Yes, you are confused. "curl", "divergence", and "gradient" apply to mathematical functions (scalar for gradient, vector functions for curl and divergence), not to oil spills!
Do you have a particular mathematical model of the oil spill in mind?
I'm reviewing Meirovitch's "Methods of Analytical Dynamics," and I don't understand the commutation of the derivative from r to dr:
$$
\mathbf{F} \cdot d\mathbf{r} = m \ddot{\mathbf{r}} \cdot d\mathbf{r} = m\mathbf{\dot{r}} \cdot d\mathbf{\dot{r}}
$$