Both Lagrangian Method and Eulerian Method determine the motion of fluid particles. To get the relationship between Lagrangian equation and Eulerian equation, we consider the equations-
S ⃗ = S (S ⃗0 , t) (From Lagrangian Method)
V ⃗ = V (S ⃗, t) (From Eulerian Method)
If we integrate the equations, we get the scalar components of x, y and z coordinates as observed for Lagrangian method. It can be written as-
x = x (x0, y0, z0, k)
y = y (x0, y0, z0, k) (For y-direction)
z = z (x0, y0, z0, k) (For z-direction)
This is only the scalar components of directions.
If we consider deriving the equation to get simultaneous differential equation, it is not easy to reach to the exact point. This is the reason Eulerian method is primarily preferred for the calculation of different factors for the motion of fluid particles.