# WORKSHEET#14
# Systems of non-linear equations in many variables
> f:=exp(y^2-z^3)-2;
2 2 3
f := exp(sin(theta) sin(phi) - cos(phi) ) - 2
> g:=sin(x+2*y+3*z^2)+x^4;
2
g := sin(cos(theta) sin(phi) + 2 sin(theta) sin(phi) + 3 cos(phi) )
4 4
+ cos(theta) sin(phi)
> h:=x^2+y^2+z^2-1;
2 2 2 2 2
h := cos(theta) sin(phi) + sin(theta) sin(phi) + cos(phi) - 1
# We want to find all simultaneous solutions of f(x,y,z)=0, g(x,y,z)=0, h(x,y,z)=0. Each of
# these equations defines a surface in 3-space. We want the common intersection of these
# surfaces. Notice that h=0 is just the unit sphere centered at the origin. The equation f=0
# is equivalent to y^2-z^3=ln(2), a cylinder in the x direction.
> fsolve({f,g,h},{x,y,z});
Error, (in fsolve) should use exactly all the indeterminates
# There is no error in the command itself, rather Maple can not find a solution this way.
# We have to be more artful. The third equation h=0 is much more suited to spherical
# coordinates.
> x:=cos(theta)*sin(phi);y:=sin(theta)*sin(phi);z:=cos(phi);
x := cos(theta) sin(phi)
y := sin(theta) sin(phi)
z := cos(phi)
> f;
2 2 3
exp(sin(theta) sin(phi) - cos(phi) ) - 2
> g;
2
sin(cos(theta) sin(phi) + 2 sin(theta) sin(phi) + 3 cos(phi) )
4 4
+ cos(theta) sin(phi)
> h;
2 2 2 2 2
cos(theta) sin(phi) + sin(theta) sin(phi) + cos(phi) - 1
> simplify(");
0
# In other words the equation h=0 is automatically satisfied when we convert to spherical
# coordinates.
> F:=simplify(f);
2 2 2 2 3
F := exp(1 - cos(phi) - cos(theta) + cos(theta) cos(phi) - cos(phi) ) - 2
> G:=simplify(g);
2
G := sin(cos(theta) sin(phi) + 2 sin(theta) sin(phi) + 3 cos(phi) )
4 4 2 4 4
+ cos(theta) - 2 cos(theta) cos(phi) + cos(theta) cos(phi)
# I don't know if I would agree that these expressions are simpler. In any case the original
# problem is now down to 2 equations, namely F(theta,phi)=0, G(theta,phi)=0.
> fsolve({F,G},{theta,phi});
{theta = .5806816709, phi = -9.984737488}
> assign(");
# This actually assigns the above values to theta and phi. We can now work out the
# corresponding values of x, y, z and at the same time check that we do have a solution.
> print([x,y,z],[f,g,h]);
-9
[.4440902047, .2913867500, -.8472766089], [0, -.47*10 , 0]
# Now we'll look for other solutions. First we have to unassign values.
> theta:='theta':phi:='phi':
> with(plots):
> graphF:=implicitplot(F,theta=0..2*Pi,phi=0..Pi,title=`F(theta,phi)=0`):
> graphF;
#
** Maple V Graphics **
> graphG:=implicitplot(G,theta=0..2*Pi,phi=0..Pi,title=`G(theta,phi)=0`):
> graphG;
#
** Maple V Graphics **
> display([graphF,graphG],title=`points of intersection`);
#
** Maple V Graphics **
# There are clearly 2 points of intersection and possibly 2 more in the upper left hand
# corner. By using the mouse and clicking on the possible intersection point we come up
# with theta=0.74 and phi=2.51.
> zoomF:=implicitplot(F,theta=0..2,phi=2.3..3):
> zoomG:=implicitplot(G,theta=0..2,phi=2.3..3):
> display([zoomF,zoomG],title=`upper left hand corner`);
#
** Maple V Graphics **
#
#
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# Yes indeed, there are 2 more points of intersection. We can find approximate coordinates
# by using the mouse in the plot window. They seem to be (theta,phi)=(0.59, 2.57) and
# (0.85,2.46). Armed with this approximation we can use Maple to find them to 10 places.
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> fsolve({F,G},{theta,phi},{theta=0.55..0.65,phi=2.5..2.7});
{theta = .5806816709, phi = 2.581633126}
> assign(");
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> print([x,y,z],[f,g,h]);
-9
[.4440902049, .2913867502, -.8472766087], [0, .61*10 , 0]
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# x,y, and z are functions of the spherical coordinates theta, phi. The values of f,g and h are
# zero at these solutions. We save these values for future use.
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> sol[1]:=[x,y,z];
sol[1] := [.4440902049, .2913867502, -.8472766087]
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> theta:='theta':phi:='phi':
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> fsolve({F,G},{theta,phi},{theta=0.7..1.0,phi=2.4..2.7});\
{theta = .8545096410, phi = 2.458305107}
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> assign(");
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> print([x,y,z],[f,g,h]);
-8
[.4145344389, .4761920619, -.7755013341], [0, -.101*10 , 0]
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> sol[2]:=[x,y,z];
sol[2] := [.4145344389, .4761920619, -.7755013341]
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> theta:='theta':phi:='phi':
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# In the same way we can find the other solutions to 10 places.
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> fsolve({F,G},{theta,phi},{theta=3.9..4.2,phi=2.0..2.5});
{theta = 4.103188417, phi = 2.364619685}
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> assign(");
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> print([x,y,z],[f,g,h]);
-8 -9
[-.4011917196, -.5749960069, -.7130391266], [0, -.100*10 , -.1*10 ]
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> sol[3]:=[x,y,z];
sol[3] := [-.4011917196, -.5749960069, -.7130391266]
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> theta:='theta':phi:='phi':
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> fsolve({F,G},{theta,phi},{theta=5.0..5.4,phi=2.0..2.5});
{phi = 2.181034644, theta = 5.233829826}
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> assign(");
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> print([x,y,z],[f,g,h]);
-8 -9
[.4082232611, -.7106003231, -.5730627801], [0, .184*10 , -.1*10 ]
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> theta:='theta':phi:='phi':
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>
