- #1
maajdl
Gold Member
- 391
- 29
Hello,
I calculated the Vector Laplacian of a uniform vector field in Cartesian and in Cylindrical coordinates.
I found different results.
I can't see why.
In Cartesian coordinates the vector field is: (vx,vy,vz)=(1,0,0).
Its Laplacian is: (0,0,0) .
That's the result I expected.
In Cylindral coordinates the same vector field becomes: (vr,vt,vz)=(cos(t),sin(t),0).
I found its Laplacian to be: (-4cos(t)/r²,-4sin(t)/r²,0) .
I used Mathematica to calculate this, using the definition for 3D space:
I expected the result would not depend on the choice of the coordinate system.
I also expected the result would be (0,0,0) in any coordinate system.
My motivation was to understand the meaning of a v/r² term appearing in the Laplacian in cylindrical coordinates.
I hoped that probing with a uniform field would help to reveal the meaning.
Would you have a clue?
Thanks,
Michel
I calculated the Vector Laplacian of a uniform vector field in Cartesian and in Cylindrical coordinates.
I found different results.
I can't see why.
In Cartesian coordinates the vector field is: (vx,vy,vz)=(1,0,0).
Its Laplacian is: (0,0,0) .
That's the result I expected.
In Cylindral coordinates the same vector field becomes: (vr,vt,vz)=(cos(t),sin(t),0).
I found its Laplacian to be: (-4cos(t)/r²,-4sin(t)/r²,0) .
I used Mathematica to calculate this, using the definition for 3D space:
I expected the result would not depend on the choice of the coordinate system.
I also expected the result would be (0,0,0) in any coordinate system.
My motivation was to understand the meaning of a v/r² term appearing in the Laplacian in cylindrical coordinates.
I hoped that probing with a uniform field would help to reveal the meaning.
Would you have a clue?
Thanks,
Michel