Why are first integrals constant along characteristic curves? My lecturer in a PDE course seems to be using this fact. I found a nice explanation of what a first integral is here: Why are they called "first integrals"?. However, I am unable to connect this with the concept of a characteristic, or understand why first integrals are constant along characteristics. For context, this was used in explaining "alternative method of solution using characteristics", that is constructing a family of characteristic curves using two linearly independent first integrals of a pde.
Maybe this is relevant: What does constant along characteristic mean, but I don't see how.
The first integral of system of ODEs is such a function $f:\mathbb R^n \to \mathbb R$ that is constant along any solution of system, i.e $f(x(t)) = const$.
Intersection of surfaces $\phi_1(x_1,...,x_n)=C_1, ..., \phi_n(x_1,...,x_n)=C_n$ defined by first integrals determines a family of surfaces where solution of system lies
The method of characteristics reduces a problem of solving PDE to a system of ODEs, a general solution is said to be constant along characteristics curves so it's some arbitrary function of first integrals: $$\Phi(\phi_1(x_1,...,x_n),...,\phi_n(x_1,...,x_n)) = 0$$