I want to find a 2D vector field with three characteristics:
- it is continuous
- all the vectors are unit length
- vectors on the unit circle point to the origin
Is this possible? I haven't been able to find an example, and I'm beginning to think that something analogous to the hairy ball theorem applies. Is it possible if one of the constraints is loosened, such as allowing the norm to vary in a range of positive values?
No, this is not possible. A unit vector field defines a map to $S^1$; your assumption that all the vectors on the unit circle point towards the origin says that this map is the antipodal map $f: S^1 \to S^1$, $x \mapsto -x$. If you could extend this over the disc $D^2$, that would define a null-homotopy of your map $S^1 \to S^1$. (If you write points in $D^2$ radially, i.e. as $(r,\theta)$, and we call your extension $v: D^2 \to S^1$, the null-homotopy is defined by $f_t: S^1 \times I \to S^1$, $f_t(\theta) = v(t,\theta)$; automatically $f_1$ is a constant map.)
However, your antipodal map represents a generator of $\pi_1(S^1) \cong \Bbb Z$, the fundamental group of $S^1$, and it is not null-homotopic.