\( \DeclareMathOperator{\abs}{abs} \)
(%i1) load("cliffordan")$
\[\mbox{}\\package\,name:\,clifford.mac\mbox{}\\author:\,Dimiter\,Prodanov\mbox{}\\version:\mathit{v20}\mbox{}\\Recommended\,location:\,share/contrib\mbox{}\\last\,update:\,20\,June\,2016\mbox{}\\package\,name:\,cliffordan.mac\mbox{}\\author:\,Dimiter\,Prodanov\mbox{}\\version:\mathit{v16}\mbox{}\\Recommended\,location:\,share/contrib\mbox{}\\last\,update:\,08\,March\,2016\]

Euclidean 3D space

(%i2) clifford(e,3);
\[\mathrm{\tt (\%o2) }\quad [1,1,1]\]
(%i5) r:cvect([x,y,z]);
  
\[\mathrm{\tt (\%o5) }\quad {{e}_{1}}\cdot x+{{e}_{2}}\cdot y+{{e}_{3}}\cdot z\]

Green function

(%i11) G:r/sqrt(-cnorm(r))^3/(4*%pi);
\[\mathrm{\tt (\%o11) }\quad \frac{{{e}_{1}}\cdot x+{{e}_{2}}\cdot y+{{e}_{3}}\cdot z}{4\cdot \pi \cdot {{\left( {{x}^{2}}+{{y}^{2}}+{{z}^{2}}\right) }^{\frac{3}{2}}}}\]
(%i19)    mvectdiff(G,r)=0;
\[\mathrm{\tt (\%o19) }\quad 0=0\]

Potential

(%i10) P:-1/sqrt(-cnorm(r))/(4*%pi);
\[\mathrm{\tt (\%o10) }\quad -\frac{1}{4\cdot \pi \cdot \sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}}\]
(%i13) mvectdiff(P,r)=G;
\[\mathrm{\tt (\%o13) }\quad \frac{{{e}_{1}}\cdot x+{{e}_{2}}\cdot y+{{e}_{3}}\cdot z}{4\cdot \pi \cdot {{\left( {{x}^{2}}+{{y}^{2}}+{{z}^{2}}\right) }^{\frac{3}{2}}}}=\frac{{{e}_{1}}\cdot x+{{e}_{2}}\cdot y+{{e}_{3}}\cdot z}{4\cdot \pi \cdot {{\left( {{x}^{2}}+{{y}^{2}}+{{z}^{2}}\right) }^{\frac{3}{2}}}}\]

Homogeneous Poisson equation

(%i20) dependsv(F,[x,y,z])$
(%i17) mvectdiff(F,r,2)=0;
\[\mathrm{\tt (\%o17) }\quad \frac{{{d}^{2}}}{d\,{{x}^{2}}}\cdot F+\frac{{{d}^{2}}}{d\,{{y}^{2}}}\cdot F+\frac{{{d}^{2}}}{d\,{{z}^{2}}}\cdot F=0\]

P solves the equation

(%i18) mvectdiff(P,r,2)=0;
\[\mathrm{\tt (\%o18) }\quad 0=0\]

Define cyclindrical coordinates

(%i21) declare( [rho, phi], scalar)$
(%i24) cyl_eq:[x=rho*cos(phi), y=rho*sin(phi)];
\[\mathrm{\tt (\%o24) }\quad [x=\mathrm{cos}\left( \phi\right) \cdot \rho,y=\mathrm{sin}\left( \phi\right) \cdot \rho]\]
(%i25) r_c:coordsubst(r, cyl_eq);
\[\mathrm{\tt (\%o25) }\quad \left( {{e}_{1}}\cdot \mathrm{cos}\left( \phi\right) +{{e}_{2}}\cdot \mathrm{sin}\left( \phi\right) \right) \cdot \rho+{{e}_{3}}\cdot z\]

Green function in cylindrical coordinates

(%i26) GG_c:coordsubst(G, cyl_eq),factor;
\[\mathrm{\tt (\%o26) }\quad \frac{{{e}_{1}}\cdot \mathrm{cos}\left( \phi\right) \cdot \rho+{{e}_{2}}\cdot \mathrm{sin}\left( \phi\right) \cdot \rho+{{e}_{3}}\cdot z}{4\cdot \pi \cdot {{\left( {{\rho}^{2}}+{{z}^{2}}\right) }^{\frac{3}{2}}}}\]
(%i28) mvectdiff(GG_c,r_c)=0;
\[\mathrm{\tt (\%o28) }\quad 0=0\]
(%i30) dependsv(F,[x,y,z,rho, phi])$

Homogeneous Poisson equation

(%i31) mvectdiff(F,r_c,2)=0;
\[\mathrm{\tt (\%o31) }\quad \frac{\frac{{{d}^{2}}}{d\,{{\phi}^{2}}}\cdot F+\rho\cdot \left( \frac{d}{d\,\rho}\cdot F\right) +{{\rho}^{2}}\cdot \left( \frac{{{d}^{2}}}{d\,{{\rho}^{2}}}\cdot F\right) +{{\rho}^{2}}\cdot \left( \frac{{{d}^{2}}}{d\,{{z}^{2}}}\cdot F\right) }{{{\rho}^{2}}}=0\]
(%i32) V:coordsubst(P,cyl_eq);
\[\mathrm{\tt (\%o32) }\quad -\frac{1}{4\cdot \pi \cdot \sqrt{{{\rho}^{2}}+{{z}^{2}}}}\]
(%i33) mvectdiff(V,r_c)=GG_c;
\[\mathrm{\tt (\%o33) }\quad \frac{{{e}_{1}}\cdot \mathrm{cos}\left( \phi\right) \cdot \rho+{{e}_{2}}\cdot \mathrm{sin}\left( \phi\right) \cdot \rho+{{e}_{3}}\cdot z}{4\cdot \pi \cdot {{\left( {{\rho}^{2}}+{{z}^{2}}\right) }^{\frac{3}{2}}}}=\frac{{{e}_{1}}\cdot \mathrm{cos}\left( \phi\right) \cdot \rho+{{e}_{2}}\cdot \mathrm{sin}\left( \phi\right) \cdot \rho+{{e}_{3}}\cdot z}{4\cdot \pi \cdot {{\left( {{\rho}^{2}}+{{z}^{2}}\right) }^{\frac{3}{2}}}}\]

V solves the equation

(%i34) mvectdiff(V,r_c,2)=0;
\[\mathrm{\tt (\%o34) }\quad 0=0\]
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