> #sfb Feb 7, 2002
> with(plots);
> # Maple 2, what is the difference? 
> p1:=plot3d(x*y,x=-3..3,y=-3..3,style=WIREFRAME,color=red,axes=BOXED):
> p2:=implicitplot3d(x*y-z,x=-3..3,y=-3..3,z=-3..3,style=WIREFRAME,color=black):
> display(p1,p2);
> #
> ?gradplot
> f:=x^2+y^2; # Valley/Local Min. grad f points outward
> vec:=gradplot(f,x=-3..3,y=-3..3):vec;
> con:=contourplot(f,x=-3..3,y=-3..3):con;
> display(vec,con,title="Something is not right (angled)");
> display(vec,con,scaling=constrained,title="Aha, that's better");
> f:=2*x+y; # Plane/Usual view. grad f and contours perpendicular lines.
> fstring:=convert(f,string):
> banner:="The Good Doctor's "||fstring: #doesn't work in Maple5
> vec:=gradplot(f,x=-3..3,y=-3..3):
> con:=contourplot(f,x=-3..3,y=-3..3):
> display(vec,con,scaling=constrained,title=banner);
> # <spot for cut 'n paste demo>
> f:=-2*x-y; # Plane/Usual view. grad f and contours perpendicular lines.
> fstring:=convert(f,string):
> banner:="The Good Doctor's "||fstring: #doesn't work in Maple5
> vec:=gradplot(f,x=-3..3,y=-3..3):
> con:=contourplot(f,x=-3..3,y=-3..3):
> display(vec,con,scaling=constrained,title=banner);
> f:=x^2-y^2; # Saddle point. grad f `flows around'
> fstring:=convert(f,string):
> banner:="The Good Doctor's "||fstring:
> vec:=gradplot(f,x=-3..3,y=-3..3):
> con:=contourplot(f,x=-3..3,y=-3..3):
> display(vec,con,scaling=constrained,title=banner);
> f:=-x^2-y^2; # Hill Top/Local Max. grad f points inward
> fstring:=convert(f,string):
> banner:="The Good Doctor's "||fstring:
> vec:=gradplot(f,x=-3..3,y=-3..3):
> con:=contourplot(f,x=-3..3,y=-3..3):
> display(vec,con,scaling=constrained,title=banner);
> f:=1-x^2-y^2+x^2*y^2; # Local Max surrounded by saddles
> L:=1.5;
> fstring:=convert(f,string):
> banner:="The Good Doctor's "||fstring:
> vec:=gradplot(f,x=-L..L,y=-L..L):
> con:=contourplot(f,x=-L..L,y=-L..L):
> display(vec,con,scaling=constrained,title=banner);
> f:=sin(x)*sin(y); # 2 Local Max, 2 Local Min, Many Saddles
> L:=Pi;
> fstring:=convert(f,string):
> banner:="The Good Doctor's "||fstring:
> vec:=gradplot(f,x=-L..L,y=-L..L):
> con:=contourplot(f,x=-L..L,y=-L..L):
> display(vec,con,scaling=constrained,title=banner);
> f:=x^2+y^2+z^2; #3D versions are not so illuminating
> L:=3;
> fstring:=convert(f,string):
> banner:="The Good Doctor's "||fstring:
> vec:=gradplot3d(f,x=-L..L,y=-L..L,z=-L..L):
> con:=implicitplot3d(f=4,x=-L..L,y=-L..L,z=-L..L,numpoints=3000,style=PATCHNOGRID):
> con2:=implicitplot3d(f=9,x=-L..L,y=-L..L,z=-L..L,style=WIREFRAME,color=red):
> display(vec,con,con2,scaling=constrained,title=banner);
> #
> #
> # Directional Derivatives (algebra)
> f;
> diff(f,x);diff(f,x,x);grad(f,[x,y,z]);
> with(linalg):grad(f,[x,y,z]);
> u:=vector([3/5,4/5,0]);fu:=dotprod(u,grad(f,[x,y,z]),orthogonal);
> # Directional Derivatives (picture/graph)
> with(plottools):
> #
> #
> # Expressions and Subs, Evals
> f:=cos(sqrt(x^2+y^2)/2);subs({x=1,y=2},f);subs({x=1/2},f);subs({y=2/5},f);
> a:=1;b:=2;c:=eval(f,{x=a,y=b});
> fxfixed:=subs({x=a},f):
> fyfixed:=subs({y=b},f):
> surface:=plot3d(f,x=-1..4,y=-1..4,axes=boxed,style=PATCHNOGRID):
> xsection:=spacecurve([x,b,fyfixed],x=-1..4,color=red):
> ysection:=spacecurve([a,y,fxfixed],y=-1..4,color=green):
> display(surface,xsection,ysection);
> #arrow([a,b,c],vector([deltaA,deltaB,deltaC],...) goes from [a,b,c] to [a+deltaA,b+deltaB,c+deltaC]
> #
>  xarrow:=arrow([a,b,0],vector([1,0,0]),.2,.6,.2,color=red):
> yarrow:=arrow([a,b,0],vector([0,1,0]),.2,.6,.2,color=green):
> fxvalue:=eval(diff(f,x),{x=a,y=b});
> fxarrow:=arrow([a,b,c],vector([1,0,fxvalue]),.2,.6,.2,color=red):
> fyvalue:=eval(diff(f,y),{x=a,y=b});
> fyarrow:=arrow([a,b,c],vector([0,1,fyvalue]),.2,.6,.2,color=green):
> display(surface,xsection,ysection,xarrow,yarrow,fxarrow,fyarrow);
> normalarrow:=arrow([a,b,c],vector([-fxvalue,-fyvalue,1]),.2,.6,.2,color=blue):
> theta:=40*Pi/180;ux:=cos(theta);uy:=sin(theta);
> fu:=subs({x=a+ux*t,y=b+uy*t},f);
> usection:=spacecurve([a+ux*t,b+uy*t,fu],t=-3..3,color=blue,thickness=3):
> display(surface,xsection,ysection,usection,title="Three Cross Sections");
> uarrow:=arrow([a,b,0],vector([ux,uy,0]),.2,.6,.2,color=blue):
> fuarrow:=arrow([a,b,c],vector([ux,uy,ux*fxvalue+uy*fyvalue]),.2,.6,.2,color=blue):
> fuAtab:=eval( dotprod(vector([ux,uy]),grad(f,[x,y])),{x=a,y=b});
> display(surface,xsection,ysection,usection,xarrow,yarrow,uarrow,fxarrow,fyarrow,fuarrow,title="Directional Arrows");
> # tangent plane
> surf2:=plot3d(f,x=-1..4,y=-1..4,contours=[0,c],axes=boxed,style=PATCHCONTOUR):
> g:=c+fxvalue*(x-a)+fyvalue*(y-b);
> surf3:=plot3d(g,x=-1..4,y=-1..4,contours=[0,c],axes=boxed,style=PATCHCONTOUR):
> display(surf2,surf3,normalarrow,fxarrow,fyarrow,fuarrow,title="Tangent Plane");
> surf5:=plot3d(f,x=-1..4,y=-1..4,contours=[-2*c,-3*c/2,-c,-c/2,0,c/2,c,3*c/2,2*c],axes=boxed,style=PATCHCONTOUR):
> surf4:=plot3d(g,x=-1..4,y=-1..4,contours=[-2*c,-3*c/2,-c,-c/2,0,c/2,c,3*c/2,2*c],axes=boxed,style=CONTOUR,color=black,thickness=3):
> display(surf5,surf4,normalarrow,fxarrow,fyarrow,fuarrow,title="Tangent Plane");
> 
>