Material derivative

By using

\[d\vec{u} = \sum_j \pder{\vec{u}}{Q^j} dQ^j,\]

the material derivative of a vector \(\vec{u}\) with respect to \(t\) (one of the variables in \(Q^j\)) leads to

\[\begin{split}\mder{\vec{u}} & = \pder{\vec{u}}{t} + \sum_j \tder{X^j}{t} \pder{\vec{u}}{X^j} \\ & = \pder{\vec{u}}{t} + \sum_j V^j \pder{\vec{u}}{X^j},\end{split}\]

where

\[V^j \equiv \tder{X^j}{t}\]

is introduced to distinguish the advected (\(U^i\)) and the advecting (\(V^i\)) velocity components.

• Covariant basis
• Normalised basis