# Tema 10
10-1 Funciones de varias variables

kill(all)$

f1(x,y):=2*(x-1)^2+2*(y-1)^2;

'f1(2,3)=f1(2,3);

V(r,h):=%pi*r^2*h;

'V(5,7)=V(5,7);

Funciones reales de varias variables

f2(x,y,z):=(x+y)^2+exp(z);
'f2(1,2,3)=f2(1,2,3);

f3(x,y,u,v):=x^2+x^3*y-2*u^2+exp(v);
'f3(1,-1,1,2)=f3(1,-1,1,2);

Operaciones con funciones reales de varias variables

kill(all)$

f1(x,y):=2*x^2+y^2+1;      f2(x,y):=x^2*sin(x+y);

'(f1(x,y)+f2(x,y))=f1(x,y)+f2(x,y);

'(a*f1(x,y))=a*f1(x,y);

expand(%);

'(f1(x,y)*f2(x,y))=f1(x,y)*f2(x,y);

expand(%);

'(f1(x,y)/f2(x,y))=f1(x,y)/f2(x,y);

Funciones vectoriales de varias variables

kill(all)$

f1(x,y,z):=x^2+y-sin(z);        f2(x,y,z):=z*exp(x+y);
f(x,y,z):=[f1(x,y,z),f2(x,y,z)];

'f(1,2,-1)=f(1,2,-1);

Ejemplo 10.1.4

kill(all)$

f1(x,y,z):=x^2+y-sin(z);        f2(x,y,z):=z*exp(x+y);
f(x,y,z):=[f1(x,y,z),f2(x,y,z)];

g1(x,y,z):=2*x^2+y^2+3*z+1;      g2(x,y,z):=z^2*sin(x+y);
g(x,y,z):=[g1(x,y,z),g2(x,y,z)];

'((f+g)(x,y,z))=f(x,y,z) + g(x,y,z);

'((af)(x,y,z))=a*f(x,y,z);

f(x,y,z).g(x,y,z);

kill(all)$

f1(r,a):=r*cos(theta);        f2(r,a):=r*sin(theta);
g(x,y):=sqrt(R^2 - x^2 - y^2);
'F(r,a)=g(f1(r,theta), f2(r,theta));

trigsimp(%);

kill(all)$

g(x,y,z):=x*y^2+exp(z);
f1(t):=2*t+1;       f2(t):=t^2+2;       f3(t):=sin(t);
f(t):=[f1(t), f2(t), f3(t)];

'(g(f(t)))=g(f1(t),f2(t),f3(t));