Question

Help with Lagrange (material) and Eulerian (spatial) Coordinates

Answer 1

\[\left[\begin{matrix}\cos \omega t & -\sin \omega t & 0 \\sin \omega \omega t & \cos \omega t & 0\\ 0 & 0 &\sqrt{t ^{2}+1}\end{matrix}\right]\left[\begin{matrix}X _{1} \\X _{2} \\X _{3}\end{matrix}\right]=\left[\begin{matrix}x _{1} \\ x _{2}\\x _{3}\end{matrix}\right]\]
Where X are material coordinates and x are spatial coordinates.

Answer 2

So I have that\[x _{1}(X _{1},X _{2},X _{3},t)=X _{1} \cos \omega t-X _{2} \sin \omega t\]\[x _{2}(X _{1},X _{2},X _{3},t)=X _{1} \sin \omega t+X _{2} \cos \omega t\]\[x _{3}(X _{1},X _{2},X _{3},t)=X _{3} \sqrt{t ^{2}+1}\]

Answer 3

Then the velocity field is:
\[v=x \prime=\left(\begin{matrix}-\omega X _{1} \sin \omega t-\omega X _{2} \cos \omega t \\ \omega X _{1} \cos \omega t-\omega X _{2} \cos \omega t \\X _{3}\frac{ t }{ \sqrt{t ^{2}+1} }\end{matrix}\right)\]

Answer 4

Now I am asked for the divergence of the velocity field.
I don't know how I would do this.
Since what I have is in terms of X's and t. So I can't take partials with respect to x's.

Answer 5

???

Answer 6

What history class r u in

Answer 7

No history. Just math. :)

Answer 8

what do you mean by material coordinates

Answer 9

They are the Lagrangian coordinates.

Answer 10

They are the positions of a particle at time t=0

Answer 11

Figured it out :)