kill(all)$

maxima :intro how construct first order pde origin (version1),part 5d
peter.vlasschaert@gmail.com,01/05/2021

ref :https://petervlaschemeng.weebly.com/
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maxima :intro how construct pde from geometrical shape (p,q),version1,part 1.
maxima :how construct pde(p,q) :eliminating arbitrary (a,b),version1,part 2.
maxima :intro how construct pde from geometrical shape (p,q,r,s,t),version1,part 3*.
maxima :intro how construct pde from arbitrary functions version1,part 4.
maxima :intro how construct pde complex theory version1,part 5a.{p,q,r,s,t}*
maxima :intro how construct first order pde theory (version1),part 5b
maxima :intro how construct first order pde theory (version1),part 5c
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load packages,option
-------------------------------

load(lrats)$  ratprint :false$

What origin :linear pde 'Lagrange'
==========================

first step : definition :{p,q,r,s,t}*
-----------------------------------------

p1:p = 'diff(z,x);
p1a:q = 'diff(z,y);
p1b:r = 'diff(z,x,2);
p1c:s = 'diff(z,x,1,y,1);
p1d:t = 'diff(z,y,2);

p2:[p1,p1a,p1b,p1c,p1d];
p2a:makelist(rhs(p2[i])=lhs(p2[i]),i,1,length(p2));

second step : sphere ,with center (a11,b11,c11)=constants
-----------------------------------------------------------------

1e) suppose unit sphere  : F(x,y,z,a11,b11,c11) = 0

general equation : unit sphere

p3:(x-a11)^2+(y-b11)^2+(z-c11)^2=1; /*unit sphere :r^2=1*/

special example :unit  sphere with (a11=0,b11=0)

p3a:subst([a11=0,b11=0],p3);

2e)  differentation p3 respect {x,y}

 depends(z,[x,y]);

p3b1:diff(p3a,x);
p3b2:diff(p3a,y);

3e)  insert definition:{p,q,r,s,t},see above

p3c1:subst(p2a,p3b1);
p3c2:subst(p2a,p3b2);

third step : elimination c11
-------------------------------------

rem : How to use maxima command :eliminate.

p3d:eliminate([p3c1,p3c2],[c11]);

p3d1:p3d[1]=0;

p3d2a:part(p3d1,1,1);
p3d2b:part(p3d1,1,2);

final result :
========

p3e:p3d1/p3d2a;

rem : G(x;y,p,q)=0

remenber : P*p +Q*q = R, proof given before.

p3e1:P*p+Q*q = R;

?  P,Q,R, 

→ratcoeff

p3e2a:P = ratcoeff(lhs(p3e),p);
p3e2b:Q = ratcoeff(lhs(p3e),q);
p3e2c:R = rhs(p3e);

→ differentation with respect to {P,Q,R}

p3f2a:P = diff(lhs(p3e),p);
p3f2b:Q = diff(lhs(p3e),q);
p3f2c:R = diff(rhs(p3e),R);

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rem :F(x,y,z,a11,b11,c11) = 0
=======================

p4:F(x,y,z,a11,b11,c11) = 0;

depends(F,[x,y,z,a11,b11,c11]);

differentation : F with respect  'x'

p4a1:diff(F,x)=rhs(p4);
p4a2:subst(p2a,p4a1);/*use definition */

→ coef :p,see other method above

diff(lhs(p4a2),p);

differentation : F with respect  'y'

p4b1:diff(F,y)=rhs(p4);
p4b2:subst(p2a,p4b1);/*use definition */

→ coef q,see other method above

diff(lhs(p4b2),q);

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