On page 213, from book nonlinear partial differential equations by Lokenath Debnath, 3rd edition. I do not understand one step in the solution. Book shows how to solve the pde $(y-z)u_x+(z-x)u_y+(x-y)u_z=0$.
I follow everything, up to the step they say that $dx+dy+dz=0$.
I do not understand, why, given that $du=0$, then $dx+dy+dz=0$?
I also do not see where $x dx+y dy+ z dz=0$ came from.
Here is screen shot of the solution. It is not long.
Update
Thanks to answer below. But I still do not see how $x dx+y dy+ z dz=0$ came about. The book seems to use some other techniques to manipulate these characteristic equations without giving an explanation. For example on page 214 book gives
The answer below shows how book obtained relation of the form $dx+dy=0$, but I do not see how the other forms shown above can be obtained as well. Book must be doing some algebra, which it does not explain.
For a characteristic curve parametrized as $(x,y,z) = (X(t), Y(t), Z(t))$, we have
$$\frac{dx}{dt} = X'(t) = Y(t) - Z(t), \\ \frac{dy}{dt} =Y'(t)= Z(t) - X(t),\\ \frac{dz}{dt} =Z'(t) = X(t)-Y(t)$$
Hence, on a characteristic,
$$\frac{dx}{dt} +\frac{dy}{dt}+ \frac{dz}{dt} = 0$$
Similarly,
$$\begin{align}x\frac{dx}{dt} +y\frac{dy}{dt}+ z\frac{dz}{dt} &= x(y-z)+y(z-x)+z(x-y)\\ &= xy -xz + yz - xy +xz - yz \\ &= 0\end{align}$$