In my book the following is written:
The change of a scalar field $du$ in an arbitrary direction, given by an infinitesimal vector with an arbitrary direction $d\vec r$ is calculated: $$du=u(\vec r + d\vec r) - u(\vec r)= d\vec r\cdot\nabla u.$$ And for the vector field $\vec v$ we have: $$d\vec v= \vec v(\vec r + d\vec r) - \vec v(\vec r)= (d\vec r \cdot\nabla) \vec v.$$
Can someone explain:
How do we derivate this!
A visualization of both of them! or a link to the asked topic.
It's a direct application of the chain rule for multi-variable calculus. Perhaps using summatory notation clarifies further the point: $$ dv^{i}=v^{i}\left(\boldsymbol{r}+d\boldsymbol{r}\right)-v^{i}\left(\boldsymbol{r}\right)\simeq\sum_{k}dr^{k}\frac{\partial v^{i}}{\partial r^{k}} $$ As to an intuitive picture of what it is, it is how much the i-th component of $\boldsymbol{v}$ varies in the direction of the small displacement vector $d\boldsymbol{r}$, so it's natural to project gradient of $v^i$ along such small vector.