Disappearing conservative field with zero divergence: is it zero in higher dimensions? I have a vector field, which I know is conservative (it is the divergence of a scalar field). It has divergence zero, and it disappears at infinite distance. 
The dimensionality of the problem is arbitrary. I am working in a Euclidean space.
If I understand correctly, then in two or three dimensions, this vector field is necessarily zero. This is because any vector-valued function is fully determined by its divergence (=0), its curl (=0) and its value on the boundary (->0).
Does this finding extend to higher dimensions? Is a disappearing conservative field with zero divergence still necessarily zero?
 A: The short answer is yes. A (sufficiently smooth) conservative, divergence free vector field is always harmonic. And the only harmonic function disappearing at infinity is zero.
To get the details, you need to have a look at Hodge theory and work with differential forms. However the gist is the following:
You can define the curl in arbitrary dimensions, as an antisymmetric matrix. 
$$(\operatorname{curl} F)_{ij} := \partial_i F_j - \partial_j F_i$$
The space of antisymmetric $3\times 3$ matrices just happens to be three-dimensional, so we can identify it with $\mathbb{R}^3$.
Now any (sufficiently smooth) conservative vector field still has curl zero, by the usual argument, since still
$$(\operatorname{curl} \operatorname{grad} \phi)_{ij} = \partial_i \partial_j \phi - \partial_j \partial_i \phi = 0.$$
But then for your conservative, divergence free vector field:
$$\Delta F_i = \sum_j \partial_j \partial_j F_i \stackrel{\operatorname{curl}=0}= \sum_j \partial_j \partial_i F_j =\partial_i\sum_j  \partial_j F_j \stackrel{\operatorname{div}=0}=0.$$
So each component $F_i$ is harmonic. As mentioned before, you can then show, for example using the maximum principle that $F=0$ if it vanishes at infinity.
A: you confusing the divergence with the gradient. If there is a dot it is divergence if there is no dot then it is gradient.
In order the field to be conservative the force must be the gradient of a scalar potential function. F= ∇f
