Exercise 4

Find the Laplacian of the following in spherical coordinates:
u=r
u=r^2
u=sin(phi)
u=r^2sin(theta)

Do any solve Laplace's Equation?


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


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


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


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


  Clear[u1,u2,u3,u4]
  u1[r_,t_,p_]:=r
  u2[r_,t_,p_]:=r^2
  u3[r_,t_,p_]:=Sin[p]
  u4[r_,t_,p_]:=r^2 Sin[t]


  u1[r,theta,phi]
  u2[r,theta,phi]
  u3[r,theta,phi]
  u4[r,theta,phi]


  newLaplacian[u1][r,theta,phi]


  Simplify[%]


  newLaplacian[u2][r,theta,phi]


  Simplify[%]


  newLaplacian[u3][r,theta,phi]


  Simplify[%]


  newLaplacian[u4][r,theta,phi]


  Simplify[%]