Vector Calculus: Is F=0 When ∇·F=0 & ∇×F=0?

[latentcorpse]

is it true that if \nabla \cdot \vec{F}=0 , \nabla \times \vec{F}=0 then \vec{F}=0?

 

[robphy]

is it true that if \nabla \cdot \vec{F}=0 , \nabla \times \vec{F}=0 then \vec{F}=0?

Suppose \vec F=3\hat x \ldots

 

[latentcorpse]

the divergence of 3x is 3 not 0 though?

 

[robphy]

the divergence of 3x is 3 not 0 though?

By \hat x, I mean what you might have seen as \hat \imath...
that is,
\vec F= (3) \hat \imath
or
\vec F= (3,0,0)
...that is, a nonzero constant vector field.