Question 1)

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

\dot{f}(\rho, L_{ij},\theta,\alpha_{,i})<br />
= 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 \alpha_{,i}

as \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

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. v=\dot{x} as usual.

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