> # sfb 7 Mar, 2002
> restart:
> with(plots);with(plottools);with(linalg);
> x=r*cos(theta);y=r*sin(theta);coordplot(polar,color=[red,black],scaling=constrained);
> a:=polarplot(1+cos(theta),theta=0..2*Pi,scaling=constrained,color=blue):a;
> b:=plot({x,x/sqrt(3)},x=-0.5..2,color=green):b;
> c:=polarplot({1, 1.2},theta=0..2*Pi,color=green):c;
> display(a,b,c);
> # dA = dx dy = r dr dtheta
> 
> time(int(int(1,y=-sqrt(1-x^2)..sqrt(1-x^2)),x=-1..1));
> time(int(int(1*r,r=0..1),theta=0..2*Pi));
> time(int(int(sqrt(1-x^2-y^2),y=-sqrt(1-x^2)..sqrt(1-x^2)),x=-1..1));
> time(int(int(sqrt(1-r^2)*r,r=0..1),theta=0..2*Pi));
> # cylindrical coordinates (r, theta, z) [Polar with z]
> x=r*cos(theta);y=r*sin(theta);z=z;coordplot3d(cylindrical);
> # cylinderplot assume r = r(theta,z);
> # dV = dx dy dz = r dr dtheta dz
> cylinderplot(1+z*sin(theta),theta=0..2*Pi,z=0..2);
> cylinderplot(2,theta=0..2*Pi,z=0..2);
> coordplot3d(spherical);
> sphereplot(2,theta=0..2*Pi,phi=0..Pi);
> a:=sphereplot(2,theta=0..2*Pi,phi=0..Pi/4,scaling=constrained):a;

> b:=cylinderplot(z,theta=0..2*Pi,z=0..sqrt(2)):b;
> display(a,b);
> #dV = rho^2 sin (phi) d pho d phi d theta
> 'int(int(int(rho^2*sin(phi),rho=0..1),phi=0..Pi),theta=0..2*Pi)'=int(int(int(rho^2*sin(phi),rho=0..1),phi=0..Pi),theta=0..2*Pi);
> 'int(int(int(1*r,z=-sqrt(1-r^2)..sqrt(1-r^2)),r=0..1),theta=0..2*Pi)'=int(int(int(1*r,z=-sqrt(1-r^2)..sqrt(1-r^2)),r=0..1),theta=0..2*Pi);
> 'int(int(int(1,z=-sqrt(1-x^2-y^2)..sqrt(1-x^2-y^2)),y=-sqrt(1-x^2)..sqrt(1-x^2)),x=-1..1)'=int(int(int(1,z=-sqrt(1-x^2-y^2)..sqrt(1-x^2-y^2)),y=-sqrt(1-x^2)..sqrt(1-x^2)),x=-1..1);
> x=rho*sin(phi)*cos(theta);y=rho*sin(phi)*sin(theta);z=rho*cos(phi);r=rho*sin(phi);
> #1. normal dot (X - X_0)
> F:=x+y^2+z^3+x*y*z;
> gF:=grad(F,[x,y,z]);
> gfp:=eval([gF[1],gF[2],gF[3]],{x=3,y=2,z=1});
> xmx0:=vector([x-3,y-2,z-1]);
> dotprod(gfp,xmx0,orthogonal)=0;
> #2 directional derivative
> F:=sin(x)*cos(y)*ln(z);
> gF:=grad(F,[x,y,z]);
> gfp:=eval([gF[1],gF[2],gF[3]],{x=Pi/4,y=Pi/3,z=3});
> u:=vector([4/6,-4/6,2/6]);
> dotprod(gfp,xmx0,orthogonal);
> Z:=(x+y)*ln(x+y);diff(Z,x);diff(Z,y);
> X:=u*sin(v);Y:=u^2+v^(-2);diff(X,u);diff(X,v);diff(Y,u);diff(Y,v);
> Z_u:=diff(Z,x)*diff(X,u)+diff(Z,y)*diff(Y,u);
> Z_v:=diff(Z,x)*diff(X,v)+diff(Z,y)*diff(Y,v);
> r:=[t*cos(2*Pi*t),t*sin(2*Pi*t)];
> diff(r,t);
> diff(r,t,t);
> plot([t*cos(2*Pi*t),t*sin(2*Pi*t),t=0..2],x=-2..2);
> 
> int(sqrt(diff(r[1],t)^2+diff(r[2],t)^2),t=0..2);
> evalf(int(sqrt(diff(r[1],t)^2+diff(r[2],t)^2),t=0..2));
> #8
> F:=x+2*y-lambda*(x^2+4*y^2-1);
> diff(F,x)=0;diff(F,y)=0;diff(F,lambda)=0;;
> #5
> f:=8*x*y-(x+y)^4/4;
> diff(f,x)=0;diff(f,y)=0;
> diff(f,x,x);diff(f,x,y);diff(f,y,y);
> f:=int(x*(x-1),x);
> plot(f,x=-1..2);
> g:=int((x-3)*(x-4)*(x-5),x);
> plot(g,x=2..6);
> plot3d(f+subs(x=y,g),x=-1..2,y=2..6);
>