Show that the path line of a particle at point x currently, and point ξ at time τ is given by
ξ(τ) = x + (τ-t)Lx
Pathline is solution to
dx/dt = u
x(t)|t=τ = X
L is the velocity gradient and is a 2nd order tensor Lij = dui/dxj
The Attempt at a Solution
I am not really sure how to start, any hints or leads on how to begin proving this?
I know how to obtain pathlines if the velocity field is given in terms of x,y,z,t (via integration) but how do I integrate the equations to obtain a general equation with a velocity gradient?