ex1
restart:
with (student):with(plots):
with(linalg):
F:=(x,y)-> x*exp(y) + cos(x +y) ;
v:=array(1..2,[3,-4]);u := v/norm(v,2); # get that unit vector
Diff(F(x,y),x)=diff(F(x,y),x);Diff(F(x,y),y)=diff(F(x,y),y);
fx:=subs(x=2,y=0,diff(F(x,y),x));
fy:=subs(x=2,y=0,diff(F(x,y),y));
grad1:=array(1..2,[fx,fy]);answer := dotprod(grad1,u); evalf(%);# answer to a
ex #1b
fx:=subs(x=2,y=0,diff(F(x,y),x));
fy:=subs(x=2,y=0,diff(F(x,y),y));
grad1:=array(1..2,[fx,fy]);
# 1c the direction of the gradient
#1d norm of the gradient
norm(grad1,2);
#1e a vector perpendicular to grad1:=array(1..2,[fx,fy]);
perp1:=array(1..2,[fy,-fx]);
dotprod(grad1,perp1);evalf(%);
ex2
restart:
with (student):with(plots):
with(linalg):
F:=(x,y,z)-> x^2 + y^2 + z -9 ;
# tangent plane at P=(1,2,4)
Diff(F(x,y,z),x)=diff(F(x,y,z),x);Diff(F(x,y,z),y)=diff(F(x,y,z),y);Diff(F(x,y,z),Z)=diff(F(x,y,z),z);
Fx:=subs(x=1,y=2,z=4,diff(F(x,y,z),x));Fy:=subs(x=1,y=2,z=4,diff(F(x,y,z),y));Fz:=subs(x=1,y=2,z=4,diff(F(x,y,z),z));
v:=array(1..3,[Fx,Fy,Fz]);Po:=array(1..3,[x-1,y-2,z-4]);
dotprod(v,Po)=0;# the tangent plane
#3
restart:
with (student):with(plots):
f:=(x,y)-> -x^2 +x*y -2*y -y^2 + 4 ;
xf:=diff(f(x,y),x);yf:=diff(f(x,y),y); solve({xf=0,yf=0},{x,y});
disc:= (x,y)-> diff(f(x,y),x,x)*diff(f(x,y),y,y)-diff(f(x,y),x,y)*diff(f(x,y),x,y);DXY:=eval(fxx*fyy-fxy*fxy);;
subs(x=-2/3,y= -4/3,disc(x,y));
subs(x=-2/3,y= -4/3,diff(f(x,y),x,x)); # says a local max
ex 4a
restart:with (student):W:= (x,y,z)-> x*y+z;
X:= (t)-> cos(t);Y:= (t)-> sin(t);Z:= (t)-> (t);
Diff(W(x,y,z),x)*Diff(X(t),t)+Diff(W(x,y,z),y)*Diff(Y(t),t) + Diff(W(x,y,z),z)*Diff(Z(t),t); diff(W(x,y,z),x)*diff(X(t),t)+diff(W(x,y,z),y)*diff(Y(t),t) + diff(W(x,y,z),z)*diff(Z(t),t);
subs(x=cos(t),y=sin(t),z=t,%);simplify(%);
W1:=t -> 2*cos(t^2);diff(W1(t),t); #ex 4b
# ex5 not using chain rule
restart:
with (student):with(plots):
:
f:=(x,y)-> x^2 + y^2 ;
X:= (u,v)-> u + v;Y:= (u,v)-> u - v;
Diff(f(x,y),x,x) + Diff(f(x,y),y,y)=diff(f(x,y),x,x)+diff(f(x,y),y,y);
Diff(f(x,y),x)*Diff(X(u,v),u)+Diff(f(x,y),y)*Diff(Y(u,v),v); diff(f(x,y),x)*diff(X(u,v),u)+diff(f(x,y),y)*diff(Y(u,v),u);
subs(x=u+v,y=u-v,%);simplify(%);
W1:=u ->4*u;diff(W1(u),u); #ex 5b not using chain rule
ex6
restart:with (student):with(linalg):
f:=(x,y,z)-> exp(3*x+4*y)*cos(5*z);
Diff(f(x,y,z),x,x) + Diff(f(x,y,z),y,y) + Diff(f(x,y,z),z,z)=diff(f(x,y,z),x,x)+diff(f(x,y,z),y,y)+diff(f(x,y,z),z,z);
ex7 it is #2
# ex 8
restart:with (student):with(linalg):with(plots):
F:=(x,y)-> x^2+y^2;plot1 :=plot3d(x^2 + y^2,x=-3..3,y=-3..3, color = red):
plot2:=plot3d(1,x=-3..3,y=-3..3,color = blue):
display(plot1,plot2,axes=boxed);
r: t -> (t,sqrt(1-t^2),1);
v: t -> (1, -t/sqrt(1-t^2),0);
subs(t=a,%);
# ex 9
restart:with(student):with(plots):
f:=(x,y)-> -x^4 +4*x*y -y^4 + 1 ;
xf:=diff(f(x,y),x);yf:=diff(f(x,y),y); solve({xf=0,yf=0},{x,y});
# UGH ! it says 3 real roots
disc:= (x,y)-> diff(f(x,y),x,x)*diff(f(x,y),y,y)-diff(f(x,y),x,y)*diff(f(x,y),x,y);DXY:=eval(fxx*fyy-fxy*fxy);;
subs(x=0,y= 0,disc(x,y)); # saddle point
subs(x=1,y= 1,disc(x,y));subs(x=1,y= 1,diff(f(x,y),x,x)); # says a local max
subs(x=-1,y= -1,disc(x,y));subs(x=1,y= 1,diff(f(x,y),x,x)); # says a local max
#EX 10
restart:
with (student):with(plots):
with(linalg):
F:=(x,y,z)-> x^2 /9+ y^2 + z^2/9 -3 ;
# tangent plane at P=(-3,1,-3)
Diff(F(x,y,z),x)=diff(F(x,y,z),x);Diff(F(x,y,z),y)=diff(F(x,y,z),y);Diff(F(x,y,z),Z)=diff(F(x,y,z),z);
Fx:=subs(x=-3,y=1,z=-3,diff(F(x,y,z),x));Fy:=subs(x=-3,y=1,z=-3,diff(F(x,y,z),y));Fz:=subs(x=-3,y=1,z=-3,diff(F(x,y,z),z));
v:=array(1..3,[Fx,Fy,Fz]);Po:=array(1..3,[x+3,y-1,z +3]);
dotprod(v,Po)=0;# the tangent plane
# ex11
restart:with (student):w:= (x,y,z)-> x^4*y + y^2*z^3;
X:= (r,s,t)-> r*s*exp(t);Y:= (r,s,t)-> r*s^2*exp(-t);Z:= (r,s,t)-> r*s*sin(t);
Diff(w(x,y,z),x)*Diff(X(r,s,t),s)+Diff(w(x,y,z),y)*Diff(Y(r,s,t),s) +Diff(w(x,y,z),z)*Diff(Z(r,s,t),s);
diff(w(x,y,z),x)*diff(X(r,s,t),s)+diff(w(x,y,z),y)*diff(Y(r,s,t),s) +diff(w(x,y,z),z)*diff(Z(r,s,t),s);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
subs(x=r*s*exp(t),y=r*s*exp(-t),z=r*s*sin(t),%);simplify(%);
ex12
restart:with(student):with(linalg):with(VectorCalculus):
T:=(x,y,z)-> 80/(1 +x^2+2*y^2+3*z^2);
grad(T(x,y,z), vector([x,y,z])); # direction of gradient
norm(%,2);
#ex13 dis in calss
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEiI0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVE0fmV4fjE0fmRpZH5pbn5jbGFzc0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRjRGLw==
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