While finding out stream lines in a problem of fluid dynamics, the text-book wrote at one stage something similar to this:
$$\frac{dx}{a}=\frac{dy}{b}=\frac{dz}{0}$$
The next step is to equate two pairs and then integrate: $$\frac{dx}{a}=\frac{dy}{b}$$ $$\frac{dy}{b}=\frac{dz}{0}$$
My question is about the second pair. The book then wrote $dz=0$. I do not understand this step at all. As far as I know, cross multiplication is not just moving the denominator of one side to the numerator of the other side. To write $dz=0$, I need to write $\frac{dy}{b}\times0=\frac{dz}{0}\times0$ and zero divided by zero is not one; it's indeterminate.
So, how can one write $dz=0$ from $\frac{dy}{b}=\frac{dz}{0}$?
Also, what is the term used for equations like this - $\frac{dx}{a}=\frac{dy}{b}=\frac{dz}{0}$ ?
The first equations you write are the Lagrange-Charpit equations for a characteristic curve. As the wikipedia page points out, they are just a shorthand (invariant) expression for $$\frac{dx}{dt} = a, \qquad \frac{dy}{dt} =b, \qquad \frac{dz}{dt}=0$$ with $t$ an arbitrary parameterization of the curve $\gamma: t\mapsto ( x(t), y(t), z(t))$.
The last equation is solved by $z = \text{const.}$ which can be also written as $dz =0$. The derivation is of this fact in your book looks quite informal.