$\frac{\partial\mathbf{u}}{\partial{t}}+(\mathbf{u}.\nabla)\mathbf{u}=-\frac{1}{\rho}\nabla{p}+v\nabla^{2}\mathbf{u}$
I need to write the component form of the Navier-Stokes equation, where
$\mathbf{u(x,}t)=u\mathbf{\mathfrak{i}}+v\mathbf{\mathfrak{j}}+w\mathbf{\mathfrak{k}}$
$\rho$ is the density, $p(\mathbf{x},t)$ is the pressure and $v$ is the kinematic viscosity. I need to write this equation in component form.
I only need to know what to do with $\mathbf{x}$. I do not have any idea how to start it, so I guess I cannot post a nonsense. Please help.
$$ \partial_t \mathbf{u} = \left(\matrix{\partial_x u\\ \partial_y v\\ \partial_z w}\right)\\ \mathbf{u}\cdot \nabla = (u\partial_x +v\partial_y + w\partial_z)\\ $$ to get $(\mathbf{u}\cdot \nabla)\mathbf{u}$ you need to apply the second equation to each component.
To get to the answer you desire you need to re-write everything in terms of a vector and then for the $x$ component you can take the first component of each vector.
You have to take care with the pressure term since this is purely a function of $x$ so $$ \nabla \rho = \left(\matrix{\partial_x \rho(x,t)\\ 0\\ 0}\right) $$