Curl

F1={1,0}

VectorPlot[F1,{x,-1,1},{y,-1,1},VectorPoints->10,VectorScale->Small,Axes->True,AxesLabel->{x,y}]

f1=Curl[{1,0,0},{x,y,z}]

Now try another vector field:

F2={Exp[-x^2],0}

VectorPlot[F2,{x,-1,1},{y,-1,1},VectorPoints->10,VectorScale->Small,Axes->True,AxesLabel->{x,y}]

f2=Curl[{Exp[-x^2],0,0},{x,y,z}]

Now try another field:

F3={Exp[-y^2],0}

VectorPlot[F3,{x,-1,1},{y,-1,1},VectorPoints->10,VectorScale->Small,Axes->True,AxesLabel->{x,y}]

f3=Curl[{Exp[-y^2],0,0},{x,y,z}]

Now try another vector field:

F4={x,y}

VectorPlot[F4,{x,-1,1},{y,-1,1},VectorPoints->10,VectorScale->Small,Axes->True,AxesLabel->{x,y}]

f4=Curl[{x,y,0},{x,y,z}]

Now try another vector field:

F5={x/(x^2+y^2),y/(x^2+y^2)}

VectorPlot[F5,{x,-1,1},{y,-1,1},VectorPoints->10,VectorScale->Small,Axes->True,AxesLabel->{x,y}]

f5=Simplify[Curl[{x/(x^2+y^2),y/(x^2+y^2),0},{x,y,z}]]

Now try another vector field:

F6={-y,x}

VectorPlot[F6,{x,-1,1},{y,-1,1},VectorPoints->10,VectorScale->Small,Axes->True,AxesLabel->{x,y}]

f6=Curl[{-y,x,0},{x,y,z}]

Now try another vector field:

F7={-y/(x^2+y^2),x/(x^2+y^2)}

VectorPlot[F7,{x,-1,1},{y,-1,1},VectorPoints->10,VectorScale->Small,Axes->True,AxesLabel->{x,y}]

f7=Curl[{-y/(x^2+y^2),x/(x^2+y^2),0},{x,y,z}]

Simplify[%]

Now try another vector field:

F8={Sin[x*y],y}

VectorPlot[F8,{x,-2,2},{y,-2,2},VectorPoints->10,VectorScale->Small,Axes->True,AxesLabel->{x,y}]

f8=Curl[{Sin[x*y],y,0},{x,y,z}]

Now try another vector field:

F10={0,1,0}

VectorPlot3D[F10,{x,-5,5},{y,-5,5},{z,-5,5},VectorPoints->5,VectorScale->Small,Axes->True,AxesLabel->{x,y,z}]

f10=Curl[{0,1,0},{x,y,z}]

Now try another vector field.

F11={x,y,z}

VectorPlot3D[F11,{x,-5,5},{y,-5,5},{z,-5,5},VectorPoints->5,VectorScale->Small,Axes->True,AxesLabel->{x,y,z}]

f11=Curl[{x,y,z},{x,y,z}]

Now try another vector field:

f12=1/((x^2+y^2+z^2)^(1/2))

Efield=Grad[-f12,{x,y,z}]

VectorPlot3D[Efield,{x,-5,5},{y,-5,5},{z,-5,5},VectorPoints->5,VectorScale->Small,Axes->True,AxesLabel->{x,y,z}]

f13=Curl[Efield,{x,y,z}]