Exercise 2

Show that r^ncos(n theta) and r^n sin(n theta) are solutions to 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[f,g]
  f[r_,t_,z_]:=r^n Cos[n t]
  g[r_,t_,z_]:=r^n Sin[n t]


  f[r,theta,z]
  g[r,theta,z]


  newLaplacian[g][r,theta,z]


  Simplify[%]


  newLaplacian[f][r,theta,z]


  Simplify[%]