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A concept in fluid dynamics requires the following derivative to be evaluated: $$ \frac{D\mathbf u}{Dt} = \frac{\partial \mathbf u}{\partial t}+\mathbf v\cdot\nabla\mathbf u$$ This notation is very confusing to me - particulary the notion of gradient of the vector field $\mathbf u$. Is there any way to write this derivative in terms of $\mathbf {grad}, \mathbf {div}$ etc. in a way that allows its application in many different coordinate systems?

EDIT: this same derivative is also often written with additional parentheses (as below). Any clarification on this point?$$ \frac{D\mathbf u}{Dt} = \frac{\partial \mathbf u}{\partial t}+\mathbf (v\cdot\nabla)\mathbf u$$

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    Does it help you writing out this material derivative? i.e $$\frac{D\mathbf u}{Dt}=\begin{pmatrix} \partial_t u_1 \\ \vdots \\ \partial_t u_n \end{pmatrix} + v_1\begin{pmatrix} \partial_1u_1 \\ \vdots \\ \partial_1u_n \end{pmatrix}+ ... +v_n \begin{pmatrix} \partial_nu_1 \\ \vdots \\ \partial_nu_n \end{pmatrix}$$ – 2017-01-04

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