DerivedScalarField::usage="Find the derivative of the scalar field u = x
2
y
+
3
z
- x y z at the point M(1, 1, 2) in the direction that forms with the coordinate axes the angles α = 60°, β = 45° and γ = 60°.References: Fedenko, A.S. Problemas de Geometría Diferencial. Editorial Mir Moscú: p.128, 1981; Personal.";
Development:
In[]:=
f[x_,y_,z_]:=x
2
y
+
3
z
-xyz;​​((Grad[f[x,y,z],{x,y,z}])/.{x1,y1,z2}).Cos
π
3
,Cos
π
4
,Cos
π
3

Out[]=
5
Graph:
In[]:=
f[1,1,2]
Out[]=
7
To draw the graph we will use a level surface with C = 7:
In[]:=
ShowContourPlot3D[x
2
y
+
3
z
-xyz==7,{x,-15,15},{y,-15,15},{z,-15,15},ContourStyle->Opacity[0.6],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality"],Graphics3D[{Black,Ball[{1,1,2},0.35]}],Graphics3DRed,Point[{1,1,2}],Arrowheads[.05],ArrowTube{1,1,2},{1,1,2}+3Cos
π
3
,Cos
π
4
,5,PlotRangeAll,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,ImageSize700
Out[]=