kill(all)$

maxima : math :surface area of a plane,version 1.0
peter.vlasschaert@gmail.com,12/09/2021

ref :schaum' s outline series:
      theory and problems of vector analysis
      Murray R.Spiegel

area plane into first octant in 3d dimension
==================================

draw  the plane (trough A,B,C , 'three points')

ordergreat(i,j,k,x,y,z);

graphics data : plane 
===========

p1:pt=[x,y,z];

→ three points : needed to form a plane (see :figure 1)

p1a:A=[1,0,0];/*along x-axis= intersect plane 'x-axis'*/
p1b:B=[0,0,1];/*along z-axis= intersect plane 'z-axis'*/
p1c:C=[0,1,0];/*along y-axis= intersect plane 'y-axis'*/

definition :vector 
=============

vec(l1,l2):=rhs(l2)-rhs(l1);

→ a=vec(BA)

p2a:vec(p1b,p1a);

→ c=vec(BC)

p2b:vec(p1b,p1c);

AA:genmatrix(zz, 3, 3);

unit(vector) 'cartesian coordinates
-----------------------------------------------

uu:[i,j,k];

un:makelist(AA[1,ii]:uu[ii],ii,1,length(uu));

→  vetor (a),vector(c)

p3a:makelist(AA[2,ii]:p2a[ii],ii,1,length(uu));
p3b:makelist(AA[3,ii]:p2b[ii],ii,1,length(uu));

 AA;

→ vector product(a,b)

p3c:BB=determinant(AA);

rem :values before [i,j,k]

p3c1:coeff(rhs(p3c),uu[1]);
p3c2:coeff(rhs(p3c),uu[2]);
p3c3:coeff(rhs(p3c),uu[3]);

p3c4:[p3c1,p3c2,p3c3];

definition :norm (square norm)
=========================

norm(l1,l2,l3):=sqrt(l1^2+l2^2+l3^3);

p4a:norm(p3c1,p3c2,p3c3);

unit vector to the plane:
--------------------------------

p4b:[p3c1,p3c2,p3c3]/p4a;

find plane through 3points( A,B,C)
-----------------------------------------------

general equation of the plane. (p4c)
========================

p4c:a1*x+a2*y+a3*z +a4 = 0;

p4c1:makelist(concat(a,i)=p3c4[i],i,1,length(uu));

p4c2:subst(p4c1,p4c);

to find ;a4 to put point on the plane

→ select point A figure :[]

rhs(p1)[1]=(p3c4)[1];

p4c3:makelist(rhs(p1)[ii]=rhs(p1a)[ii],ii,1,3);

p4c4:subst(p4c3,p4c2);

p4c5:solve(p4c4,a4);

final :result:plane see figure 1
========================

p4c6:p4c2-a4;

p4c7:subst(p4c5,p4c6);

plane ( xOy)

plane x0y  (use :p4c7,above)
========
unit vector =[i,j]

p5:subst([z=0],p4c7);

p5a:solve(p5,y)[1];

computation : area (above: xy plane)
=============================

depends(z,[x,y])$

for every point (x0y plane) exit point on the  surface(=plane)'z=f(x,y)'
use :p4c7

p6:solve(p4c7,z);

p6a:[rhs(p1)[1],rhs(p1)[2],p6[1]];

method 1: area (A,B,C),see figure 1
============================

a) differentation z with respect to(x,y)
=============================

rx:'diff(p6a,x)=diff(p6a,x);
rx1:[1,0,rhs(rhs(rx)[3])];
ry:'diff(p6a,y)=diff(p6a,y);
ry1:[0,1,rhs(rhs(ry)[3])];

b) determinant (cross product:(rx1,ry1))
============

p7:matrix(un,rx1,ry1);

p7a:determinant(p7);

see: 'above :(p3c4)

p7b:p3c4;

see: norm 'definition':see p4a

p7c:p4a;

c) integration double integral 
======================

p8a:xrange=[0,1];
p8b:yrange=[0,p5a];

p8c:area(A,B,C) = integrate(integrate(p7c,y,rhs(p8b)[1],rhs(p5a)),x,rhs(p8a)[1],rhs(p8a)[2]);

method 2: area (A,B,C),see figure 1
============================

a),surface above :x0y plane: angle(normal to the plane ,k)
------------------------------------------------------------------------------

p9a:p4b;

k :thrid component

p9b:[0,0,1];

dot(product)

p9c:p9a*p9b;

p9c1:p9c[3];

reverse :

p9c2:1/p9c1;

b) integration:p8c (change : p7c to p9c2);
-------------------------------------------------------

p10a:area(A,B,C) = integrate(integrate(qq,y,rhs(p8b)[1],rhs(p5a)),x,rhs(p8a)[1],rhs(p8a)[2]);

p10b:subst([qq=p9c2],p10a);