# Bilineare Abbildung des Master-Quadrates auf das Einheitsdreieck:
> with(linalg):
Warning, the protected names norm and trace have been redefined and
unprotected

> M:=matrix(4,4,[[1,-1,-1,1],[1,1,-1,-1],[1,1,1,1],[1,-1,1,-1]]);
> 

                           [1    -1    -1     1]
                           [                   ]
                           [1     1    -1    -1]
                      M := [                   ]
                           [1     1     1     1]
                           [                   ]
                           [1    -1     1    -1]

> Inverse:=(1/4)*inverse(M/4):
> R[a]:=matrix(1,4,[0,1,1/2,0]);
> 

                     R[a] := [0    1    1/2    0]

> A:=matrix(1,4,[a[1],a[2],a[3],a[4]]);
> 

                 A := [a[1]    a[2]    a[3]    a[4]]

> Loesung[a]:=transpose(linsolve(M,transpose(R[a])));
> 

              Loesung[a] := [3/8    3/8    -1/8    -1/8]

> xi:=a[1]+a[2]*x+a[3]*y+a[4]*x*y;
> 

               xi := a[1] + a[2] x + a[3] y + a[4] x y

> xi:=factor(subs({a[1]=3/8,a[2]=3/8,a[3]=-1/8,a[4]=-1/8},%));
> 

                               (-3 + y) (1 + x)
                       xi := - ----------------
                                      8

> R[b]:=matrix(1,4,[0,0,1/2,1]);
> 

                     R[b] := [0    0    1/2    1]

> B:=matrix(1,4,[b[1],b[2],b[3],b[4]]);
> 

                 B := [b[1]    b[2]    b[3]    b[4]]

> Loesung[b]:=transpose(linsolve(M,transpose(R[b])));
> 

              Loesung[b] := [3/8    -1/8    3/8    -1/8]

> eta:=b[1]+b[2]*x+b[3]*y+b[4]*x*y;
> 

               eta := b[1] + b[2] x + b[3] y + b[4] x y

> eta:=factor(subs({b[1]=3/8,b[2]=-1/8,b[3]=3/8,b[4]=-1/8},%));
> 

                               (1 + y) (-3 + x)
                      eta := - ----------------
                                      8

> JACOBI_Matrix:=matrix(2,2,[[diff(xi,x),diff(eta,x)],[diff(xi,y),diff(e
> ta,y)]]):
> J:=JACOBI_Determinante=det(%);
> 

                                               x      y
             J := JACOBI_Determinante = 1/8 - ---- - ----
                                               16     16

