VolumeTetrahedron::usage="Demonstrate that the planes tangent to the surface x y z =
3​
a
form with the planes of the coordinates a tetrahedron of constant volume. References:: A.S. Fedenko. Problemas de Geometría Diferencial. Editorial Mir: Moscú, p. 79, 1981; Personal.";
Development:
Tangent plane to the surface:
F[x_,y_,z_]:=xyz-
3
a
;​​
F
x
[x_,y_,z_]=D[F[x,y,z],x];​​
F
y
[x_,y_,z_]=D[F[x,y,z],y];​​
F
z
[x_,y_,z_]=D[F[x,y,z],z];​​(x-a)
F
x
[a,a,a]+(y-a)
F
y
[a,a,a]+(z-a)
F
z
[a,a,a]==0​​
2
a
(-a+x)+
2
a
(-a+y)+
2
a
(-a+z)0;​​x+y+z-3a==0;
Intersection with the X axis:
{3a,0,0}
Intersection with the Y axis:
{0,3a,0}
Intersection with the Z axis:
{0,0,3a}
Graph of the function and the tangent plane, considering a = 2.
In[]:=
ResourceFunction["TangentPlane"][xyz==8,{x,2},{y,2},{z,2}]
Out[]=
StandardFormEquation04(-2+x)+4(-2+y)+4(-2+z),Slopesx4,y4,z4,Plot

Tetrahedron volume:
The volume of a pyramid is one third of the base area by the height:
Base area:
In[]:=
L=3
2
a;​​A=
9
2
a
2
3
9
2
a
2
-3
2
a
In[]:=
Simplify
3
a
3
9a
2
-3
2
a
1/4
2

In[]:=
9
2
3
4
a
=
9
2
3
2
a
;​​base=
9
2
3
2
a
;
​​height=
Abs[1(0)+1(0)+1(0)-3a]
3
;​​Tetrahedron=
1
3
base*height
9
2
2
a
Abs[a]=
9
2
3
a
It can also be calculated like this:
In[]:=
V=
1
6
Det[{{3a,0,0},{0,3a,0},{0,0,3a}}]
Out[]=
9
3
a
2
Or also in the form:
In[]:=
V=
1
3!
Det[{{3a,0,0,1},{0,3a,0,1},{0,0,3a,1},{0,0,0,1}}]
Out[]=
9
3
a
2
The volume is constant and that’s what we wanted to demonstrate.
For the graph of the tangent plane, the coordinate planes and the tetrahedron are also a = 2:
In[]:=
Manipulate[ClearAll;​​Show[{ContourPlot3D[x+y+z==6,{x,-10,10},{y,-10,10},{z,-10,10},ContourStyle->{Blue,Opacity[0.1]},MeshFalse,PerformanceGoal"Quality",BoundaryStyleDirective[Black,Thick]],​​ContourPlot3D[x==0,{x,-10,10},{y,-10,10},{z,-10,10},ContourStyle->{Yellow,Opacity[0.1]},MeshFalse,PerformanceGoal"Quality",BoundaryStyleDirective[Black,Thick]],​​ContourPlot3D[y==0,{x,-10,10},{y,-10,10},{z,-10,10},ContourStyle->{Yellow,Opacity[0.1]},MeshFalse,PerformanceGoal"Quality",BoundaryStyleDirective[Black,Thick]],​​ContourPlot3D[z==0,{x,-10,10},{y,-10,10},{z,-10,10},ContourStyle->{Yellow,Opacity[0.1]},MeshFalse,PerformanceGoal"Quality",BoundaryStyleDirective[Black,Thick]],​​Graphics3D[{Gray,Ball[{6,0,0},0.08]}],Graphics3D[{Gray,Ball[{0,6,0},0.08]}],Graphics3D[{Black,Ball[{0,0,6},0.08]}],Graphics3D[{Black,Ball[{0,0,0},0.08]}],Graphics3D[{Green,Opacity[0.3],Tetrahedron[{{0,0,0},{3a,0,0},{0,3a,0},{0,0,3a}}]}]/.aa1},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange6,ImageSize900,ViewPoint{-10,-20,50}],Style["Tetrahedron constant volume:",Bold,Large],{{a1,0.1,"Value (a1)"},0.01,2,0.01},ControlPlacementTop]
Out[]=
​
Tetrahedron constant volume:
Value (a1)
The plane tangent to the surface in blue, the coordinated planes in yellow, the tetrahedron in green, the two front vertices in black and the two posterior ones in gray.