plot3d(x^2+y^2,x=-1..1,y=-1..1);

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plot3d(x^2+y^2,x=-1..1,y=-(1-x^2)^(1/2)..(1-x^2)^(1/2));

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with(plots):
implicitplot3d(z=x^2+y^2,x=-3..3,y=-3..3,z=0..6);

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plot3d([cos(t),2*sin(t),z],t=0..2*Pi,z=-1..2,scaling=constrained);

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animate3d([cos(t*u),2*sin(t*u),z],t=0..2*Pi,z=-1..2,u=0..1,scaling=constrained);

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plot3d([u*cos(t),u*sin(t),t/2],t=0..2*Pi,u=-2..2,scaling=constrained);

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animate3d([u*cos(t*v),u*sin(t*v),t*v/2],t=0..2*Pi,u=-2..2,v=0..1,scaling=constrained);

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animate3d([(3+1.5*cos(t))*cos(u*v),(3+1.5*cos(t))*sin(u*v),1.5*sin(t)],t=0..2*Pi,v=0..2*Pi,u=0..1,style=patch,orientation=[45,58],scaling=constrained);

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catenaria:=animate3d([t,(exp(t)+exp(-t))*cos(v*u)/2,(exp(t)+exp(-t))*sin(v*u)/2],t=-2..2,v=0..2*Pi,u=0..1,scaling=constrained,color=v,style=patch,frames=16):
eixorot1:=spacecurve([u,0,0,u=-3..3],color=black):
display([catenaria,eixorot1]);

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restart:

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f:=(x,y)->[x,y,4-x-y]; fx:=diff(f(u,v),u); fy:=diff(f(u,v),v);
with(linalg):
c:=crossprod(fx,fy);
dS:=sqrt(innerprod(c,c));
with(student):
Doubleint(dS,y=0...sqrt(4-x^2),x=0..2);
value(%);

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dS1:=sqrt(3)*r;
Doubleint(dS1,r=0..2,t=0...Pi/2);
value(%);

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F:=(x,y,z)->[x,y,2*z];
g:=(u,v)->(u,v,1-u^2-v^2);
F(f(u,v));

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gu:=diff([u,v,1-u^2-v^2],u); gv:=diff([u,v,1-u^2-v^2],v);
guxgv:=crossprod(gu,gv);

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Fgugv:=innerprod(F(g(u,v)),guxgv);
Doubleint(Fgugv,v=-(1-u^2)^(1/2)...(1-u^2)^(1/2),u=-1..1);
value(%);

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H:=(x,y,z)->[x/(sqrt(x^2+y^2+z^2))^3,y/(sqrt(x^2+y^2+z^2))^3,z/(sqrt(x^2+y^2+z^2))^3];
diverge(H(x,y,z),[x,y,z]);
simplify(%);
curl(H(x,y,z),[x,y,z]);
curl([-y,x,0],[x,y,z]);

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Tripleint(diverge([x,y,z],[x,y,z]),z=0..1,y=0..1,x=0..1);
value(%);

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f:=(x,y)->8*x^3+y^3-3*x*y;
fx:=diff(f(x,y),x);fy:=diff(f(x,y),y);
solve({fx=0,fy=0},{x,y});
Hf:=hessian(f(x,y), [x,y]);
det(Hf);

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delta:=(x,y)->288*x*y-9;
delta(0,0);

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delta(1/4,1/2);

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f:=(u,v)->[R*sin(v)*cos(u),R*sin(v)*sin(u),R*cos(v)]; fu:=diff(f(u,v),u); fv:=diff(f(u,v),v);
with(linalg):with(student):
c:=crossprod(fu,fv);
dS:=sqrt(simplify(innerprod(c,c)));
simplify(%);
Doubleint(dS,u=0...2*Pi,v=0..Pi);
value(%);

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