f[u_,v_]=x^2 y/(x^2+y^2);
cp3D=ContourPlot3D[z==f[x,y],{x,-4.1,4},{y,-4,4},{z,-2,2}]

fx=D[f[x,y],x]
fy=D[f[x,y],y]

n=Simplify[Cross[{1,0,fx},{0,1,fy}]]
Simplify[{-fx,-fy,1}];

dA=FullSimplify[Sqrt[(-((2 x (y^3))/((((x^2)+(y^2)))^2)))^2+(((-(x^4)+(x^2) (y^2))/((((x^2)+(y^2)))^2)))^2+1]]

grad[x_,y_] = Grad[f[x,y],{x,y}]
normal[x_,y_] = -Simplify[grad[x,y]]

Manipulate[ContourPlot[f[x,y],{x,-2,2},{y,-2,2}, Epilog->Arrow[{pt,pt+normal@@pt}], PerformanceGoal->"Quality" ,Contours->30,PlotRange->{{-2,2},{-2,2},{-2,2}}], {{pt,{.01,-.1}}, Locator}, FrameLabel->"Clicke auf einen Punkt um den Normalenvektor zu sehen",SaveDefinitions->True]

NIntegrate[dA,{x,0,2},{y,0,2}]

NIntegrate[dA,{x,-0.05,0.05},{y,0,2}]

a=NIntegrate[dA,{x,0,2},{y,0,x}]
b=NIntegrate[dA,{x,0,2},{y,x,2}]
a+b

kegel=ParametricPlot3D[{u Cos[v],u Sin[v],u},{u,0,1},{v,0,2 Pi}]

vp=VectorPlot3D[{x,-y^2 x^3,0},{x,-1,1},{y,-1,1},{z,-.5,1.3},VectorPoints->10]

Curl[{x,-y^2 x^3,0},{x,y,z}]

vp=VectorPlot3D[{0,0,-3 (x^2) (y^2)},{x,-1,1},{y,-1,1},{z,-.5,1.3},VectorPoints->10]

Show[kegel,vp]

x[u_,v_]:=u Cos[v]
y[u_,v_]:=u Sin[v]
z[u_,v_]:=u

RotFmaln={0,0,-3x[u,v]^2 y[u,v]^2}.{-u Cos[v],-u Sin[v],u}

Integrate[RotFmaln,{u,0,1},{v,0,2 Pi}]

Fmalr={x[1,v],-y[1,v]^2 x[1,v]^3,0}.{-Sin[v],Cos[v],0}

Integrate[Fmalr,v]
Integrate[Fmalr,{v,0,2Pi}]

ParametricPlot3D[{v,2Sin[u+v],u},{u,0,2π},{v,-π/2,π},Mesh->3]
flaeche=ParametricPlot3D[{v,2Sin[u+v],u},{u,0,2π},{v,-π/2,π},PlotStyle->Opacity[0.35],Mesh->3];

vp0=VectorPlot3D[{x y z,x,y},{x,-1.5,3},{y,-1.5,3},{z,-.5,6},VectorPoints->{6,7,10}]

Curl[{x y z,x,y},{x,y,z}]

vp2=VectorPlot3D[{1,x y,1-x z},{x,-1.5,3},{y,-1.5,3},{z,-.5,6}]

Show[flaeche,vp2]

nvec=Cross[{0,2Cos[u+v],1},{1,2Cos[u+v],0}]

Fn={1,v 2Sin[u+v],1-u v}.nvec

Integrate[Fn,{u,0,2π},{v,-π/2,π}]

pplot1=ParametricPlot3D[{v,2Sin[0+v],0},{v,π,-π/2},PlotStyle->{Blue,Thickness[0.015]}];
pplot2=ParametricPlot3D[{-π/2,2Sin[u-π/2],u},{u,0,2π},PlotStyle->{Black,Thickness[0.015]}];
pplot3=ParametricPlot3D[{v,2Sin[2π+v],2π},{v,-π/2,π},PlotStyle->{Red,Thickness[0.015]}];
pplot4=ParametricPlot3D[{π,2Sin[u+π],u},{u,2π,0},PlotStyle->{Green,Thickness[0.015]}];
Show[flaeche,pplot1,pplot2,pplot3,pplot4]

IBlue=Integrate[{0,v,2Sin[0+v]}.D[{v,2Sin[0+v],0},v],{v,π,-π/2}]

IBlack=Integrate[{-π/2 2Sin[u-π/2]u,-π/2 ,2Sin[u-π/2]}.D[{-π/2,2Sin[u-π/2],u},u],{u,0,2π}]

IRed=Integrate[{v 2Sin[2π+v]2π,v,2Sin[2π+v]}.D[{v,2Sin[2π+v],2π},v],{v,-π/2,π}]

IGreen=Integrate[{π 2Sin[u+π]u,π ,2Sin[u+π]}.D[{π,2Sin[u+π],u},u],{u,2π,0}]

IBlue+IRed

Show[vp0,pplot1,pplot2,pplot3,pplot4]