> #sfb 14 Feb 2002
> with(plots):with(plottools):with(linalg):
> #1
> u:=vector([2,4]);v:=vector([5,1]);
> u1:=arrow([0,0],u,.1,.2,.1):u2:=arrow(v,u,.1,.2,.1):
> v1:=arrow([0,0],v,.1,.2,.1):v2:=arrow(u,v,.1,.2,.1):
> uplusv:=arrow([0,0],matadd(u,v,1,1),.1,.2,.1):display(u1,u2,v1,v2,uplusv);
> #2
> solve({3*x-2*y+5*z=12,x=3*t,y=-2*t,z=5*t});
> n:=arrow([18/19,-12/19,30/19],[3,-2,5],.1,.3,.2):
> plane:=plot3d((12+2*y-3*x)/5,x=-3..3,y=-3..3,scaling=constrained,axes=normal,title="Plane 3x-2y+5y=12 and normal"):
> display(plane,n);
> #3
> e:=exp(1);
> f:=3*(1-x)^2*e^(-x^2-(y+1)^2)-10*(x/5-x^3-y^5)*e^(-x^2-y^2)-(1/3)*e^(-(x+1)^2-y^2);
> 
> f_x:=diff(f,x);f_y:=diff(f,y);
> f_xx:=diff(f_x,x);f_xy:=diff(f_x,y);f_yx:=diff(f_y,x);f_yy:=diff(f_y,y);
> #Question is f_xy=f_yx?
> f_xy-f_yx;
> #Answer, yes.
> #4
> curve1:=[1/5,y,subs({x=1/5},f)];curve2:=[x,3/2,subs({y=3/2},f)];
> p1:=spacecurve(curve1,y=-3..3,thickness=3,color=blue):
> p2:=spacecurve(curve2,x=-3..3,thickness=3,color=red):
> p3:=plot3d(f,x=-3..3,y=-3..3,title="The Demo Function with x=1/5 and y=3/2 cross sections",numpoints=3000):display(p1,p2,p3,style=contour);
> #5
> u:=vector([1,2,3]);v:=vector([3,1,3]);
> scalarproj:=dotprod(u,v,orthogonal)/norm(v,2);
> vectorproj:=scalarmul(v,scalarproj/norm(v,2));
> u1:='u1';v1:='v1';u2:='u2';v2:='v2';
> u:=vector([u1,u2,u3]);v:=vector([v1,v2,v3]);
> scalarproj:=dotprod(u,v,orthogonal)/norm(v,2);
> vectorproj:=scalarmul(v,scalarproj/norm(v,2));
> #
> #
> # Some examples from previous classes, Hunting for Global Max/Min
> f:=8*y^3+12*x^2-24*x*y;
> plot3d(f,x=-3..3,y=-3..3,title=convert(f,string),style=patchcontour,axes=boxed);
> subs(x=-3,f);subs(x=3,f);subs(y=-3,f);subs(y=3,f);
> display(plot([subs(x=3,f),subs(x=-3,f)],y=-3..3),plot([subs(y=3,f),subs(y=-3,f)],x=-3..3));
> f:=1-x^2-y^2+x^2*y^2;
> subs(x=-3,f);subs(x=3,f);subs(y=-3,f);subs(y=3,f);
> plot(subs(y=3,f),x=-3..3);
> plot3d(f,x=-3..3,y=-3..3,title=convert(f,string),style=patchcontour,axes=boxed);
> f:=x^2+2*y^2;#Region is x^2+y^2<=4
> p1:=plot3d([r*cos(t),r*sin(t),subs({x=r*cos(t),y=r*sin(t)},f)],r=0..2,t=0..2*Pi):
> p2:=plot3d([r*cos(t),r*sin(t),0],r=0..2,t=0..2*Pi,color=green):
> display(p1,p2,axes=boxed);
> plot(subs({x=2*cos(t),y=2*sin(t)},f),t=0..2*Pi,title="Boundry Curve");
> #
> f:=x^2*(y+1)^3+y^2;
> plot3d(f,x=-3..3,y=-3..3);
> subs(y=x,f);solve({diff(f,x)=0,diff(f,y)=0},{x,y});subs({x=0,y=0},diff(f,x,x)*diff(f,y,y)-(diff(f,x,y))^2);subs({x=0,y=0},diff(f,x,x));
> #
> # Critical Point Extreme Machine
> cpem:=proc(f)
>   local f_x, f_y, f_xx, f_yy, f_xy, bigD, critical, dd, xx, i;
>   f_x:=diff(f,x);f_y:=diff(f,y);
>   f_xx:=diff(f_x,x);f_xy:=diff(f_x,y);f_yy:=diff(f_y,y);
>   bigD:=f_xx*f_yy-(f_xy)^2;
>   critical:=solve({f_x=0,f_y=0},{x,y});
>   for i from 1 to nops([critical]) do
>     if ( nops([critical]) = 1 ) then
>       dd:=eval(bigD,critical); xx:=eval(f_xx,critical);
>     if ( dd < 0 ) then
>       print("The critical point ",critical," is a saddle ", dd,xx);
>     elif ( dd = 0 ) then
>       print("The test fails for ",critical, dd,xx);
>     elif ( xx > 0 ) then
>       print("The critical point ",critical," is a local min ", dd,xx);
>     else
>       print("The critical point ",critical," is a local max ", dd,xx);
>     fi;
>     else
>       dd:=eval(bigD,critical[i]); xx:=eval(f_xx,critical[i]);
>     if ( dd < 0 ) then
>       print("The critical point ",critical[i]," is a saddle ", dd,xx);
>     elif ( dd = 0 ) then
>       print("The test fails for ",critical[i], dd,xx);
>     elif ( xx > 0 ) then
>       print("The critical point ",critical[i]," is a local min ", dd,xx);
>     else
>       print("The critical point ",critical[i]," is a local max ", dd,xx);
>     fi
>     fi;
>   od;
>   #print(f);print(critical);
> end;
> f:=x^2+y^2;cpem(f);
> f:=8*y^3+12*x^2-24*x*y;cpem(f);
> cpem(1-x^2-y^2+x^2*y^2);
> cpem(sin(x)*sin(y));
> f:=x^2*(y+1)^3+y^2;cpem(f);
> solve({diff(1-x^2-y^2+x^2*y^2,x)=0,diff(1-x^2-y^2+x^2*y^2,y)=0},{x,y});
>