When the divergence of a function is zero, what does it say about the curl

Question

If a function $T$ has divergence zero, does that mean that the curl of the function is equal to the gradient of the function ?

Answer 1

As stated by Ninad, If T has a divergence it must be a vector field. And vector fields don't have gradients. But I think I see what you are looking for.

If you have a vector field with divergence 0, it means your function T can be expressed as the curl of some other function (locally). Why is that? It helps to notice that:

$\nabla \cdot \mathbf{T}=0$ can be rewritten as $\nabla \cdot(\nabla \times \mathbf{A})=0 $

As you might know that this triple product is always 0