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## Homework Statement

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**

## Homework Equations

Pathline is solution to

d

**x**/dt =

**u**

x(t)|t=τ =

x

**X**

Lis the velocity gradient and is a 2nd order tensor L

L

_{ij}= du

_{i}/dx

_{j}

## 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?

Thanks