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I'm in Continuum Mechanics, and I stumbled on an equation that is supposed to be a total differentiation of speed vector (or so, i don't know how to call it in English). The equation is $$\frac{\textrm{D} \vec{v}}{\textrm{D} t} = \left.\frac{\partial \vec{v}}{\partial t} \right\vert_{\vec{r}} + \left(\vec{v} \cdot \textrm{grad}\right) \vec{v} = \left.\frac{\partial \vec{v}}{\partial t} \right\vert_{\vec{r}} + \textrm{grad}\left(\frac{v^2}{2}\right) - \vec{v} \wedge \textrm{rot} \ \vec{v}.$$

I have no idea how to prove it because I didn't understand how does the gradient operator act on a vector (instead of a scalar). I'd like some hints or references if possible.

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The expression $(\vec{A} \cdot \textrm{grad}) \vec{B}$ is not meant as the gradient acting on a vector. It is in fact an abuse of notation meaning $$\begin{align} (\vec{A} \cdot \textrm{grad}) \vec{B} &= \left(A_x \frac{\partial}{\partial x} + A_y \frac{\partial}{\partial y} + A_z \frac{\partial}{\partial z}\right) \vec{B}, \\ &= A_x \frac{\partial \vec{B}}{\partial x} + A_y \frac{\partial \vec{B}}{\partial y} + A_z \frac{\partial \vec{B}}{\partial z}. \end{align}$$

Hence, you are not taking the gradient of a vector. Instead, you first "take the scalar product inside the parentheses" (where I'm using commas because we write it in this way just to exploit notation) and then you just take the usual derivatives of a vector field.

Recall that the meaning of $\frac{\textrm{D}}{\textrm{D} t}$ is the rate of change when you are following a bit of ink moving through the fluid, for example. The changes you see can be either due to time alone (for example, the fluid is getting faster as a whole) or due to being on a different place (for example, you are getting close to a waterfall, where the fluid flows faster). Your goal on the first $=$ sign is to relate these two views.

As for the second equation, I suggest just opening up all of the expressions. There is no secret in doing it, it is just a bit lengthy.

As for references, most books on Fluid Mechanics and Electrodynamics cover the basics of these expressions (Electrodynamics usually won't use the first equality though). I recommend Paterson's A First Course in Fluid Dynamics and Chapter 1 of Griffiths' Introduction to Electrodynamics. It's been a while since I read them, but they should help you out.

As for nomenclature, $\frac{\textrm{D}}{\textrm{D} t}$ is often called material derivative or total derivative or a bunch of other names.

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Think of $v\cdot \nabla$ as $v_{x}\partial_{x} + v_{y}\partial_{y} + v_{z}\partial_{z}$

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