What is the vector field shown on this sphere?
The source should be at $P(0,0,0)$ and the sink should be at $P(1,1,1).$
How would I derive this vector field?
It looks like the vectors are all tangent to great circles.
So would the vector field be something like: $$ F=\{ 2x+y^2+z^2,2y+x^2,+z^2,2z+x^2+y^2\}? $$
I'm sure that is wrong though.
The explanation will be simpler if the source is the South Pole and the sink the North pole on the unit sphere centered in $O(0,0,0)$.
We are going to work on spherical coordinates at first : let $\theta, \phi$ be the longitude and latitude resp.
The depicted vectors have visibly all of the same norm, orthogonal to unit vectors that are their "attach point".
Consider the generic point $P$ of the sphere characterized by $(\theta, \phi, r=1)$, i.e., with cartesian coordinates
$$P=(\cos \theta \cos \phi, \sin \theta \cos \phi, \sin \phi).\tag{1}$$
The great circles passing through the north and south poles being integral lines of the vector fiels, the generic vector belonging to the vector field at point $P$ is proportional to the derivative of (1) with respect to $\phi$ ($\theta $ being kept constant).
Thus the general formula for this vector field is :
$$\vec{V}=k(-\cos \theta \sin \phi, -\sin \theta \sin \phi, \cos \phi).\tag{2}$$
Please note that $\|\vec{V}\|=k$. Nothing give us information on the real value of $k$.
One can pass from this particular case to the general case by a rotation and an homothety (the sphere you are dealing with has radius $1/(2 \sqrt{3})$).