gradient pola skalarnego

g[x_,y_]:=Sqrt[x^2+2 y^2 Sqrt[y^2/(1+x^2)]];
Simplify[{D[g[a,b],{a,1}],D[g[a,b],{b,1}]}/.{a->x,b->y}]

v[x_,y_]=Simplify[{D[g[a,b],{a,1}],D[g[a,b],{b,1}]}/.{a->x,b->y}];
xa=-3;xb=3;
ya=-3;yb=3;
aa=ContourPlot[g[x,y],{x,-3,3},{y,-3,3}];
bb=DensityPlot[g[x,y],{x,-3,3},{y,-3,3}];
cc=VectorPlot[v[x,y],{x,-3,3},{y,-3,3},VectorStyle->Orange,StreamPoints->10,StreamScale->Full];

Show[{aa,cc}]

Show[{bb,cc}]