I solved an exercise and I need to confront some other point of view. Given this equation
$$u_x + u_y + u_z=0$$
I solved by using characteristics, obtaining
$$\frac{dx}{1}=\frac{dy}{1}=\frac{dz}{1}$$
i.e. the system
$$ \begin{cases} dx=dy\\ dy=dz \end{cases} $$
$$ \begin{cases} x=y+K_1\\ y=z+K_2 \end{cases} $$
$$ \begin{cases} K_1=x-y\\ K_2=y-z \end{cases} $$
concluding as
$$ u(x,y,z)=\Phi(K_1+K_2)=\Phi(\Phi_1(x-y)+\Phi_2(y-z)) $$
with $\Phi, \Phi_1, \Phi_2: \mathbb{R}^3\rightarrow\mathbb{R} $
In order to prove the solutions, I substituted them in the equality.
I computed the partial derivatives
$$u_x =\frac{\partial u}{\partial x}=\Phi'(\Phi_1(x-y)+\Phi_2(y-z))\cdot(\Phi'_1(x-y))$$
$$u_y =\frac{\partial u}{\partial y}=\Phi'(\Phi_1(x-y)+\Phi_2(y-z))\cdot(-\Phi'_1(x-y)+\Phi'_2(y-z))$$
$$u_z =\frac{\partial u}{\partial z}=\Phi'(\Phi_1(x-y)+\Phi_2(y-z))\cdot(-\Phi'_2(y-z))$$
Computing the summations, result is zero, so the solutions seem correct.
Was I right in applying the method? Are the computations reported correct? Is there some theoretical passages I missed? Thanks.