Volume 2 -- Várias Variáveis

Gabaritos

1) [> F:=(x,y)->[x^2+y^2,x*y];
    fieldplot(F(x,y),x=0..2,y=0..8,grid=[4,8],arrows=SLIM);

2)
[> P:=x->(x,x^3):
  campo:=fieldplot(F(x,y),x=0..2,y=0..8,grid=[6,6],arrows=SLIM):
  curva:=plot([P(x),x=0..2]):
  display(campo,curva);

3)
[> dP:=unapply(diff([P(x)],x),x); dP(x);
  FdP:=x->innerprod(F(P(x)),dP(x)); FdP(x);

4)
[> Int('FdP(x)',x=0..2)=int(FdP(x),x=0..2);

5)
[> F:=(x,y,z)->[y,z,x];
  P:=t->(t,t^2,t^3);

[> campo:=fieldplot3d(F(x,y,z),x=0..2,y=0..4,z=0..8,
                     grid=[3,3,3],arrows=SLIM,color=red):
  curva:=spacecurve([P(t)],t=0..2,color=blue,thickness=2):
  display(campo,curva,axes=normal,orientation=[10,60]);

[> dP:=unapply(diff([P(t)],t),t); dP(t);
  FdP:=t->innerprod(F(P(t)),dP(t)); FdP(t);

[> Int('FdP(t)',t=0..2)=int(FdP(t),t=0..2);

6)
[> L:=(x,y,z)->6*x*y^3+2*z^2;
  M:=(x,y,z)->9*x^2*y^2;
  N:=(x,y,z)->4*x*z +1;

7)
[> Diff('L(x,y,z)',y)=diff(L(x,y,z),y);
  Diff('M(x,y,z)',x)=diff(M(x,y,z),x);
  Diff('L(x,y,z)',z)=diff(L(x,y,z),z);
  Diff('N(x,y,z)',x)=diff(N(x,y,z),x);
  Diff('M(x,y,z)',z)=diff(M(x,y,z),z);
  Diff('N(x,y,z)',y)=diff(N(x,y,z),y);

8)
[> Int('L(x,y,z)',x)=int(L(x,y,z),x);

[ > f:=(x,y,z)->3*x^2*y^3+2*z^2*x+k(y,z);

[ > Diff('f(x,y,z)',y)=diff(f(x,y,z),y);
  'M(x,y,z)'=M(x,y,z);

[> f:=(x,y,z)->3*x^2*y^3+2*z^2*x+k(z);
  Diff('f(x,y,z)',z)=diff(f(x,y,z),z);
  'N(x,y,z)'=N(x,y,z);

[> Diff(k(z),z)=1;
  k:=z->z+c;
  f(x,y,z);

[> `Trabalho`=f(1,0,2)-f(-2,1,3);

9)
[> r:=theta->1+sin(theta);
  plot(r(theta),theta=0..2*Pi,coords=polar,scaling=constrained);
  r:='r':

10)
[> I3:=rhs(I1)=Doubleint(integrand(rhs(I1)),r=0..1+sin(theta),
                        theta=0..2*Pi);
  evalf(value(rhs(I3)));

11)
[> r:=theta->sin(2*theta);
  plot(r(theta),theta=0..Pi/2,coords=polar,scaling=constrained);
  r:='r':

12)
[> L:=(x,y)->x*y;
  M:=(x,y)->x^2+y^2;
  Doubleint(Diff('M(x,y)',x)-Diff('L(x,y)',y),x,y,R) =
  Doubleint(diff(M(x,y),x)-diff(L(x,y),y),x,y,R);

13)
[> I4:=Doubleint(x,x,y,R)=changevar({x=r*cos(theta),y=r*sin(theta)},
      Doubleint(x,x,y,R1),[r,theta]);
  I5:=rhs(I4)=Doubleint(integrand(rhs(I4)),r=0..sin(2*theta),
                        theta=0..Pi/2);
  evalf(value(rhs(I5)));