I have the following equation that I'm struggling to understand within a proof for Kelvin's Circulation Theorem within Rotating Shallow Water:
$\frac{d}{dt}\oint_{c(u)}\textbf{v}\cdot d\textbf{x} = \oint_{c(u)}(\partial_t\textbf{v} + \textbf{u}\cdot\nabla\textbf{v} + v_j\nabla u^j)\cdot d\textbf{x}$
where $\textbf{v} = \textbf{v}(t,x(t)$) and $\frac{d}{dt}$ is the material derivative (I know it's normally capitalised by my professor for some reason hasn't).
I understand the chain rule is coming into play here, but shouldn't the last term correspond to $\textbf{v}_\textbf{x}\textbf{x}_t$, ie
$\nabla v_j \cdot u^j$ ?
I'm just really unsure where the last term has appeared.