I was told to varify if the following two forms of the navier-stokes equation are equal (which the obviously should)
$$\rho \dfrac{d \vec{v}}{d t} = \rho g-\nabla p+\eta \nabla^2 \vec{v}$$
$$\rho \left(\dfrac{\partial \vec{v}}{\partial t} + (\vec{v}\cdot \nabla)\vec{v}\right)= \rho g-\nabla p+\eta \nabla^2 \vec{v}$$
Therefor I had to verify that: $$\dfrac{\partial \vec{v}}{\partial t} + (\vec{v}\cdot \nabla)\vec{v} = \dfrac{d \vec{v}}{d t}$$ Using that $\vec{v} = (u,v,w)^T$ $$\dfrac{\partial \vec{v}}{\partial t} + (\vec{v}\cdot \nabla)\vec{v} = \dfrac{\partial \vec{v}}{\partial t} +\dfrac{\partial \vec{v}}{\partial x} u +\dfrac{\partial \vec{v}}{\partial y} v+\dfrac{\partial \vec{v}}{\partial z} w$$ $$\Rightarrow(\vec{v}\cdot \nabla)\vec{v} = \dfrac{\partial \vec{v}}{\partial x} u +\dfrac{\partial \vec{v}}{\partial y} v+\dfrac{\partial \vec{v}}{\partial z} w$$
Using $\Rightarrow(\vec{v}\cdot \nabla)\vec{v} = \begin{bmatrix}\dfrac{\partial v}{\partial x} & \dfrac{\partial u}{\partial y} & \dfrac{\partial w}{\partial z} \end{bmatrix} \cdot \begin{bmatrix}u \\ v\\w \end{bmatrix}$:
$$\begin{bmatrix}u \left(\dfrac{\partial v}{\partial x} + \dfrac{\partial u}{\partial y} + \dfrac{\partial w}{\partial z} \right)\\ v \left(\dfrac{\partial v}{\partial x} + \dfrac{\partial u}{\partial y} + \dfrac{\partial w}{\partial z} \right)\\w \left(\dfrac{\partial v}{\partial x} + \dfrac{\partial u}{\partial y} + \dfrac{\partial w}{\partial z} \right)\end{bmatrix} = \begin{bmatrix} \dfrac{\partial u}{\partial x} u+ \dfrac{\partial u}{\partial y} v+ \dfrac{\partial u}{\partial z} w\\ \dfrac{\partial v}{\partial x} u+ \dfrac{\partial v}{\partial y} v+ \dfrac{\partial v}{\partial z} w\\ \dfrac{\partial w}{\partial x} u+ \dfrac{\partial w}{\partial y} v+ \dfrac{\partial w}{\partial z} w \end{bmatrix}$$
And I do not see how these two should be equal as long as i am not allowed to change whatever is inside the derivative.
I am happy for any help!