Derive an expression for \(D T / D t\) in Cartesian coordinates.

Also show that the same relation holds for a vector; i.e.,

\[\frac{D \boldsymbol{v}}{D t}=e_{x} \frac{D v_{x}}{D t}+e_{y} \frac{D v_{y}}{D t}+e_{z} \frac{D v_{z}}{D t}\]

where \(D v_{z} / D t\) etc. have the same form as if they were acting on a scalar quantity.