Given a vector field $\vec{E}:\Bbb R^3\rightarrow \Bbb R^3$ (all kind of regularity are assumed), my book claims that the equation for field lines is $$\frac{dx}{E_x(x,y,z)}=\frac{dy}{E_y(x,y,z)}=\frac{dz}{E_z(x,y,z)}\quad\quad\quad(1)$$
I'm struggling to get this result from the general definition of field lines, i.e. a line in $\Bbb R^3$ which is tangent to the vector field:
$$\frac{d\vec{(x,y,z)}(t)}{dt}=\vec{E}(x(t),y(t),z(t)) \quad\quad\quad(2)$$
In vector components $$\frac{dx}{dt}=E_x(x,y,z), \frac{dy}{dt}=E_y(x,y,z), \frac{dz}{dt}=E_z(x,y,z) $$
I'm stuck here, dealing with this differential equation. Using some (unrigousus, sorry I study physics :) ) algebraic manipulation I get the following: $$\frac{dx}{E_x(x,y,z)}=dt, \frac{dy}{E_y(x,y,z)}=dt, \frac{dz}{E_z(x,y,z)}=dt $$ thus from this sistem of equations you get (1).
Can anyone show me how to get (1) (which I assume is a sistem of differential form equations) from (2) in a (at least more) rigorous way?