Atividades Tutoriais de Cálculo

Volume 2 -- Várias Variáveis

Gabaritos

1)
[> I1:=Doubleint('Fpv(x,y)',x,y,D)=changevar({x=r*cos(theta),
                y=r*sin(theta)},Doubleint(Fpv(x,y),x,y,D1),[r,theta]);

2)
[> I2:=rhs(I1)=Doubleint(integrand(rhs(I1)),r=0..2,theta=0..Pi/2);

3)
[> value(rhs(I2));

4)
[> npv:=(x,y)->norm(pv(x,y),2); npv(x,y);

5)
[> Doubleint('npv(x,y)',x,y,D)=value(Doubleint(npv(x,y),
            y=0..sqrt(4-x^2),x=0..2));

6)
[> assume(theta>0);additionally(theta<Pi/2);
  I3:=Doubleint('npv(x,y)',x,y,D)
     =changevar({x=r*cos(theta),y=r*sin(theta)},
                Doubleint(npv(x,y),x,y,D1),[r,theta]);

[ > I4:=rhs(I3)=Doubleint(simplify(integrand(rhs(I3))),r=0..2,theta=0..Pi/2);

[ > value(rhs(I4));

7)
[> I5:=Tripleint(1,x,y,z,Q)=changevar({x=x,y=r*cos(t),z=r*sin(t)},
                Tripleint(1,x,y,z,Q1),[x,r,t]);

[ > I6:=rhs(I5)=Tripleint(integrand(rhs(I5)),x=0..2,r=0..1,t=0..2*Pi);
   value(rhs(I6));

8)
[> q1:=(x,t)->(x,cos(t),sin(t));

9)
[> q1_x:=unapply(diff([q1(x,t)],x),(x,t));
  q1_t:=unapply(diff([q1(x,t)],t),(x,t));

10)
[> q1v:=(x,t)->crossprod(q1_t(x,t),q1_x(x,t)); q1v(x,t);

11)
[> Fq1v:=(x,t)->innerprod(F(q1(x,t)),q1v(x,t)); Fq1v(x,t);

12)
[> Doubleint('Fq1v(x,t)',x,t,D1)=
  value(Doubleint(Fq1v(x,t),x=0..2,t=0..2*Pi));

13)
[> q2:=(r,t)->(2,r*cos(t),r*sin(t));
  q2_r:=unapply(diff([q2(r,t)],r),(r,t));
  q2_t:=unapply(diff([q2(r,t)],t),(r,t));
  q2v:=(r,t)->simplify(crossprod(q2_r(r,t),q2_t(r,t))); q2v(r,t);
  Fq2v:=(r,t)->innerprod(F(q2(r,t)),q2v(r,t)); Fq2v(r,t);
  Doubleint('Fq2v(r,t)',r,t,D2)=value(Doubleint(Fq2v(r,t),
            r=0..1,t=0..2*Pi));

14)
[> q3:=(r,t)->(0,r*cos(t),r*sin(t));
  q3_r:=unapply(diff([q3(r,t)],r),(r,t));
  q3_t:=unapply(diff([q3(r,t)],t),(r,t));
  q3v:=(r,t)->simplify(crossprod(q3_t(r,t),q3_r(r,t))); q3v(r,t);
  Fq3v:=(r,t)->innerprod(F(q3(r,t)),q3v(r,t)); Fq3v(r,t);
  Doubleint('Fq3v(r,t)',r,t,D2)=value(Doubleint(Fq3v(r,t),
            r=0..1,t=0..2*Pi));

15)
[> p:=(r,t)->(r*cos(t),r*sin(t),1);

16)
[> S:=plot3d([p(r,t)],r=0..1,t=0..2*Pi):
  display(S,axes=normal,orientation=[20,60]);

17)
[> p_r:=unapply(diff([p(r,t)],r),(r,t));
  p_t:=unapply(diff([p(r,t)],t),(r,t));
  pv:=(r,t)->simplify(crossprod(p_r(r,t),p_t(r,t))); pv(r,t);

18)
[> curl(F(x,y,z),[x,y,z]);

[ > rotF:=(x,y,z)->[3*y^2, 3, 2*x]; rotF(p(r,t));

19)
[> rotFpv:=(r,t)->innerprod(rotF(p(r,t)),pv(r,t)); rotFpv(r,t);

20)
[> Doubleint(rotFpv(r,t),r,t,D)=
  value(Doubleint(rotFpv(r,t),r=0..1,t=0..2*Pi));