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.