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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

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# I Vector Laplacian: different results in different coordinates

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