kill(all)$

maxima :intro how construct first order pde theory (version1),part 5c
peter.vlasschaert@gmail.com,30/04/2021

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*
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load packages,option
-------------------------------

load(lrats)$  ratprint :false$

depends , see maxima code.

*What meaning : u=u(x,y,z) =c1 ,v= v(x,y,z) = c2 , niveau 2 : Figure 1
{c1,c2} = constants

reason : understanding this    "auxiliary equations"

a),simple surface (si*,definition) = plane equation , i=1
---------------------------------------------------------------------------
representation : u or v

→ first step : interpretation of u,v 
=========================

general representation : plane.
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 def :A*x+B*y+C*z+D = 0

p1a:A*x+B*y+C*z+D = 0;
p1b:p1a-D;  /* -D is c1 or c2*/

→ second step : differentation with respect {x,y,z}
======================================

p1c:lhs(p1a);

p1c1:'diff(p1c,x)=lhs(diff(p1a,x));
p1c2:'diff(p1c,y)=lhs(diff(p1a,y));
p1c3:'diff(p1c,z)=lhs(diff(p1a,z));

→ thrid step:Gradient concept (show:(A,B,C))
====================================
reason :vector = [A,B,C], perpendicular to plane.

Def : Gradient 
--------------------

p1d:Gradient(F):=[diff(F,x),diff(F,y),diff(F,z)];

 => Gradient of the plane

p1d1:Gradient(lhs(p1a));

proof : perpendicularity
--------------------------------

1e)suppose point  (x0;y0,z0) on the plane:'p1a'

p1d2a:p1a;

p1d2b:subst([x=x0,y=y0,z=z0],p1a);

2e) p1d2a-p1d2b

p1d2c:p1d2a-p1d2b;

p1d2c1:lhs(p1d2c);

p1d2c1a:(factorsum(p1d2c1)=rhs(p1d2c))*(-1);

3e) dot product = 0 ,perpendicular both vectors

→ first vector :

p1e1:p1d1;

→ second vector : 'in plane'

p1e2:[ratcoeff(lhs(p1d2c1a),A),ratcoeff(lhs(p1d2c1a),B),ratcoeff(lhs(p1d2c1a),C)];

→ result :first vector perpendicular to second vector.

p1e2:p1e1*p1e2=rhs(p1d2c1a);

b) other surface (si*,definition) = paraboloid equation , i=2
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representation : u or v

p2a:z=x^2+y^2;  /* paraboloid */
p2b:rhs(p2a)-lhs(p2a)=0;

→ first step : computation 'Gradient :see Def above'

p2c1:Gradient(lhs(p2b));

rem : gradient @ (x0,y0,z0)

p2c2:subst([x=x0,y=y0,z=z0],p2c1);

→ second step : tangent plane when (x0,y0,z0) is on 
use :p1d2c1a & p1e1

p2c3:map("=",p1e1,p2c1);

@(x0,y0,z0) on paraboloid

p2c3a:subst([x=x0,y=y0,z=z0],p2c3);

p2c4:expand(subst(p2c3a,p1d2c1a));

paraboloid :A=Ap, B=Bp,C=Cp

p2c4a:Ap=ratcoeff(lhs(p2c4),x);
p2c4b:Bp=ratcoeff(lhs(p2c4),y);
p2c4c:Cp=ratcoeff(lhs(p2c4),z);
p2c4d:Dp=subst([x=0,y=0,z=0],lhs(p2c4));

p2c4:[p2c4a,p2c4b,p2c4c,p2c4d];

p2c5:subst([A=Ap,B=Bp,C=Cp,D=Cp],p1d2c1a);

rem :tangent plane {p2c4 & p2c5}

c) intersection : paraboloid  & plane (z=c1)// xy plane
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p3a:z=c1;  /*s1: u(x,y,z)=c1 */
p3b:p2a;  /*s2: v(x,y,z)=c2 */

s1 intersection s2 = circle

p3c:subst([p3a],p3b);

rem  :general equation : suppose z =c1=1

result : circle @ z=1 , above xy plane.