Expanding a time derivative in terms of other variables /// Material derivative

Thomas154321

Question 1)

I need to expand $\displaystyle \dot{f}(\rho, L_{ij},\theta,\alpha_{,i})$ as follows:

$\displaystyle \dot{f}(\rho, L_{ij},\theta,\alpha_{,i}) = f_{\rho}\dot{\rho} + f_{L_{ij}}\dot{L_{ij}} + f_{\theta}\dot{\theta} + [\text{a term involving }f_{\alpha_{,i}}]$

The paper I am reading seems to write the final term corresponding to $\displaystyle \alpha_{,i}$

as $\displaystyle \frac{1}{2}f_{\alpha{,i}}\dot{\alpha_{,j}} + f_{\alpha+{,j}}\dot{\alpha_{,i}}$

However I do not see why this is the case. Why can I not write it as

$\displaystyle f_{\alpha{,i}}\dot{\alpha_{,i}}$ ?

----------------------------------------------------------------------------
Question 2)

I have an equation in a journal that is not quoted from anywhere and is just "observed", involving a material derivative. $\displaystyle v=\dot{x}$ as usual.

$\displaystyle \dot{\alpha_{,i}}=(\dot{\alpha})_{,i}-v_{j,i}\alpha_{,j}$