(*Visualizing the Gradient on the Level Surface F(x,y,z)=0 of F(x,y,z)=z-x^2+y^2*)
Manipulate[Show[
ParametricPlot3D[{x,y,x^2-y^2},{x,-3,3},{y,-3,3},Mesh->None,ImageSize->{500,500},PlotRange->{{-9,9},{-9,9},{-9,9}},AxesOrigin->{0,0,0},Boxed->False],
Graphics3D[{Opacity[v],Arrow[{{s,t,s^2-t^2},{-s,3*t,s^2-t^2+1}}]}]
],
{{s,0,"a"},-3,3},{{t,0,"b"},-3,3},{{v,1,"∇F(a,b,(a^2)-(b^2))"},{0,1}}]

(*Tangent Surfaces*)
Show[
ContourPlot3D[3x^2+2y^2+z^2==9,{x,-3,10},{y,-3,10},{z,-3,10},ContourStyle->Orange,Mesh->False],ContourPlot3D[x^2+y^2+z^2-8x-6y-8z+24==0,{x,-3,10},{y,-3,10},{z,-3,10},ContourStyle->Blue,Mesh->False]
]

(*Intersection of Surfaces*)
Manipulate[Show[
ContourPlot3D[x^2-y^2+z^2==6,{x,-6,6},{y,-6,6},{z,-6,6},ContourStyle->{Orange,Opacity[h]},Mesh->False,AxesOrigin->{0,0,0}],ContourPlot3D[x^2+y^2+z^2==14,{x,-6,6},{y,-6,6},{z,-6,6},ContourStyle->{Blue,Opacity[s]},Mesh->False],ParametricPlot3D[{(Sqrt[10])*Cos[t],2,(Sqrt[10])*Sin[t]},{t,0,2*π},PlotStyle->{Red,Thick,Opacity[int]}],ParametricPlot3D[{(Sqrt[10])*Cos[t],-2,(Sqrt[10])*Sin[t]},{t,0,2*π},PlotStyle->{Red,Thick,Opacity[int]}]
],{{h,1},0,1},{{s,1},0,1},{{int,0},0,1}]