I don't see how. Curl and divergence are essentially "opposites" – essentially two "orthogonal" concepts. The entire field should be able to be broken into a curl component and a divergence component and if both are zero, the field must be zero.
I'm visualizing it like a vector in $\mathbb{R}^2$. A vector cannot have a zero $x$ component and a zero $y$ component and still be non-zero.
EDIT: Here's a slightly more formal formulation of my thoughts: The way I see it, the curl and divergence form a "basis" – they are essentially orthogonal vectors. So how can a non-zero vector not be in their span?
Please don't just give me a counterexample. Please explain why my logic is incorrect.
Best Answer
You've had some complex analysis, so you know what a harmonic function is. Take the gradient of any harmonic function. They also have harmonic functions in three dimensions, same example.
You said not to do that. Life is tough.
Two dimensional, we can take harmonic function $x^2-y^2,$ which is the real part of $(x+yi)^2,$ to get vector field $$ (2x, -2y). $$ This has divergence zero and "curl" (as used in Green's Theorem) zero. It really is the curl, we just write it as a scalar.
No more difficult in three dimensions, we may take function $x^2 + y^2 - 2 z^2,$ giving vector field $$ (2x, 2y,-4z). $$ Again, zero divergence and zero curl.