F = 1 - x^2 - 4 y^2;
Vector[ pt_, v_ ] := Graphics[Arrow[ {pt, pt + v}]];
GradF[{x_, y_}] := Vector[ {x, y}, (1/10){-2x, -8 y} ];
P = ContourPlot[F, {x, -3, 3}, {y, -2, 2}, AspectRatio -> Automatic];
Manipulate[ Show[P, GradF[pt], ImageSize -> 500], { {pt, {2, 0}}, Locator}]

V = VectorPlot[ {-2x, -8y}, {x, -3, 3}, {y, -2, 2}, VectorPoints->12, VectorScale-> Small, VectorStyle -> {Black} ];
Show[P, V]

IntegralCurve[{x_, y_}] := ParametricPlot[{Exp[-2 t] x, Exp[-8 t]y }, {t, 0,10}];
Manipulate[ Show[P, IntegralCurve[pt], ImageSize -> 500], { {pt, {2, 0}}, Locator}]

P3 = Plot3D[F, {x, -2, 2}, {y, -1, 1}, BoxRatios -> Automatic]

IntegralCurveParam[ x_, y_, t_] := Module[ {u, v},
u = Exp[-2 t] x;
v = Exp[-8 t]y;
Return[ {u, v, 1 - u^2 - 4 v^2}];
];

IntegralCurve3D[{x_, y_}] := ParametricPlot3D[IntegralCurveParam[ x, y, t] , {t, 0,10}, PlotStyle -> {Thick, Black}];


Manipulate[Show[ P3, IntegralCurve3D[{2Cos[t], 2Sin[t]}]], {t, 0, 2Pi}]