Exercise 3

Show that ln r and 1/Sqrt[r^2+z^2] satisfy Laplace's equation in cylindrical coordinates.


  Clear[x,y,z,r,theta,z]


  x[r_,t_,z_]:=r Cos[t]
  y[r_,t_,z_]:=r Sin[t]
  z[r_,t_,z_]:=z


  x[r,theta,z]
  y[r,theta,z]
  z[r,theta,z]


  Clear[a,b,c]
  {a,b,c}=Simplify[scalefactors[x,y,z][r,theta,z]]


  Clear[u1,u2]
  u1[r_,t_,z_]:=Log[r]
  u2[r_,t_,z_]:=1/Sqrt[r^2+z^2]


  newLaplacian[u1][r,theta,z]


  Simplify[%]


  newLaplacian[u2][r,theta,z]


  Simplify[%]