Mathematics 41, section 1 -- A First constrained max-min problem:

Find the largest and smallest values of the function f(x,y) = sin(x)sin(y)

at the points on the circle of radius 2, center at (1,0).

> with(plots): with(DEtools):

> f := (x,y) -> sin(x)*sin(y);

> CoP:=contourplot(f(x,y),x=-Pi..Pi,y=-Pi..Pi,color=blue,contours=20):

> eqns:={diff(x(t),t)=cos(x(t))*sin(y(t)), diff(y(t),t)=sin(x(t))*cos(y(t))};

> DP:=DEplot(eqns,{x(t),y(t)},t=0..1,[[x(0)=0,y(0)=0]],x=-Pi..Pi,y=-Pi..Pi,arrows=SLIM):

> CiP:=plot([1+2*cos(t),2*sin(t),t=0..2*Pi],color=black):

> display({CiP,DP,CoP});

> plot(f(1+2*cos(t),2*sin(t)),t=0..2*Pi);

>