We defined a molar flux relative to the molar average velocity as:

\[\overrightarrow{\mathbf{J}}_{i}=-c_{t} D_{i j} \vec{abla} x_{i}\]

We could just as easily have defined the flux relative to the volume average velocity, $v^{v}$.

\[\overrightarrow{\mathbf{J}}_{i}^{v}=-c_{i}\left(\overrightarrow{\mathbf{v}}_{i}-\overrightarrow{\mathbf{v}}^{v}\right)=-D_{i j}^{v} \vec{abla} c_{i}\]

Show that the two diffusivities are equal even if the molar concentration, $c_{t}$, is not constant.