Recently the material derivative was mentioned in my fluid dynamics class. The definition given is $$\operatorname{MD}=\partial_t +\mathbf{u}~\boldsymbol{\cdotp}\nabla$$ Where of course $\nabla$ is taken to be the covariant derivative using the Levi Civita connection. However, I'm a little confused as to what to do with that dot product in the general setting, especially when dealing with higher order tensors in arbitrary coordinate systems. Let's say I am using some arbitrary coordinate system $\{x_i\}$ with corresponding metric tensor $\mathbf{g}$. Now suppose I have some $(r,s)$ tensor $\mathbf{T}$. I want to know what the components of that "dot product" will be. Do I just do
$$(\mathbf{u}~\boldsymbol{\cdot}\nabla\mathbf{T})^{i_1,...,i_r}_{j_1,...,j_s}=g_{\mu\nu}u^\mu T^{i_1,...,i_r}_{j_1,...,j_s~;~\nu}$$
Or is it something else completely? My professor really didn't talk much about the general case for higher order tensors, nor did he talk about what to do in an arbitrary coordinate system - he just gave formulas for the material derivative of scalars/vectors in the standard and cylindrical coordinate cases.
Also, if the components of $\mathbf{u}$ are given with respect to a normalized basis, e.g in plane polars $\hat{\mathbf{e}}_r=\cos\theta \mathbf{e}_x+\sin\theta\mathbf{e}_y$, $\hat{\mathbf{e}}_\theta=-\sin\theta\mathbf{e}_x+\cos\theta\mathbf{e}_y$ do I need to unnormalize them before I try to take the dot product? I would really like some clarification on this matter.
Thanks.