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Laplacian

Laplacian[f]
gives the Laplacian, , of the scalar function or vector field f in the default coordinate system.

gives the Laplacian of f in the coordinate system coordsys.
Laplacian of a function in Cartesian coordinates:
Verify that a function is harmonic:
Expression for the Laplacian in cylindrical coordinates:
Needs["VectorAnalysis`"]
Laplacian of a function in Cartesian coordinates:
In[2]:=
Click for copyable input
Out[2]=
 
Needs["VectorAnalysis`"]
Verify that a function is harmonic:
In[2]:=
Click for copyable input
Out[2]=
 
Needs["VectorAnalysis`"]
Expression for the Laplacian in cylindrical coordinates:
In[2]:=
Click for copyable input
Out[2]=