GrowthCharacterScalarField::usage="Determine the growth character of the scalar field u = 5
2
x
y z - 7 x
2
y
z + 5 x y
2
z
in the direction of the vector a = 8i - 4 j + 8 k at the point M(1, 1, 1): Find the rate of variation of the given field.References: Fedenko, A. S. Problemas de Geometría Diferencial. Editorial Mir Moscú: p.129, 1981; Personal.";
In[]:=
f[x_,y_,z_]:=5
2
x
yz-7x
2
y
z+5xy
2
z
;​​(Grad[f[x,y,z],{x,y,z}]/.{x1,y1,z1}).{8,-4,8}
Out[]=
144
Rate of variation of the given field:
In[]:=
v=Norm[Grad[f[x,y,z],{x,y,z}]/.{x1,y1,z1}]
Out[]=
12
See the graph:
In[]:=
f[1,1,1]
Out[]=
3
To draw the graph we will use a level surface with C = 3:
In[]:=
ShowContourPlot3D[5
2
x
yz-7x
2
y
z+5xy
2
z
==3,{x,-5,5},{y,-5,5},{z,-5,5},ContourStyle->Opacity[0.6],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality"],Graphics3D[{Black,Ball[{1,1,1},0.15]}],Graphics3DRed,Point[{1,1,1}],Arrowheads[.05],ArrowTube{1,1,1},{1,1,1}+
1
12

2
3
,-
1
3
,144,PlotRangeAll,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,ImageSize700
Out[]=