in a contractible set not containing the origin.

For instance, if you restrict the domain to the open "spherical shell" S define by 1<r<2 around the origin, then B still isn't the curl of any A. But if you restrict to the open "spherical shell" S around (3,0,0), then B is the curl of some A, because S is contained in the ball of radius 2 centered at (3,0,0) which is contractible.

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- #5

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In the meantime I searched the scriptures

Thanks for the help!

* Tom Apostol's

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