Fluid flow velocity field is usually described by the following vector field:
$$\mathbf{u}(x(t),y(t),z(t),t)=u_x(x(t),y(t),z(t),t)\mathbf{i}+u_y(x(t),y(t),z(t),t)\mathbf{j}+u_z(x(t),y(t),z(t),t)\mathbf{k}$$
where $x,y,z$ are positions of bundle at time $t$. However I sometimes see that $\mathbf{u}$ is defined by
$$\mathbf{u}(x(t),y(t),z(t),t)=\frac{dx}{dt}\mathbf{i}+\frac{dy}{dt}\mathbf{j}+\frac{dz}{dt}\mathbf{k}$$
So it follows that
$$u_x(x(t),y(t),z(t),t)=\frac{dx}{dt}(x(t),y(t),z(t),t)$$ $$u_y(x(t),y(t),z(t),t)=\frac{dy}{dt}(x(t),y(t),z(t),t)$$ $$u_z(x(t),y(t),z(t),t)=\frac{dz}{dt}(x(t),y(t),z(t),t)$$
Is that right? It is weird to me that $\frac{dx}{dt}$ is a function of $x(t),y(t),z(t),t$. I am struggling to understand which quantities $u_x,u_y,u_z,x,y,z$ depend on.