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

Sep 2006
77
15
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

\(\displaystyle f_{\alpha{,i}}\dot{\alpha_{,i}}\) ?



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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.

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