kill(all)$

maxima :intro how construct pde from geometrical shape (p,q,r,s,t),version1,part 3.
peter.vlasschaert@gmail.com,27/04/2021

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.
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load packages,option
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load(lrats)$  ratprint :false$

1e) given : x^2+y^2 =1
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simple circle @ origin, radius =1

p1:x^2+y^2=1;

rem : this not function of z ,because definition 2e) see below
reason : only two dimension, z : thrid dimension

→  z=f(x,y)

p1a:z=lhs(p1)-rhs(p1);

→ dependencies variable.

depends(z,[x,y]);

2e) partial differentials
--------------------------------

p2:p = diff(z,x);
p2a:q = diff(z,y);
p2b:r = diff(z,x,2);
p2c:s = diff(z,x,1,y,1);
p2d:t = diff(z,y,2);

rem:p2c,about s =definition, (s1,s2)

p3a:s1=diff(z,x,1,y,1);
p3b:s2=diff(z,y,1,x,1);

s1 :first partial differentation to x and then to y
s2:first partial differentation to y and then to x

→ s1

first with respect x,differentation

p3a1:diff(lhs(p1),x);

second with respect y,differentation

p3a2:diff(p3a1,y);

→ s2

first with respect y,differentation

p3b1:diff(lhs(p1),y);

second with respect x,differentation

p3b2:diff(p3b1,x);

reason only one definition :s (definition)

p3c:is(p3a2=p3b2);

other no problem other definition,example

p4a:subst([p1a],p2);
p4a1:ev(p4a,diff);

p4b:subst([p1a],p2a);
p4b1:ev(p4b,diff);

p4c:subst([p1a],p2b);
p4c1:ev(p4c,diff);

p4d:subst([p1a],p2c);
p4d1:ev(p4d,diff);

p4e:subst([p1a],p2d);
p4e1:ev(p4e,diff);

rem :symmetric properties because of x →y, y→x
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→ p and q
→ r = t
→ s1 = s2 = s, 'general'

general rules :

formation of pde
=============
1e)  by elminating arbitrary constants.(part 1,2,see :above).
2e)  by eliminating arbitrary functions

→ z =f(x,y)   result :{p,q,r,s,t}
first rule      :  '≤'
=========
number of arbitrary constants  ≤ number of independent variable use (p,q)
second rule :'>'
==========
number of arbitrary constants  > number of independent variable use (p,q,r,s,t)

rem : see part 1,2
===============
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arbitrary constants    :{a,b,c,r}
depent variable          : z                   ,depends(z,[x,y]) 'see above'
independent variable :{x,y}
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rem :part 4:form pde by eliminating the arbitrary functions.