kill(all)$

Ejemplo 11.1.1

f1(x,y):=exp(x);        f2(x,y):=exp(x+y);
f(x,y):=[f1(x,y), f2(x,y)];

Eq11:u=f1(x,y);     Eq12:v=f2(x,y);

JF(x,y)=Jfxy: jacobian ( [f1(x,y), f2(x,y)] , [x,y]  );

Dxf1(x,y):=%e^x$          Dyf1(x,y):=0$
Dxf2(x,y):=%e^(x+y)$    Dyf2(x,y):=%e^(x+y)$

Jf(0,0)=Jf00:matrix(  [Dxf1(0,0), Dyf1(0,0)] , [Dxf2(0,0), Dyf2(0,0)]);

'(detJf(0,0))=determinant(Jf00);

Mxy=Mxy:matrix([x],[y]);

'f(0,0)=f00:matrix( [f1(0,0)] , [f2(0,0)]) ;

Muv=Muv:matrix([u],[v]);

Muv = f00 + Jf00.Mxy;

Eq11_lin:u=1+x;
Eq12_lin:v=1+x+y;

'Jf_inv(1,1)=inv_Jf00:invert(Jf00);

'h=h:matrix([u-1],[v-1]);

f_inv(1,1)=finv11: matrix( [0], [0] );

Mxy = finv11 + inv_Jf00.h;

Eq21_lin:x=u-1;
Eq22_lin:y=v-u;

Ejemplo 11.1.2

a) Coordenadas polares planas

kill(all)$

f1(r, theta):=r*cos(theta);        f2(r, theta):=r*sin(theta);
f(r,theta):=[f1(r, theta), f2(r, theta)];

Jf(r, theta)=Jfcpp: jacobian ( [f1(r,theta), f2(r,theta)] , [r,theta]  );

'(detJf(r,theta))=determinant(Jfcpp);

trigsimp(%);

b) Coordenadas cilíndricas

kill(all)$

f1(r, theta, z):=r*cos(theta);  f2(r, theta, z):=r*sin(theta);   f3(r, theta, z):=z;                 
f(r,theta,z):=[f1(r, theta), f2(r, theta) , f3(r, theta)];

Jf(r, theta,z)=Jfcc: jacobian 
( [f1(r, theta,z), f2(r, theta,z) , f3(r, theta,z)] , [r,theta,z]  );

'(detJf(r,theta))=determinant(Jfcc);

trigsimp(%);

c) Coordenadas esféricas

kill(all)$

f1(r,phi,theta):=r*cos(phi)*cos(theta);  
f2(r,phi,theta):=r*cos(phi)*sin(theta);   
f3(r,phi,theta):=r*sin(phi);                 
f(r,phi,theta):=[f1(r,phi,theta), f2(r,phi,theta) , f3(r,phi,theta)];

Jf(r,phi,theta)=Jfce: jacobian 
( [f1(r,phi,theta), f2(r,phi,theta) , f3(r,phi,theta)] , [r,phi,theta]  );

'(detJf(r,theta))=determinant(Jfce);

trigsimp(%);

Ejemplo 11.1.3

kill(all)$

f(x,y):=x^2 + 3*y^2;

x=x:r*cos(theta)$   y=y:r*sin(theta)$
F(r,theta)=f(x,y);

trigsimp(%);

factor(%);

Ejemplo 11.1.4

kill(all)$

r(x,y):=+sqrt(x^2 + y^2);       th(x,y):=atan(y/x);

diff(r(x,y), x);      diff(r(x,y), y);

Dxr(x,y):=x/sqrt(y^2+x^2);     Dyr(x,y):=y/sqrt(y^2+x^2);

ratsimp(diff(th(x,y), x));    ratsimp(diff(th(x,y), y));

Dxth(x,y):=-y/(y^2+x^2);        Dyth(x,y):=x/(y^2+x^2);

Dxf=Dxf:DrF(r,theta)*Dxr(x,y) + DthF(r,theta)*Dxth(x,y);

Dyf=Dyf:DrF(r,theta)*Dyr(x,y) + DthF(r,theta)*Dyth(x,y);

T1:(x + 2*y*sqrt(x^2+y^2))*Dxf;

T1:expand(%);

T2:(y - 2*x*sqrt(x^2+y^2))*Dyf;

T2:expand(%);

T1+T2=0;

factor(%);

%*(1/sqrt(x^2+y^2));