If we have a function f(x,y,z) and have this equation
$af_{x}+bf_{y}+cf_{z} = 0$
I have 2 questions :How do we transform the variables x,y,z to ξ,n,k to solve the differential equation?
I mean $1)\frac{dx}{ds} = a , 2)\frac{dy}{ds} = b , 3)\frac{dz}{ds} = c$ if we had 2 variables we would divide $\frac{1)}{2)}$ to find a relationship between $dx$ and $dy$ in order to find the characteristic equation, now what do we do ?
When we find the Jacobian to see if the transformation from x,y,z to ξ,n,k is valid in case of my example now do we need to find the determinant of a 3x3 matrix?