attuca6_gab.html

Atividades Tutoriais de Cálculo

Volume 2 -- Várias Variáveis

Gabaritos

Respostas dos Exercícios - attuca6

1)
[> f:=(x,y)->2-(x^2+y^2);
q:=(x,y)->[x,y,f(x,y)];

2)
[> iq:=plot3d(q(x,y),x=-1..1,y=-1..1,grid=[17,16]):
display(iq,scaling=constrained,style=hidden,
orientation=[20,70],title= `Figura 2`);

3)
[> cq1:=spacecurve(q(x,1/3),x=-1..1,color=red,thickness=2):
cq2:=spacecurve(q(1/4,y),y=-1..1,color=blue,thickness=2):
display(cq1,cq2,axes=normal,scaling=constrained,orientation=[20,70]);

4)
[> dq1:=unapply(diff(q(x,y),x),(x,y));
dq2:=unapply(diff(q(x,y),y),(x,y));

5)
[> vq1:=V(q(1/4,1/3),q(1/4,1/3)+dq1(1/4,1/3)):
vq2:=V(q(1/4,1/3),q(1/4,1/3)+dq2(1/4,1/3)):
display(cq1,cq2,vq1,vq2,axes=normal,scaling=constrained,
orientation=[20,70]);

6)
[> cq1:=spacecurve(q(x,1/3),x=-1..1,color=red,thickness=2):
cq2:=spacecurve(q(1/4,y),y=-1..1,color=blue,thickness=2):
vq1:=V(q(1/4,1/3),q(1/4,1/3)+dq1(1/4,1/3)):
vq2:=V(q(1/4,1/3),q(1/4,1/3)+dq2(1/4,1/3)):
display(cq1,cq2,vq1,vq2,axes=normal,scaling=constrained,
orientation=[20,70]);

7)
[> h:=0.4: k:=0.3:
iqhk:=plot3d(q(x,y),x=1/4..1/4+h,y=1/3..1/3+k,grid=[5,5]):
display(iqhk,cq1,cq2,vq1,vq2,scaling=constrained,
orientation=[20,70],title=`Figura 6`);

8)
[> h:=0.4: k:=0.3:
vq1h:=V(q(1/4,1/3),q(1/4,1/3)+h*dq1(1/4,1/3)):
vq2k:=V(q(1/4,1/3),q(1/4,1/3)+k*dq2(1/4,1/3)):
display(iqhk,vq1h,vq2k,scaling=constrained,orientation=[20,70],
title=`Figura 7`);

9)
[> iqthk:=polygonplot3d([q(1/4,1/3),q(1/4,1/3)+h*dq1(1/4,1/3),
                    q(1/4,1/3)+h*dq1(1/4,1/3)+k*dq2(1/4,1/3),
                    q(1/4,1/3)+k*dq2(1/4,1/3)],
                    style=wireframe,color=red,thickness=2):
display(iqhk,iqthk,vq1h,vq2k,scaling=constrained,
        orientation=[20,70],title=`Figura 8`);

10)

11)
[> Ap:=Doubleint(npv(theta,phi),theta,phi,Dp)
=Doubleint(npv(theta,phi),theta=0..2*Pi,phi=0..Pi);
value(rhs(Ap));

12)
[> R:=1:
ip:=plot3d(p(theta,phi),theta=0..2*Pi,phi=0..Pi,grid=[17,16]):
ipS:=plot3d(p(theta,phi),theta=Pi/4..Pi/2,phi=Pi/6..Pi/3,
grid=[17,16],color=red):
display(ip,ipS,scaling=constrained,style=hidden,orientation=[20,70]);
R:='R':

13)
[> assume(R>0):
ApS:=Doubleint(npv(theta,phi),theta,phi,DpS)=Doubleint(npv(theta,phi),
theta=Pi/4..Pi/2,phi=Pi/6..Pi/3);
value(rhs(ApS));

14)
[> iq:=plot3d(q(x,y),x=-1..1,y=-1..1,grid=[17,16]):
iqS:=plot3d(q(x,y),x=1/4..1/2,y=1/6..1/3,grid=[17,16],color=red):
display(iq,iqS,scaling=constrained,style=hidden,orientation=[20,70]);

15)
[> qv:=(x,y)->crossprod(dq1(x,y),dq2(x,y)); qv(x,y);
nqv:=(x,y)->norm(qv(x,y),2); nqv(x,y);

16)
[> Aq:=Doubleint(nqv(x,y),x,y,Dq)=Doubleint(nqv(x,y),x=-1..1,y=-1..1);
evalf(rhs(Aq));

17)
[> AqS:=Doubleint(nqv(x,y),x,y,DqS)
=Doubleint(nqv(x,y),x=1/4..1/2,y=1/6..1/3);
evalf(rhs(AqS));

[> 'AqS'= 100*(0.05644160828/7.446256723),`% de Aq`;

18)
[> sol1:=solve((t*x)^2+(t*y)^2=R^2,t);

19)
[> simplify(subs({x=R*cos(theta)*sin(phi),y=R*sin(theta)*sin(phi)},{sol1}));

[ > t:=phi->1/(-R^2*(-1+cos(phi)^2))^(1/2)*R; t(phi);

20)
[> P:=(theta,phi)->[t(phi)*R*cos(theta)*sin(phi),
t(phi)*R*sin(theta)*sin(phi),R*cos(phi)];
P(theta,phi);

21)
[> R:=1:
ip:=plot3d(p(theta,phi),theta=0..2*Pi,phi=0..Pi,grid=[17,16],
             style=hidden):
iP:=plot3d(P(theta,phi),theta=0..2*Pi,phi=0..Pi,style=wireframe,
             color=cyan,grid=[17,16]):
linha:=spacecurve([0,t,sqrt(3/4)],t=0..1,color=black,thickness=2):
texto:=textplot3d({[0,1/2,sqrt(3/4),`P`],[0,1,sqrt(3/4),`P'`]},
                    color=black):
display(ip,iP,linha,texto,scaling=constrained,orientation=[20,70]);
R:='R':

22)
[> R:=1:
ip:=plot3d(p(theta,phi),theta=0..2*Pi,phi=0..Pi,grid=[17,16],
             style=hidden):
iP:=plot3d(P(theta,phi),theta=0..2*Pi,phi=0..Pi,style=wireframe,
             color=cyan,grid=[17,16]):
ipS:=plot3d(p(theta,phi),theta=Pi/4..Pi/2,phi=Pi/6..Pi/3,grid=[4,4],
              color=red):
iPS:=plot3d(P(theta,phi),theta=Pi/4..Pi/2,phi=Pi/6..Pi/3,grid=[5,3],
            color=blue):
display(ip,ipS,iPS,iP,scaling=constrained,orientation=[20,70]);
R:='R':

23)
[> assume(R>0); assume(theta>0,phi>0); additionally(theta<2*Pi,phi<Pi);
dP1:=unapply(diff(P(theta,phi),theta),(theta,phi));
dP2:=unapply(diff(P(theta,phi),phi),(theta,phi));

24)
[> Pv:=(theta,phi)->simplify(crossprod(dP1(theta,phi),dP2(theta,phi)));
nPv:=(theta,phi)->simplify(norm(Pv(theta,phi),2));
Pv(theta,phi);
nPv(theta,phi);

25)
[> AP:=Doubleint(nPv(theta,phi),theta,phi,DP)
=Doubleint(nPv(theta,phi),theta=0..2*Pi,phi=0..Pi);
value(rhs(AP));

26)
[> APS:=Doubleint(nPv(theta,phi),theta,phi,DPS)
=Doubleint(nPv(theta,phi),theta=Pi/4..Pi/2,phi=Pi/6..Pi/3);
value(rhs(APS));

Desafios

1)
[> Q:=(theta,phi)->[t(phi)*R*cos(theta)*sin(phi),
t(phi)*R*sin(theta)*sin(phi),t(phi)*R*cos(phi)];
Q(theta,phi);

2)
[> R:=1:
ip:=plot3d(p(theta,phi),theta=0..2*Pi,phi=0..Pi,grid=[17,16],
style=hidden):
iQ:=plot3d(Q(theta,phi),theta=0..2*Pi,phi=Pi/6..5*Pi/6,
style=wireframe,color=cyan,grid=[17,16]):
display(ip,iQ,scaling=constrained,orientation=[20,70]);
R:='R':

3)
Justificativa : basta observar que o cilindro acima possui área maior do que aquele obtido pela projeção de Peters.

4)
[> R:=1:
ip:=plot3d(p(theta,phi),theta=0..2*Pi,phi=0..Pi,grid=[17,16],
             style=hidden):
iQ:=plot3d(Q(theta,phi),theta=0..2*Pi,phi=Pi/6..5*Pi/6,
             style=wireframe,color=cyan,grid=[17,16]):
ipS:=plot3d(p(theta,phi),theta=Pi/4..Pi/2,phi=Pi/6..Pi/3,
              grid=[4,4],color=red):
iQS:=plot3d(Q(theta,phi),theta=Pi/4..Pi/2,phi=Pi/6..Pi/3,
              grid=[5,3],color=blue):
display(ip,ipS,iQS,iQ,scaling=constrained,orientation=[20,70]);
R:='R':

5)
[> assume(R>0); assume(theta>0,phi>0); additionally(theta<2*Pi,phi<Pi);
dQ1:=unapply(diff(Q(theta,phi),theta),(theta,phi));
dQ2:=unapply(diff(Q(theta,phi),phi),(theta,phi));
Qv:=(theta,phi)->crossprod(dQ1(theta,phi),dQ2(theta,phi));
nQv:=(theta,phi)->simplify(norm(Qv(theta,phi),2)); nQv(theta,phi);

6)
[> AQS:=Doubleint(nQv(theta,phi),theta,phi,DQS)
=Doubleint(nQv(theta,phi),theta=Pi/4..Pi/2,phi=Pi/6..Pi/3);
value(rhs(AQS));

[> evalf(value(rhs(AQS))/((1/8)*(sqrt(3)-1)*R^2*Pi));