Exercise 3
Exercise 3
In[69]:=
Clear[x,y,z,r,theta,z]
In[70]:=
x[r_,t_,z_]:=r Cos[t]
y[r_,t_,z_]:=r Sin[t]
z[r_,t_,z_]:=z
In[71]:=
x[r,theta,z]
y[r,theta,z]
z[r,theta,z]
Out[71]=
r Cos[theta]
Out[72]=
r Sin[theta]
Out[73]=
z
In[74]:=
Clear[a,b,c]
{a,b,c}=Simplify[scalefactors[x,y,z][r,theta,z]]
Out[74]=
2
{1, Sqrt[r ], 1}
In[75]:=
Clear[u1,u2]
u1[r_,t_,z_]:=Log[r]
u2[r_,t_,z_]:=1/Sqrt[r^2+z^2]
In[76]:=
newLaplacian[u1][r,theta,z]
Out[76]=
2
1 Sqrt[r ]
-------- - --------
2 2
Sqrt[r ] r
-------------------
2
Sqrt[r ]
In[77]:=
Simplify[%]
Out[77]=
0
In[78]:=
newLaplacian[u2][r,theta,z]
Out[78]=
2 2 2 2 2
3 r Sqrt[r ] 3 Sqrt[r ] z r
(------------- + ------------- - --------------------- -
2 2 5/2 2 2 5/2 2 2 2 3/2
(r + z ) (r + z ) Sqrt[r ] (r + z )
2
2 Sqrt[r ] 2
------------) / Sqrt[r ]
2 2 3/2
(r + z )
In[79]:=
Simplify[%]
Out[79]=
0
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