kill(all)$

Elektrochemistry :appendix :Δψ (spherical coordinates Δψ(r),version 1.0)
peter.vlasschaert@gmail.com,25/02/2019

see  website :https://petervlaschemeng.weebly.com/
                          topic :see : SPECIAL TOPICS  
                          =>    PDE  :    numerical
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new: 21/12/2017: Laplace equation 3D ( = Δu)
                                 (spherical coordinate system: rho=r,theta=th,phi=ph),analyse
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p1:Δu =(('diff(u,ρ,2))*ρ^2+2*('diff(u,ρ,1))*ρ)/ρ^2+(('diff(u,θ,2))*sin(θ)+('diff(u,θ,1))*cos(θ))/(sin(θ)*ρ^2)+'diff(u,φ,2)/(sin(θ)^2*ρ^2);

replace : ρ  by  r
                  u by  ψ
                Δu by  Δψ

p2:[ρ = r,u = ψ,Δu = Δψ];
p2a:subst(p2,p1);

depends(ψ,[r,θ,φ]);

part : partial ψ to  r
================

p3:Δψ(r) = part(rhs(p2a),3);

p3a:lhs(p3) = 1/r^2*'diff(r^2*'diff(ψ,r),r);

simplification : 'nouns' : more general than 'diff'

p3b1:lhs(p3a) = ev(rhs(p3a),nouns);

p3b2:lhs(p3a) = ev(rhs(p3a),diff);

check : relation between   'p3'  ? 'p3a'
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is(rhs(p3) = rhs(p3b2));

part : partial ψ to  θ
================

p4:Δψ(θ) =  part(rhs(p2a),2);

p4a:lhs(p4) = 1/(r^2*(sin(θ)))*'diff(sin(θ)*'diff(ψ,θ),θ);

simplification : 'nouns' : more general than 'diff'

p4b:lhs(p4) = ev(rhs(p4a),diff);

check : relation between   'p4'  ? 'p4a'
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is(rhs(p4) = rhs(p4b));

part : partial φ to  θ
================

p5:Δψ(φ) =  part(rhs(p2a),1);

p5a:lhs(p5) = ev(rhs(p5),diff);

like you  find formula ( =  in a book  about vector analysis)
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p6:Δψ =subst([p3a,p4a,p5a],Δψ(r) + Δψ(θ) +Δψ(φ));

electrochemistry (when  'diff(ψ,θ,1) = 0,   'diff(ψ,φ,1) = 0 ) depends only on  'r'
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p7:[diff(ψ,θ,1) = 0,diff(ψ,φ,2) = 0 ];

rem : diff(ψ,φ,2)=diff(diff(ψ,φ),φ)

Δψ = Δψ(r)  ?
=========

p7a:subst(p7,p6);

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